Introduction

TE materials have been researched and developed for 200 years since the Seebeck effect was discovered in 1821. Due to the promising applications of TE materials in waste heat power generation and solid-state refrigeration, persistent efforts have been made to improve TE performance1. Metals were studied as TE materials firstly, but they are no longer considered as good TE materials because of their small Seebeck coefficients and low zT values (zT = S2σT/κ, where S, σ, T, and κ are the Seebeck coefficient or thermopower, electrical conductivity, absolute temperature, and total thermal conductivity, respectively). Instead, the best TE materials at present, such as Bi2Te32, PbTe3, GeTe4, and CoSb35, are all heavily-doped narrow-bandgap semiconductors. Thus, the currently common consensus in the TE field is that the first-rank TE material should be a heavily-doped narrow-bandgap semiconductor6, which can achieve the optimal power factor S2σ by balancing the Seebeck coefficient and electrical conductivity7. As a result, an optimized zT, which determines the TE conversion efficiency, is realized.

After about 70 years of development, the only commercialized TE material is the Bi2Te3-based material with a peak zT around the unity. In addition, reliably reproducible zT is around 2.0, the reported zT values above 2.0 are often in debate, and all belong to heavily-doped narrow-bandgap semiconductors1,8,9. It seems desperately needed to break through the limitation of narrow-bandgap semiconductors for discovering high-performance TE materials. Many attempts have been made to explore non-semiconductor TE materials. High spin entropy was once believed to enhance the TE performance of a few lamellar cobalt oxides10,11,12. For instance, NaxCo2O4 with Co3+/Co4+-determined spin entropy achieves Seebeck coefficients about 100 and 200 μV K−1 at 300 K and 800 K, repectively13,14. Up to now, the maximum zT value of this type of oxide approaches ~1.0 at 800 K after a long-time endeavor15, halted in thermodynamics by the spin entropy limit from d-band degeneracy.

Over the years, a few rare-earth 4f-electron-based heavy-fermion systems were found to show metallic electrical conductivities and relatively large Seebeck coefficients16 at extremely low temperatures. There also exist a few exceptions to the recognized heavy-fermions as YbAl317 and CePd318, showing thermopower as high as ~100 μV K−1 at room temperature. Arguably, this was also considered as hybridizing f electrons with conduction band, resulting in a resonant peak near the Fermi surface19,20 as in normal materials. The peak zT values of CePd318 and YbAl317 reach about 0.2 and 0.3 at 300 K, respectively. Nevertheless, the optimal Seebeck coefficients for both systems reach only <120 μV K−1 and impede further enhancement of their TE performance. Very recently, the spin fluctuation was reported to enhance the thermopower of weak itinerant ferromagnetic alloys Fe2V0.9Cr0.1Al0.9Si0.1 and Fe2.2V0.8Al0.6Si0.421 around the Curie temperature TC. A broad shoulder-like hump on the S(T) ~ T curve was observed around TC, leading to only a 15% to 20% enhancement of the Seebeck coefficient at TC. The well-known NFL22 superconducting oxides, such as YBa2Cu4O823 and La2-xSrCuO424, also show certain intriguing TE behavior, but these oxides always have negligible zT values, several orders of magnitude lower than the state-of-the-art TE semiconductors.

Conceptually, there is no substantial breakthrough in the search for non-semiconductor TE materials. In this work, we find that a few Heusler-like materials, TiFexCu2x−1Sb (x = 0.70, 0,75, 0.80) and TiFe1.33Sb with excess Fe/Cu occupying the vacant sites of the half-Heusler lattice, show peculiar NFL metallic transport together with promising TE performance. A high power factor of 20.8 μWcm−1 K−2 and a zT value of 0.75 are achieved in TiFe0.7Cu0.4Sb at 973 K, and the other systems also show zTs higher than 0.3, all comparable with the best half-Heusler TE semiconductors.

Results and discussion

Unconventional metallic transport properties

Figure 1 plots the resistivity ρ(T) of TiFexCu2x−1Sb and TiFe1.33Sb. They all show quasi-linear dependence in the range from near-zero temperature up to 500 K, indicating that both materials are unconventional metals. This temperature dependence of resistivity completely precludes that TiFexCu2x−1Sb and TiFe1.33Sb are intrinsic or doped semiconductors, which should have the maximum resistivities at 0 K. Another typical characteristic of metallicity is the vanishing small Seebeck coefficients, S(T), when approaching 0 K (the inset of Fig. 1). The Seebeck coefficients of the TiFexCu2x−1Sb and TiFe1.33Sb gradually increase and become comparable to those of heavily-doped narrow-bandgap semiconductors at 300 K and higher. Moreover, the quasi-linear- or linear-T dependence of ρ(T) (particularly, the resistivity of TiFe1.33Sb is almost perfectly linear with temperature in 2–100 K) in the above materials is valid in a wide temperature range (0–700 K in Fig. 1; see also Supplementary Fig. 1a for a wider range of 0–973 K) while stably keeping the thermopower as high as approaching 200 μV K−1. By fitting to the low-temperature resistivity with the Bloch–Grüneisen model25 (Supplementary Note 1 and Supplementary Fig. 2 and Table 1), the parameter of Debye-temperature term is estimated to be as low as ~40 K, much lower than the phonon-mediated Debye temperature of 389 K estimated from the measured sound velocity (Supplementary Note 2 and Supplementary Table 2). X-ray photoemission spectroscopy (XPS) experiments reveal that the Fe element in both TiFe0.7Cu0.4Sb and TiFe1.33Sb has a valence of zero, and their Fe-2p XPS spectra are almost identical to the metallic Fe, FeSe and FeTe but are obviously different from FeO and Fe2O3 (Supplementary Fig. 3). The resistivity of normal or conventional metals shows the ρ(T) ~ T2 dependence in Landau-Fermi-liquid paradigm, in contrast, the linear and quasi-linear temperature dependence of resistivity [i.e., ρ(T) ~ T] is usually considered as a canonical signature of the NFL metals. Notice also that the resistivities of the above materials, ρ(T) ~ 10–6–10–5 Ω m, are one to two orders of magnitude higher than the constantan26 (Cu0.6Ni0.4, an alloy of copper and nickel), the best TE metal. The electrical mean-free paths could be estimated to be as low as 0.47 nm in TiFe0.7Cu0.4Sb and 0.73 nm in TiFe1.33Sb (Supplementary Note 3 and Supplementary Table 3), which are close to or even smaller than the lattice constants. All the above observations point to the bad metallicity and unconventional metallic transport in TiFe0.7Cu0.4Sb and TiFe1.33Sb.

Fig. 1: Temperature-dependent electrical resistivities and Seebeck coefficients of TiFexCu2x-1Sb and TiFe1.33Sb samples.
figure 1

The data of the Cu0.6Ni0.4 alloy (solid orange line) are included for comparison27. ρ0 is the resistivity at 0 K, which is obtained by fitting to the low-temperature resistivity using the Bloch–Grüneisen model (Supplementary Methods and Supplementary Fig. 2 and Table 1). The resistivity of Cu0.6Ni0.4 shown in this figure is the raw resistivity without eliminating the ρ0 part.

Full size image

Exceptional thermopower and TE performance

Despite the metallicity, the above materials exhibit promising TE properties (Fig. 2). The thermopower increases continuously with the rising of temperature up to 600 K. The Seebeck coefficient of TiFe0.7Cu0.4Sb reaches 194 μV K−1 at 700 K, the highest among all known TE metals, with the reported peak value of ~60 μV K−1 for constantan27, ~−120 μV K−1 for YbAl317, and ~110 μV K−1 for CePd318. There appears to be a nonlinear turnover above 600 K, which could be ascribed to a structural transition (Supplementary Fig. 4), but all the materials still maintain high values of thermopower (Supplementary Fig. 1b) while simultaneously a reasonably low resistivity (Supplementary Fig. 1a) at a temperature close to 1000 K. Coupled with the very low thermal conductivity (Supplementary Fig. 5), the metallic TiFexCu2x−1Sb and TiFe1.33Sb show remarkable TE performances, and the TiFe0.7Cu0.4Sb reaches a zT value of 0.75 at 973 K, comparable with or even better than those of typical half-Heusler TE semiconductors26,28,29,30,31. This value of zT is recognized to be the highest one among all the ternary Heusler or Heusler-like materials including those half-Heusler TE semiconductors optimized only at the transition metal (Fe/Co/Ni) site26,29,30,31 (Supplementary Fig. 6 and Table 4).

Fig. 2: Temperature-dependent zT values and Seebeck coefficients of TiFexCu2x-1Sb and TiFe1.33Sb, all in lines with symbols.
figure 2

The zT values of CePd3, Yb1.05Al3, ZrCo0.9Ni0.1Bi, TiFe0.6Ni0.4Sb, TiFe0.4Ni0.6Sb, ZrCo0.85Ni0.15Sb, NbFe0.94Ir0.06Sb, and NbFe0.94Co0.06Sb, and the Seebeck coefficients of CePd3, Yb1.05Al3, La1.95Sr0.05CuO4, La1.925Sr0.075CuO4, La1.9Sr0.1CuO4, and YBa2Cu4O8 are taken from the literature for comparison17,18,23,24,26,29,30,31, all in lines without symbols.

Full size image

Logarithmic temperature-dependent electronic specific heat

To elucidate the origin of the exceptional TE properties of the metallic TiFexCu2x−1Sb and TiFe1.33Sb materials, low-temperature specific heat and resistivity were also measured (Fig. 3). Interestingly, the temperature-dependent electronic specific heat Cel of both TiFe0.7Cu0.4Sb and TiFe1.33Sb follow the relationship Cel/T ~ −lnT (see Supplementary Note 4 and Supplementary Fig. 7 and Table 5 for details of Cel extraction), while the resistivities of those systems still show nearly linear dependence on the temperature. The observed coexistence of the quasi-linear temperature dependence of resistivities and the logarithmic temperature-dependent specific heats strongly hints at the typical features of NFL behavior32,33 and quantum critical phenomenon as will be discussed later.

Fig. 3: Temperature-dependent electronic specific heat capacities and resistivities of the TiFe0.7Cu0.4Sb and TiFe1.33Sb samples.
figure 3

Note that the electronic specific heat coefficients are at least one order-of-magnitude higher than those of pure metals (for example, Cu49 and CuNi50) and even approach those of rare-earth-related heavy Fermi systems51.

Full size image

Random distribution of Fe/Cu

All the TiFexCu2x−1Sb and TiFe1.33Sb samples crystallize in a single-phase Heusler-like structure with the space group \(F\bar 43m\) even with excess Fe/Cu atoms as revealed by the X-ray powder diffraction (XRD) patterns (Supplementary Figs. 8 and 9). The structure of TiFe1.33Sb agrees with that of Tavassoli et al.34 who reported the phase relation and TE properties of Ti1+xFe1.33-xSb. All the samples exhibit polycrystalline characteristics with irregular grain sizes, typically on the order of several hundred nanometers (Supplementary Fig. 10 and Table 6). The measurements on the TiFe1.33Sb samples sintered under different pressures indicate that only the mass density has a slight effect on the thermoelectric transport properties, while the grain size up to several hundred nanometers shows no obvious effect on the magnetic and thermoelectric properties (Supplementary Figs. 11–14). In addition, the sample does not show any preferred growth orientation (Supplementary Figs. 15 and 16).

Both the high-angle annular dark-field scanning transmission electron microscopy (HAADF-STEM) image and energy-dispersive X-ray spectroscopy (EDS) maps (Fig. 4a) clarify a uniform distribution of compositions in the samples. The aberration-corrected high-resolution HAADF image along the [110] direction reveals that both the 4c and 4d Wyckoff sites in the half-Heusler lattice are partially occupied for TiFe0.7Cu0.4Sb and TiFe1.33Sb (left panels of Fig. 4b, c), and the corresponding intensity profiles of HAADF reveal no obvious difference in the atomic occupancies of the 4c or 4d sites (right panels of Fig. 4b, c). The integrated differential phase contrast (iDPC) image also indicates that the 4c and 4d sites of TiFe0.7Cu0.4Sb are nearly randomly occupied (Fig. 4d). All the above observations about structures clearly conclude that the 4c and 4d sites in the Heusler lattice are nearly randomly occupied by the Fe/Cu atoms without noticeable preference. This is reasonable because the 4c and 4d sites in a Heusler lattice could be considered to be equivalent to each other from a pure crystallography point of view, with identical both the nearest neighbors and cubic symmetry.

Fig. 4: Microstructures of TiFe0.7Cu0.4Sb and TiFe1.33Sb.
figure 4

a HAADF-STEM image and corresponding EDS maps for TiFe1.33Sb. b, c Aberration correction high-resolution HAADF image along a [110] direction and the corresponding intensity profiles across the horizontal lines for TiFe1.33Sb and TiFe0.7Cu0.4Sb, respectively. d iDPC image along a [110] direction of the TiFe0.7Cu0.4Sb sample.

Full size image

Origin of the bad metallicity and quantum criticality

The TiFexCu2x−1Sb and TiFe1.33Sb samples show a canonical signature of the NFL metals, i.e., the linear temperature dependence of resistivity [ρ(T) ~ T]. While the measured electrical transport implies a breakdown of the quasiparticle-based Fermi-liquid picture, it would be more exciting for the discovery of the NFL TE metals with high zTs at (or above) room temperature and its proximity to a quantum critical point with logarithmic-in-T specific heat, which may provide a basis for a possible paradigm shift of research. To get insight into the origin of the exotic physical properties, density-functional-theory (DFT) calculations are carried out to understand the electronic structures of the materials. To include the 4c-4d random occupation, the Fe4c-Fe4d disordering is simulated in a superstructure for TiFe1.33Sb, in which 216 tetrahedral sites (108 4c and 108 4d) are randomly occupied by 144 Fe ions (Supplementary Fig. 17) based on the above experimental observations. It should also be addressed that different configurations of the superstructure with quasi-random occupancy of the Fe sites lead to almost identical conclusions (Supplementary Fig. 18).

The calculated electronic structures predict TiFe1.33Sb as a metal, and the band-edge states around the Fermi level (EF) show a mixture of Fe-3d, Ti-3d, and noticeable Sb-5p electrons (Fig. 5a). This is understandable because the current materials could be considered as embedding Fe/Cu atoms into the closely packed Ti-Sb matrix. While the interaction in the Ti-Sb matrix is determined by Ti-3d and Sb-5p orbitals, Fe atoms at the 4c-4d sites bond through Fe-3d-orbitals with both Ti-3d and Sb-5p orbitals in the matrix. Due to the well-kept cubic symmetry, the Fe-3d states close to the EF are dominantly the eg states with 2-fold degeneracy and partially unfilled. Spin-polarized calculations reveal that Fe atoms have a strong magnetic instability (Fig. 5b) to save considerable energies (>40 meV/Fe atom) compared with the spin-unpolarized case. The Fe-3d-based local magnetic moments are relatively diverse, covering from ~zero to values as high as ~1.0 μB or even higher (Fig. 5b), and fluctuate from site to site. The nominally empty Ti-3d orbitals have small electron occupation and are passively polarized through the hybridization with the Fe-3d orbitals, and show relatively small magnetic moments fluctuating around 0 ~ ±0.4 μB. It should also be addressed that the band-edge Ti-3d electrons in TE half-Heusler semiconductors with Ti-Sb matrix are dominantly itinerant as understood before30,34. Sb-5p electrons show itinerant characteristics and are nearly uniformly distributed. Figure 5b, c depict the whole picture of the distributions of both fluctuating local magnetic moments and dual-type itinerant electrons around the EF.

Fig. 5: Bad metallicity, antiferromagnetism, transport networks of itinerant electrons, and spin fluctuations in TiFe1.33Sb from theoretical simulations.
figure 5

a Electronic density of states. b Spin density of a (001) plane containing Fe4c and Fe4d atoms with the local magnetic moments denoted by the number below the atoms. A redder (bluer) color denotes a more positive (negative) value, and the rule also applies to subplots (c). c Electron density in logarithmic scale on a (110) plane from EF − 0.1 to EF + 0.1, where EF is the Fermi level (spin-up channel only, See Supplementary Fig. 17 for spin-down channel).

Full size image

There are a few specific aspects that make the current materials intriguing and interesting. Firstly, it is the Fe-3d-eg-states with fluctuating local magnetic moments that embed in a metallic environment. The bandwidth of the eg-states close to EF is found to be substantially narrow, only about ~1.5 eV, due to the much expanded nearest-neighboring Fe-Fe distance (~3 Å in TiFe1.33Sb, compared with 2.48 Å in the BCC Fe) with the constraint from the Ti-Sb matrix. The on-site Coulomb interaction U for the eg electrons is evaluated by the constraint-random-phase-approximation (CRPA) and is around 1.7 eV, clearly implying the role of strong electron correlation. They point to the special role of the localized Fe-3d-eg electrons, and naturally refer to the fact of magnetic impurities in an itinerant electron environment. This has been studied for a long time that the localized states, Fe-3d-eg in the current case, with magnetic moment introduce strong correlation effects and Kondo effect into itinerant electron system and thus the NFL behavior35,36. Secondly, there exist symmetry-enforced 2-fold degenerate Fe-3d-eg states coupled with the dual-type itinerant electrons, mixing of the Sb-5p and Ti-3d, close to the EF. This is also a strong indication of the multi-channel Kondo effect that has also been investigated early for multiple-orbital magnetic impurities in a metallic environment37,38. Both points could lead to quantum criticality. Thus, the magnetic quantum criticality, suggested by the logarithmic-in-T specific heat, also finds microscopic signatures from the simulated electronic properties.

These materials show two types of AFM interactions, i.e., Fe4c-Fe4d AFM at the nearest-neighbor 4c-4d sites and Fe-Ti AFM, unaffected by the structural complexity due to the 4c/4d occupation disordering. The Fe4c-Fe4d AFM (Fig. 5b) tends to form a checkerboard-like magnetic ordering, although being constantly disrupted by the disordered vacancies. Calculations prove that Ti-3d orbitals always develop the opposite spin densities around the nearby Fe-3d ions, providing antiferromagnetically-polarized electrons to form a spontaneous spin compensation. The two AFM interactions also frustrate each other in a Fe4c-Fe4d-Ti triangular local geometry. The effective Fe-Ti exchange interaction has a relatively strong strength of 30 meV or even higher due to the effective overlap between the itinerant Ti-3d orbitals and the Fe-3d states. Because of the good itinerancy of Ti-3d states, the discussion about the multi-channel Kondo effect takes Ti-3d electrons as heavy itinerant ones, which is considered to be reasonable based on the transport data from many Co-based half-Heusler semiconductors39,40,41,42,43.

Notice that both TiFexCu2x−1Sb and TiFe1.33Sb contains strong disordering of the magnetic atoms, which significantly enhances the fluctuations of the local magnetic moments of Fe atoms. It is thus reasonable to infer that the observed NFL and quantum criticality come from the integrated effects of magnetic-impurity-induced scattering, Fe4c-Fe4d AFM ordering at the nearest-neighbor sites, and the multi-channel Kondo effect, and all are strongly mediated by the random distribution of magnetic atoms. It is therefore expected that the NFL and quantum criticality should manifest localization character to some extent, even hardly quantified for now. Low-temperature magnetic susceptibility indicates that the TiFe1.33Sb sample is AFM coupling with a Curie-Weiss temperature TCW of −56 K by fitting with the Curie-Weiss law (Supplementary Note 5 and Supplementary Fig. 19). Nevertheless, the magnetic susceptibility below 100 K offsets from the fitting curve, implying that the normal Curie-Weiss law is not applicable to the sample and distinct magnetic and spin fluctuations may exist. Interestingly, the magnetic susceptibility of the sample can be well described in the temperature ranges of both 2−100 K and 2−300 K (Fig. 6) by the model x(T) = 1/(θ + BTα) established for the system with local quantum criticality44. The power index α, reflecting the dependence of the magnetic susceptibility on the temperature, is 0.79, agreeing surprisingly well with the value of 0.75 as theoretically derived in the literature44.

Fig. 6: Magnetic properties of TiFe1.33Sb.
figure 6

Temperature-dependent magnetic susceptibility of TiFe1.33Sb.

Full size image

In summary, we discover that the NFL metal showing a quantum critical behavior may be a type of high-performance TE material. The random filling of Fe/Cu atoms on the equivalent 4c/4d sites of the Heusler metals TiFexCu2x−1Sb and TiFe1.33Sb, leads to the structure disorder, strong magnetic fluctuation, and quantum critical behavior of these non-conventional metals. The Seebeck coefficient of TiFexCu2x−1Sb and TiFe1.33Sb increases continuously with the increase of temperature, while a low resistivity is maintained at the same time. Combined with the very low thermal conductivity due to the disordered structure, the TiFexCu2x−1Sb and TiFe1.33Sb samples achieve the TE performance comparable to or even better than the traditional TE semiconductors. The TiFe0.7Cu0.4Sb sample shows a Seebeck coefficient of 194 μV/K at 700 K and a zT value of 0.75 at 973 K, demonstrating the potential to search for high-performance TE materials along this direction.

Methods

Sample synthesis

Polycrystalline TiFexCu2x−1Sb samples (x = 0.7, 0.75, and 0.8) and TiFe1.33Sb were prepared by high-energy ball milling and spark plasma sintering (SPS). The starting materials (Ti shots 99.6%, Fe pieces 99.99%, Cu shots 99.99%, and Sb shots 99.999%) were weighed according to the nominal composition of the sample and loaded into the Ar-protected stainless-steel jar, which were ball-milled for 30 h using the SPEX 8000M Mixer/Mill. Then, the ball-milled powders were put into a graphite die with an inner diameter of 12.7 mm and consolidated into pellets under 60 MPa by SPS at 973 K for 20 min. The relative mass densities of the samples are between 94% and 98% (Supplementary Table 7).

Structure characterization

Phase identification and crystal structure analysis were carried out with high-resolution powder XRD patterns collected by a Rigaku SmartLab-II diffractometer with Cu-Kα radiation. The microstructures of the samples were examined by a high-resolution TEM (JEM-F200, JEOL, Japan) and a probe Cs-corrected TEM (Themis ETEM, Thermo Fisher Scientific, USA). TEM specimens were prepared by mechanical slicing, polishing, and dimpling, followed by ion-milling. Energy-dispersive X-ray spectroscopy was used to determine the distribution of elements at the nanoscale.

Transport properties measurements

High-temperature electrical and low-temperature thermal transport properties were performed on a rod-like sample cut directly from the SPS particle. Electrical conductivity (σ) and Seebeck coefficient (S) above room temperature were measured by a four-probe method using a ZEM-3 system (ULVAC-RIKO, Japan). Low-temperature TE properties (5–350 K) including σ, S, and κ were measured using a physical property measurement system (PPMS, Quantum Design, USA) with the thermal transport option (TTO). The thermal conductivity at high temperature is calculated by κ = λρdCP, where λ is the thermal diffusivity, ρd is the density of the sample, and CP is the specific heat capacity. λ was tested by a laser flash method (LFA 467, NETZSCH, Germany), CP was estimated by Dulong-Petit law, and ρd was measured by the Archimedes method. Room temperature sound velocity was measured by the ultrasonic material characterization System (UMS-100, TECLAB, France). Low-temperature specific heat capacity (CP) was measured by the PPMS with a dilution refrigerator option (PPMS-DR, Quantum Design, USA).

Density functional theory simulation methods

The vacancy disordering effect is considered in a 3 × 3 × 3 superstructure, in which 216 atomic sites are randomly occupied by 144 Fe ions. Crystal structures are fully relaxed. The DFT calculations are carried out using the SCAN (Strongly Constrained and Appropriately Normed) meta-GGA45. The implementation in the VASP46,47 and FHI-aims48 codes are crossly checked, and both codes give very similar results. For the VASP simulations, the energy cutoff is 500 eV, and a mesh of 2 × 2 × 2 is used for the Brillouin zone integration.