Introduction

Doping is crucial for modulating semiconductors’ electrical, optical, and thermoelectric (TE) properties. For TE energy conversion applications, both p-type and n-type doped semiconductors are desired for assembling TE devices. However, intrinsic defects could lead to the doping asymmetry problem in TE semiconductors, where one type of conduction (either n-type or p-type) is more difficult to achieve than the other due to the charged intrinsic defects that can act as compensating centers. For instance, inherent Ge-deficiency is the dominant defect type in GeTe. These ionized VGe in the as-prepared GeTe ingot pin the Fermi level deeply in the valence band and result in high hole concentrations (3–8 × 1020 cm−3)1, making n-type doping extremely difficult2,3, if not impossible. Similarly, in the Zintl-phase compounds Mg3Sb2 and XMg2Sb2 (X = Yb, Eu, Ca, Sr, Ba), the negatively charged cation vacancies (VMg and VX) have lower formation energies near the conduction band minimum (CBM)4, hindering n-type doping in these materials5,6.

Half-Heusler (HH) semiconductors with a valence electron count of 18 are very promising TE materials in the mediate-to-high temperature range. These compounds crystallize in a cubic structure (space group \(F\bar{4}3\) m) with a chemical formula of ABC, where A, B, and C occupy the Wyckoff positions 4a (0, 0, 0), 4c (1/4, 1/4, 1/4), and 4b (1/2, 1/2, 1/2), respectively. The 4d (3/4, 3/4, 3/4) sublattice remains vacant (Fig. 1a). The existing four interpenetrating face-centered cubic (fcc) sublattices offer abundant selections of dopants, as well as the formation of various intrinsic point defects, such as vacancies and interstitials.

Fig. 1: Schematic diagrams of dopants and intrinsic defects in HH compounds.
figure 1

a The crystal structure of HH compounds and (b), schematic diagram of performance asymmetry in ZrNiSn, NbCoSn, and NbFeSb HH compounds15,23,26,28,31,58.

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Typical HH compounds with high TE performance also encounter the doping asymmetry issue, as shown in Fig. 1b. For instance, the representative n-type ZrNiSn-based HH compounds achieved a TE figure of merit zT > 1 at about 1000 K7, establishing them as the prior choice for n-type TE legs in high-temperature power generation of HH materials8,9. However, achieving comparable p-type TE performance in ZrNiSn has been challenging over the past decade and remains unresolved10 Doping in the Zr site with Y11 or Sc12,13, Ni site with Co14 or Ir15, and Sn site with In16 were previously studied, but these endeavors have failed to achieve a maximum zT value greater than 0.3. The increase in the temperature-dependent electrical conductivity σ(T) in the above experiments suggests that the samples maintain the non-degenerate behavior11,12,13,14,15,16. Thus, the optimized p-type TE performance in ZrNiSn, where the σ(T) should generally present a degenerate behavior17, is far from reached. As another example, NbCoSn has attracted research interest due to its favorable p-type band structure18, characterized by a very high valence band degeneracy19. Theoretical studies based on band structure analysis have predicted that NbCoSn could potentially achieve a zT value up to approximately 220. However, experimental attempts using conventional p-type dopants have failed to yield a maximum zT higher than 0.1421,22,23,24, whereas the maximum zT for n-type NbCoSn reaches about 0.725,26,27. Luckily, such doping asymmetry does not happen in all the HH TE materials28,29,30,31. NbFeSb-based HH compounds, for instance, exhibit effective modulation of carrier concentrations across varied doping levels, demonstrating a wide doping tunability ranging from p-type28,32 to n-type31. Building upon the above-discussed doping challenges and asymmetry, unraveling the factor determining p-type dopability in HH compounds is highly desirable, given their high valence band degeneracy and thus p-type TE potential19.

It is worth noting that there have already been some previous efforts to understand the challenges of p-type dopability in HH compounds, particularly by focusing on the 4d interstitial defects23,25,32,33,34,35,36,37,38,39,40,41,42. Owing to the structural symmetry, the Wyckoff positions 4c and 4d, occupied by B elements and the vacancies (Fig. 1a), respectively, share a similar local chemical environment. Consequently, transition metals B could also occupy the 4d sites and form interstitials Bi, namely, Fei, Coi, and Nii, in NbFeSb, NbCoSn, and ZrNiSn, respectively. These interstitials at the 4d sites have been characterized through various techniques such as composition characterizations23,25,32,33,36,40, spectral analysis33,34,35,40, and transmission electron microscopy analysis38,42. Theoretical studies have attributed the difficulty in achieving p-type doping to the presence of these Bi interstitials37,39,41. The introduction of holes via extrinsic acceptors tends to shift the Fermi level towards the valence band, which lowers the formation energy of Bi, creating intrinsic “hole killers” that hinder the realization of p-type doping39. The presence of Bi introduces in-gap states, narrowing the bandgap and impacting the electronic transport properties. However, the differences in the p-type dopability among the typical HH materials are still elusive. The understanding of this issue will help find feasible strategies to achieve effective p-type doping in HH compounds.

In this work, by combining the defect calculations and the experimental studies, we try to address the issue by focusing on three representative HH materials, i.e., NbFeSb, NbCoSn, and ZrNiSn. Comprehensive first-principles studies on the intrinsic defects, dopants, and dopant/defect complexes were carried out. Our findings reveal that, in the undoped compounds, Bi and A vacancies (VA) are the primary defects with low formation energy. Furthermore, these two defects establish the dopability limit, determining the maximum doping level that can be achieved when introducing dopants. The p-type dopability of NbFeSb, NbCoSn, and ZrNiSn is systematically examined by employing different aliovalent dopants at three different Wyckoff positions. Taking NbCoSn as an example, p-type dopability has been experimentally scrutinized, validating the calculation results.

Results

Intrinsic defects in typical HH semiconductors

To comprehend the transport properties of NbFeSb, NbCoSn, and ZrNiSn, defect calculations of intrinsic defects were performed, and the results are shown in Fig. 2a–f. The band gaps in all defect formation energy figures are derived from calculations in primitive cells: 0.54 eV for NbFeSb, 0.99 eV for NbCoSn, and 0.48 eV for ZrNiSn. The chemical potential condition is determined by considering all elements and binary compounds with nonpositive energy above the Hull (detailed results are shown in Figure S4). Here, 4c/4d Frenkel pair VBBi (a point defect complex involving B vacancy and B interstitial) was excluded due to the high energy required for its formation39, which remains essentially unchanged regardless of the distance between VB and Bi. Experimentally, the as-prepared HH samples often show an excess of B43,44, making the B-rich condition depicted in Fig. 2a–c more reflective of practical scenarios. Therefore, we focus our discussion on the B-rich condition. Under B-rich condition, A vacancies VA (blue line) and B interstitials Bi (red line) have the lowest formation energies, demonstrating their stability and dominance in defect formation. Such results are reasonable considering the natural off-stoichiometry observed in experiments where there is an excess of B elements and a slight deficiency of A elements43,44.

Fig. 2: Intrinsic defects in typical Fe/Co/Ni-based HH compounds.
figure 2

The defect formation energy of intrinsic defects for a, d NbFeSb, b, e NbCoSn and c, f ZrNiSn under B-rich and B-poor conditions, respectively; g Epin/Eg and experimental Seebeck coefficients at room temperature11,14,16,23,24,46,47,48 in NbFeSb, NbCoSn, and ZrNiSn. In panels a-f, red and blue arrows indicate the p-type and n-type ‘dopability limit’, respectively, while red stars and red vertical dashed lines mark the pinned Fermi level.

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Bi defects act as electron donors (or hole traps), with the formation energy increasing as the Fermi level EF approaches the CBM. In ZrNiSn, the (−1/+1) charge transition level of Nii is located near the CBM, making them readily ionizable, which explains the experimentally observed high n-type carrier density (1.84 × 1020 cm−3 at room temperature)16. The (−3/+1) charge transition level of Fei for NbFeSb and the (0/+1) charge transition level of Coi for NbCoSn fall within the conduction band, suggesting that these defects could also be efficient electron donors.

The EF at which ΔHf of the dominant defects is zero represents the ‘dopability limit’45, as denoted by red and blue arrows in Fig. 2a–f. The defect concentration (nd) is determined by ΔHf and follows the relation nd ≈ N exp(−ΔHf / kBT), where N is a coefficient determined by vibrational entropy, kB is the Boltzmann constant, and T is the absolute temperature. When ΔHf becomes negative, the exponential term becomes positive, leading to the spontaneous formation of a high concentration of defects, which can destabilize the matrix. The ‘dopability limit’ points determined by Bi and VA explain the differences in the doping asymmetry for these three HH compounds. The ‘dopability limit’ point for NbFeSb is located within the valence and conduction band (Fig. 2a), which means that as the Fermi level approaches the valence or conduction band, the material can maintain structural stability, allowing for a significant doping range. Contrastingly, the p-type ‘dopability limit’ point for NbCoSn is approximately 0.4 eV above the VBM. This suggests that during p-type doping in this system, many Coi defects are likely to form, which could destabilize the structure and act as ‘hole killers’, thereby limiting the effectiveness of the p-type doping. The n-type ‘dopability limit’ point is about 0.1 eV near the CBM, indicating that n-type doping in NbCoSn can be more effective than p-type doping, which is consistent with experimental results23,24,26,27. Based on the differences in the relative positions of the ‘dopability limit’ point and the VBM in the three systems, the relative difficulty of achieving p-type doping is as follows: NbFeSb < ZrNiSn < NbCoSn.

The intrinsic conduction behavior can be indicated by another critical quantity, pinned Fermi level Epin, marked by red stars and vertical dash lines in Fig. 2a–f. It is derived from the charge neutrality condition ∑{D,q} q[Dq] + p – n = 0, where [Dq] is the concentration of defect D in charge state q. p and n are the concentrations of hole and electron carriers, respectively. The position of Epin relative to the CBM or VBM directly influences the carrier density and whether the material exhibits n-type or p-type behavior. The Epin for NbFeSb is slightly closer to the VBM compared to the CBM, suggesting a slight p-type behavior with a low hole concentration. In contrast, NbCoSn and ZrNiSn exhibit n-type behavior with relatively high electron concentration due to their Epin being closer to the CBM. As evident experimentally, the electron concentration for undoped NbCoSn ranges from 1.7 × 1020 cm−3 to 2.4 × 1020 cm−3 at room temperature23,25, while it ranges from 5 × 1019 cm−3 to 1.84 × 1020 cm−3 for undoped ZrNiSn16,36.

As mentioned earlier, the carrier type and concentration are directly influenced by the Epin. However, data on carrier concentration are not always provided in the literature. Therefore, we choose the Seebeck coefficients S at room temperature11,14,16,23,24,46,47,48 (summarized in Fig. 2g) to validate the consistency between theoretical predictions and experimental results. S at room temperature is widely documented in the literature and can reflect the carrier type, making it a reliable metric for verifying the predictions from Epin. For NbFeSb, the Epin is located near the middle of the forbidden gap. Consequently, in polycrystalline NbFeSb prepared by high-temperature melting, the measured S at 300 K shows only a minimal positive value47,48, probably owing to the coexistence of two types of charge carriers. In contrast, the Epin for NbCoSn and ZrNiSn is closer to the CBM, which indicates n-type behavior. Furthermore, the larger normalized Epin in ZrNiSn compared to NbCoSn is consistent with the experimentally observed higher absolute value of S in ZrNiSn.

In the equation of defect formation energy (Eq. (1)), the two independent variables, i.e., EF and μi, are not intrinsic properties but external variables, which implies their tunability. While the former has already been discussed, the focus now shifts to the μi. The formation energy of Bi increases under B-poor conditions while that of VB decreases, as shown in Fig. 2d–f. This pushes the Epin to approach near the valence band, thereby increasing the hole concentration. Meanwhile, the increase of formation energy of Bi causes the point of ‘dopability limit’ to shift towards the valence band, making it easier to achieve p-type doping. Therefore, manipulating chemical potential is another important method for tuning doping in HH semiconductors.

Extrinsic defects and complexes

Based on the above-discussed intrinsic defects and their effect on the dopability of HH semiconductors, here we focus on the extrinsic dopant’s role in determining doping effectiveness. A good p-type dopant should meet two main criteria based on defect formation energy calculations: first, the dopant should have a low formation energy, making it thermodynamically favorable for incorporation into the host material; second, the charge transition level should be close to the valence band to minimize thermal activation energy and facilitate efficient carrier generation.

For each site in the three compounds, we consider p-type dopants from the nearest neighboring column within the same period of the periodic table, such as Zr doping at the Nb site or Sn doping at the Sb site in NbFeSb. The Epin value under B-rich conditions for different doping conditions provides information on carrier types and concentrations, whose relationship with the experimentally reported S at room temperature14,16,24,28,48 is shown in Fig. 3a. As Epin shifts from VBM (Epin/Eg = 0) to CBM (Epin/Eg = 1), the material’s S decreases from positive to negative values, presenting a p-type to n-type transition. The consistency between the Epin and S at room temperature allows us to use Epin as a predictor of dopability in different doping scenarios. To directly associate Epin with p-type dopability, we divided Epin into three regions (Fig. 3a). The highly dopable region is marked by light red, where the Epin is close to the VBM, indicating a high hole concentration. Zr and Sn doping in NbFeSb fall into this region and exhibit a relatively large positive S. The white area represents the slightly dopable region, where the introduction of dopants provides a hole concentration that is significantly lower than the desirable level. Zr doping in NbCoSn and Co doping in ZrNiSn yield S around 50 μV/K at room temperature and a non-degenerate semiconductor behavior, indicating that these dopants are not effective. When Epin moves closer to the CBM, p-type doping is unachievable, which we refer to as an undopable region (marked by light blue). Indium doping in ZrNiSn falls in this region, suggesting that Indium is not a viable p-type dopant for ZrNiSn.

Fig. 3: Extrinsic doping and point defect complexes in typical Fe/Co/Ni-based HH compounds.
figure 3

a Experimental Seebeck coefficients14,16,24,28,48 at room temperature versus normalized pinned Fermi level in p-type ZrNiSn, NbCoSn, and NbFeSb with different dopants; Defect formation energies for (b), Zr-doped NbFeSb, (c) Zr-doped NbCoSn and (d) In-doped ZrNiSn. The red dashed line represents the self-consistent Epin at 300 K.

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To understand what causes the differences in the dopability, a representative example from each region is selected, and their defect formation energy and charge transition levels are shown in Fig. 3b–d. The defect formation energies of all nine doping scenarios can be found in Figures S5 and S6. As aforementioned, a good dopant should have low formation energy and (−1/0) charge transition level near the VBM. ZrNb in either NbFeSb or NbCoSn has negative formation energy, while for InSn in ZrNiSn, the formation energy is positive within the band gap. This corresponds with experimental reports where the doping concentration of the former reaches up to 20%24,28, whereas the latter is only about 1%16. The high formation energy of InSn makes it a poor candidate for p-type doping in ZrNiSn. Conversely, with a low formation energy and (−1/0) charge transition level near VBM, ZrNb exhibits high carrier concentration and is easily ionized. Therefore, Zr is supposed to be a favorable p-type dopant in both NbFeSb and NbCoSn.

The analysis of dopants supports the conclusion that Zr is an effective dopant for NbFeSb and NbCoSn, corresponding with findings from previous studies37. However, this result does not account for the observed difference in doping efficiency, as indicated by the Seebeck coefficient for Zr-doped NbFeSb and NbCoSn24,28. Specifically, the Seebeck coefficients for the former are approximately twice those of the latter, as shown in Fig. 3a. This discrepancy arises because previous studies did not consider the ‘dopability limit’ determined by intrinsic defects. The difference in ‘dopability limit’ governed by Bi in NbFeSb and NbCoSn explains the significant variation in Zr doping efficiency between the two systems. In NbFeSb, the formation energy of Fei is relatively high, and the ‘dopability limit’ falls within the valence band, while in NbCoSn, the ‘dopability limit’ lies within the bandgap. As a result, the Coi in NbCoSn has a much stronger compensating effect, compared to the Fei in NbFeSb, which leads to less effective Zr doping in NbCoSn.

The formation of defect complexes, where a dopant pairs with an intrinsic defect, can significantly impact the electronic properties and the overall doping effectiveness. For instance, the SeIVCu defect complex-induced shallow acceptor could be attributed to the observed increase in hole density in Se-doped CuI film49. Since HH compounds are heavy-band materials, the concentration of dopants (can be up to 10% and even above) required for optimal carrier concentration in HH TE compounds is very high50, the physical distance between dopants and Bi can be much closer, in contrast to the low-doping scenarios. Additionally, the negatively charged dopants and positively charged Bi are likely to experience Coulombic attraction, making their interaction more probable. Thus, we also consider the formation of the dopant/interstitial complex. Here, NbCoSn is taken as an example to show how much the dopant/interstitial complex can affect doping (Figure S7). The orange area represents the region where this defect can form, with its formation energy lower than the sum of the individual defect formation energies, situated near the valence band. In Co-rich conditions, the dopability limit is far from the valence band maximum, preventing complex formation. Thus, the Epin remains unchanged regardless of complex considerations (Figure S7a). In Co-poor conditions, the complex defects result in the pinned Fermi level slightly shifting deeper into the bandgap (only about 0.02 eV, Figure S7b), suggesting a negligible effect of complex defects on the dopability.

In short, the effectiveness of p-type doping in HH compounds is closely linked to both the characteristics of the dopants and the Bi presented in the host material. A suitable p-type dopant should possess low formation energy and have its (−1/0) charge transition level near the valence band, which facilitates high hole concentration. However, the Bi can compensate for holes and set the ‘dopability limit’. Therefore, it is essential to consider both the dopants and the ‘dopability limit’ in the host material to enable an effective experimental doping design.

Dopant selection in NbCoSn

Due to the advantageous valence band structure of NbCoSn18, there has been research interest in exploring its p-type doping potential21,22,23,24. However, these efforts failed to yield the expected improvements21,22,23,24. Through discussions on intrinsic defects and dopants, we have identified Coi as a primary factor limiting successful p-type doping. To overcome this, two strategies can be employed: reducing the formation of Coi and selecting more effective dopants. With these factors in mind, we perform defect calculations on Ti and Hf, elements from the same group as Zr, examining their effects under both Co-rich and Co-poor conditions.

As shown in Fig. 4a–c, the formation energy of Hf is close to Zr and lower than Ti, indicating that Hf is more thermodynamically favorable than Ti in NbCoSn. Moreover, the (-1/0) charge transition state of Zr is located within the band gap, whereas for Hf, it is in the valence band, suggesting that Hf is more easily ionized. Additionally, as a heavy element, Hf can further enhance thermoelectric performance by reducing the lattice thermal conductivity47,51,52. Considering these factors, Hf is identified as the suitable p-type dopant for NbCoSn. Besides, an increase in the formation energy of Coi can reduce their formation, which is modeled in the calculations by adjusting the chemical potential of Co. Figure 4a–c represent Co-rich conditions, while Fig. 4d–f illustrate Co-poor conditions. The red dashed line indicates the calculated Epin, and the gray shaded region shows the tunable range of Epin based on the change in the Co chemical potential. From Co-rich to Co-poor conditions, the Epin shifts towards the valence band, suggesting that providing a Co-deficient environment could enhance p-type doping in NbCoSn.

Fig. 4: Defect formation energy of extrinsic defects and complexes in aliovalence-doped NbCoSn.
figure 4

a, d Ti-doped, b, e Zr-doped, c, f Hf-doped NbCoSn under B-rich and B-poor conditions, respectively. The red dashed line represents the self-consistent Epin at 300 K.

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Hf doping in NbCoSn

Based on the above calculations, a series of Nb1-xHfxCoSn (0 ≤ x ≤ 0.2) samples were synthesized. The X-ray diffraction (XRD) patterns and background electron (BSE) images are presented in Figures S8 and S9. The actual composition, determined by EPMA is shown in Table S1. All examples exhibit an excess of Co relative to the nominal composition, which is consistent with the calculation results and indicates the dominance of Co interstitials. Additionally, the actual Hf content closely matches the nominal doping contents, suggesting the effective incorporation of Hf into the host material. The calculated density of states and the band structure for Nb31HfCo32Sn32 are shown in Figures S10 and S11.

The changes in the sign of Seebeck coefficients validate the effectiveness of Hf as a p-type dopant (Fig. 5a). The undoped NbCoSn exhibits a negative Seebeck coefficient above room temperature, indicating n-type behavior attributed to the presence of ionized Coi. Upon Hf doping, the Seebeck coefficients approach zero at room temperature and shift to positive values with increasing magnitudes at elevated temperatures. Electrical conductivity provides insight into the relationship between Hf doping content and effective hole doping. As shown in Fig. 5b, the electrical conductivity of all Hf-doped samples increases at elevated temperatures, demonstrating the behavior of nondegenerate semiconductors. This is consistent with the theoretical result, i.e., Epin is far from VBM (Fig. 2). In addition, as an aliovalent dopant with heavy atomic mass, Hf effectively suppresses the lattice thermal conductivity of NbCoSn. Consequently, a peak zT of 0.35 is obtained at 1023 K for p-type Nb0.8Hf0.2CoSn (Fig. 5c), making it the highest one achieved for p-type NbCoSn to date.

Fig. 5: Electrical properties in Nb1-xHfxCoSn (0 ≤ x ≤ 0.2) and Nb0.85Hf0.15Co1+ySn (−0.1 ≤ y ≤ 0.1).
figure 5

Temperature dependence of (a), Seebeck coefficient (S), (b) electrical conductivity (σ), and (c), the figure of merit zT for Nb1-xHfxCoSn. Nominal Co content dependence of (d), σ, and (e), S for Nb0.85Hf0.15Co1+ySn.

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Based on the above theoretical analysis, modulating the chemical potential of Co can also promote p-type doping in NbCoSn, which can be achieved by varying the nominal Co content in experiments. To examine this, we synthesized a series of samples with nominal compositions Nb0.85Hf0.15Co1+ySn (−0.1 ≤ y ≤ 0.1) to investigate p-type doping efficiency. The Nb0.85Hf0.15CoSn is selected as the matrix instead of Nb0.8Hf0.2CoSn to avoid the impurity phases of Nb3Sn and Sn observed in the latter that might complicate the analysis. The XRD patterns of Nb0.85Hf0.15Co1+ySn (Figure S12a) indicate that the material maintains a single phase even when y reaches 0.1, whereas Co-deficient samples exhibit impurity phases of Nb6Sn5 and Sn. All the Nb0.85Hf0.15Co1+ySn samples exhibit positive Seebeck coefficients above room temperature (Figure S12), which allows for a direct comparison of the changes in the electrical conductivity as nominal Co content varies. As shown in Fig. 5d, the suppression of the formation of Coi, as the nominal Co content shifts from excess to deficiency, enhances p-type doping and improves electrical conductivity above room temperature. The Seebeck coefficient serves as a valuable reference at high temperatures, where the Hf dopant is largely activated and contributes holes as majority carriers. The Seebeck coefficients at 973 K decrease with increasing degree of Co deficiency, confirming that Co deficiency promotes p-type doping (Fig. 5e), although it is still not reaching the optimal doping, as indicated by the non-degenerating behavior of electrical conductivity (Figure S12c).

Discussion

In this work, we theoretically investigate the p-type dopability in half-Heusler thermoelectric semiconductors through comprehensive first-principles calculations of intrinsic defects, dopants, and complexes by focusing on three representative systems, i.e., NbFeSb, NbCoSn, and ZrNiSn. We uncover that the formation of Bi significantly impacts the effectiveness of p-type doping. By systematically investigating the extrinsic doping at the three Wyckoff positions in NbFeSb, NbCoSn, and ZrNiSn, respectively, we highlight that the pinned Fermi level serves as an indicator of p-type dopability. Experimentally, we found that Hf doping successfully transforms NbCoSn from n-type to p-type. A peak zT of 0.35 is obtained at 1023 K for Nb0.8Hf0.2CoSn, the highest reported value to date for p-type NbCoSn. Our work provides insight into the challenges of p-type doping in HH compounds and offers guidance for the experimental selection of host materials and dopants to facilitate p-type doping.

Methods

Theoretical calculations

First-principle calculations were performed using Density Functional Theory (DFT) as implemented in the Vienna Ab initio Simulation Package (VASP)53,54. Projector augmented wave (PAW) pseudopotentials55, the generalized gradient approximation (GGA) for the exchange-correlation functional introduced by Perdew, Burke, and Ernzerhof (PBE)56 and a plane wave basis set with wavefunction cutoff energy of 300 eV were employed. The crystal structure of considered materials was optimized using k-points generated via VASPkit57 to create a k-mesh with a spacing value of 0.03, and the optimization ensured that the Hellmann–Feynman forces acting on each atom were kept below 10−2 eV per atom.

A single defect or defect complex was put at the center of the periodic supercell to calculate the formation energy. Here, we adopted 2 × 2 × 2 cubic supercells containing 96 atoms since the dielectric constant of half-Heusler compounds is high and the size of 2 × 2 × 2 is sufficient to screen the interactions between defects, ensuring that these defects can be considered as isolated. A small random displacement of 0.1 Å was applied to a nearest-neighbor atom of the defect before geometry optimization to break the initial symmetry, allowing the system to fully relax into its most stable configuration which may potentially exhibit Jahn-Teller distortions. The atomic positions were then relaxed while the crystal lattice was fixed. A single Γ point was used for supercell calculations. For charged defects, we considered possible spin arrangements of spin pairing or parallel alignment and identified the state with the lowest energy as the ground state for subsequent calculations. Temperature-dependent effects, including vibrational entropy and thermal expansion, are neglected as they don’t qualitatively affect our conclusion (see Supplementary note 1 and Figures S1 and S2 for details).

To determine the defect formation energy and defect transition levels, total energy E(Dq) for a supercell containing relaxed defect D in charge state q, total energy E(host) of the host for the same supercell in the absence of defect as well as the total energies of the involved elemental solids at their stable phases were calculated. From these quantities, the defect formation energy ΔHf(Dq) can be given as a function of the electron Fermi energy EF and the atomic chemical potential μi:

$$\Delta {H}_{{\rm{f}}}({D}^{q})=E({D}^{q})-E({\rm{host}})+{\sum }_{i}{n}_{i}{\mu }_{i}+q{E}_{{\rm{F}}}+{E}_{{\rm{corr}}}$$
(1)

EF is referenced to the host’s valence band maximum (VBM). μi is the chemical potential of constituent i referenced to elemental solid/gas with E(i), whose range depends on the formation enthalpy of all the competing phases. ni is the number of elements added into or removed from the host. Ecorr accounts for the finite-size corrections within the supercell approach. To eliminate the electrostatic potential shift, the potential of the atom farthest from the defect in the defect-containing supercell is aligned with that of the corresponding atom in the defect-free supercell.

The defect charge transition level ϵ(q/q’) is the Fermi energy at which the formation energy ΔHf(Dq) of defect D in charge state q is equal to that of another charge state q’ of the same defect:

$${\epsilon }(q/q{\text{'}})=[E({D}^{q})-E({D}^{q{\text{'}}})\!\left.\right)]/(q{\text{'}}-q)$$
(2)

Synthesis

The ingots with nominal composition Nb1−xHfxCoSn (0 ≤ x ≤ 0.2) and Nb0.85Hf0.15Co1+ySn (−0.1 ≤ y ≤ 0.1) were prepared by levitation melting of the stoichiometric amount of Nb (foil, 99.80%), Hf (slice, 99.99%), Co (block, 99.99%), and Sn (particle, 99.99%) under an argon atmosphere for about 3 min. Each ingot was remelted four times to ensure homogeneity and no raw material remained. The ingots were grounded manually into powders and mechanically milled (SPEX-8000D, PYNN Corporation) for 60 min under argon protection. The fine powders were compacted by spark plasma sintering SPS (LABOX-650F, Sinter Land Inc.) at 1123 K under 65 MPa under vacuum for 10 min.

Characterization and measurement

The phase structure of the sintered samples was evaluated by XRD on PANalytical Aries DY866 using Cu-Kα radiation (λ0 = 1.5406 Å), and the lattice parameters were obtained accordingly (Figure S3). The chemical compositions of all sintered samples were checked by the electron probe microanalyzer (EPMA, JEOL JXA-8100) with a wavelength dispersive spectroscope (WDS). EPMA analysis was conducted at five randomly chosen positions for each sample and then calculate the average result. The Seebeck coefficient S and electrical conductivity σ were measured on a commercial Linseis LSR-3 system with accuracy of ±5% and ±3%, respectively. The thermal conductivity κ was calculated by using κ = DρCp, where ρ is the sample density estimated by the Archimedes method, and D is the thermal diffusivity measured by a laser flash method on Netzsch LFA457 instrument with a Pycoceram standard. The specific heat at constant pressure Cp was calculated using Cp = Cph, H + CD, where Cph, H, and CD can be calculated by sound velocity, thermal expansion coefficient, and density. Normal and shear ultrasonic measurements were performed at room temperature using input from a Panametrics 5052 pulser/receiver with a filter at 0.03 MHz. The response was recorded via a Tektronix TDS5054B-NV digital oscilloscope.