Thermoelectric modules, consisting of multiple pairs of n- and p-type legs, enable converting heat into electricity and vice versa. However, the thermoelectric performance is often asymmetrical, in that one type outperforms the other. In this paper, we identified the relationship between the asymmetrical thermoelectric performance and the weighted mobility ratio, a correlation that can help predict the thermoelectric performance of unreported materials. Here, a reasonably high ZT for the n-type ZrCoBi-based half-Heuslers is first predicted and then experimentally verified. A high peak ZT of ~1 at 973 K can be realized by ZrCo 0.9 Ni 0.1 Bi 0.85 Sb 0.15 . The measured heat-to-electricity conversion efficiency for the unicouple of ZrCoBi-based materials can be as high as ~10% at the cold-side temperature of ~303 K and at the hot-side temperature of ~983 K. Our work demonstrates that the ZrCoBi-based half-Heuslers are highly promising for the application of mid- and high-temperature thermoelectric power generation.

The asymmetrical thermoelectric performance of previously reported materials has been merely viewed as an observation, while a more general relationship between the asymmetry and intrinsic material parameters is seldom discussed. Therefore, it would be highly meaningful to quantify the asymmetrical thermoelectric performance and identify the relationship between the asymmetry and the materials’ parameters. This would be highly beneficial for understanding the asymmetrical thermoelectric performance. Once the relationship can be identified, it would be possible to predict the thermoelectric performance of an unreported type of a given material by using the material’s intrinsic parameters. Such a strategy could possibly be helpful for identifying promising thermoelectric materials.

However, there are quite a few materials that can only be synthesized as one of the types, e.g., MgAgSb ( 5 – 7 ), SnTe ( 8 , 9 ), GeTe ( 10 – 12 ), and Zn 4 Sb 3 ( 13 ). This can be attributed to the presence of native defects that pin the Fermi level to either the conduction or the valence band, and therefore, the materials show persistent n- or p-type conduction ( 14 ). Among the materials for which the ambipolar doping can be realized, most show high thermoelectric performance (i.e., ZT > 1) in only one of the types: e.g., Mg 2 Sn-based ( 15 – 18 ), Mg 3 Sb 2 -based ( 19 – 22 ), ZrNiSn-based ( 23 , 24 ), and NbFeSb-based ( 25 – 27 ) compounds. Even for the state-of-the-art thermoelectric materials, e.g., Bi 2 Te 3 -based materials ( 28 , 29 ), lead chalcogenides ( 30 – 35 ), skutterudites ( 36 , 37 ), and SiGe ( 38 , 39 ), in which high thermoelectric performance can be achieved in both types, the thermoelectric performance is often asymmetrical ( Fig. 1 ). The specific reason for such an asymmetry varies among the different compounds. It could be attributed to the difference in the electronic structures between the conduction and valence bands, e.g., density-of-state effective mass ( m d * ), inertial effective mass (m I ), and band degeneracy (N). It can also be ascribed to the difference in carrier mobility due to the different degree of carrier scattering ( 33 ). In addition, disparity in the dopability, e.g., difficulty in identifying an efficient dopant for one of the types, can also lead to asymmetrical thermoelectric performance.

Solid-state energy conversion from heat to electricity and vice versa can be realized by a thermoelectric module using a thermoelectric material ( 1 , 2 ). The conversion efficiency of the module is governed by the Carnot efficiency and by the material’s figure of merit (ZT). ZT = [S 2 σ/(κ L + κ e )]T, where S, σ, κ L , κ e , and T are the Seebeck coefficient, electrical conductivity, lattice thermal conductivity, electronic thermal conductivity, and absolute temperature, respectively ( 3 ). Generally, the thermoelectric module consists of multiple pairs of n- and p-type legs that are connected thermally in parallel and electrically in series ( 4 ). To achieve a high-energy conversion efficiency, high thermoelectric performance for both types of legs is desired. In addition, to minimize the thermal stress that originates from the mismatch of thermal expansion coefficients between n- and p-type legs, it is preferable to adopt the same base compound for both types of legs.

RESULTS

Asymmetrical thermoelectric performance and materials’ parameters Generally, the lattice thermal conductivity of optimized p- and n-type compositions of a given material system are similar. Therefore, we mainly focus on the electronic transport properties of the materials. The electronic contributions to thermoelectric performance are given by a power factor, which can be optimized by chemical doping. The thermoelectric performance of a given compound can be estimated by the dimensionless material quality factor B (40, 41) B = 5.745 × 10 − 6 μ ( m d * ) 3 / 2 κ L T 5 / 2 (1)where μ is the carrier mobility. This formula captures the essential aspect that heavy effective mass (for the thermopower), high carrier mobility (which typically accompanies light mass and is needed for conductivity), and low lattice thermal conductivity are all needed for thermoelectric performance. When asymmetrical thermoelectric performance is observed for a certain compound, this asymmetry (n-type versus p-type) can then be estimated by B n B p = 5.745 × 10 − 6 μ n ( m d , n * ) 3 / 2 κ L , n T n 5 / 2 5.745 × 10 − 6 μ p ( m d , p * ) 3 / 2 κ L , p T p 5 / 2 ≈ μ n ( m d , n * ) 3 / 2 μ p ( m d , p * ) 3 / 2 = A (2)where n denotes the n-type and p denotes the p-type. κ L and the temperature for peak ZT are considered similar for both n- and p-types of the compound. A is the dimensionless n-type to p-type weighted mobility ratio. The density-of-state effective mass is related to the band effective mass (m b ) via the relation of m d * = N 2 / 3 m b . In addition, in the case when carriers are mainly scattered by acoustic phonons, the carrier mobility depends on the band effective mass as well as the inertial effective mass according to the relation of μ ∝ m I −1m b −3/2. For an isotropic single parabolic band system, these masses are identical, and in the general case, they are decoupled and are also dependent on doping and temperature. Equation 2 can thus be rewritten as A ≈ N n m I , p N p m I , n (3) It can be understood from Eq. 3 that a large band degeneracy (30) and a low inertial effective mass (32) will be greatly beneficial for high thermoelectric performance. As a result, the difference in the band degeneracy and inertial effective mass between the conduction and valence bands will lead to a noticeable asymmetrical thermoelectric performance. According to Eq. 2, the asymmetrical thermoelectric performance can be estimated by the weighted mobility ratio. The weighted mobility ratio for different materials could, in principle, be calculated by Eq. 2. For estimation of A, this can be done at similar electron and hole carrier concentration. The relationship between the n-type to p-type ZT ratio (ZT n /ZT p ) and the n-type to p-type weighted mobility ratio is shown in Fig. 2A. There is a general trend that ZT n /ZT p increases with the weighted mobility ratio, i.e., log(ZT n /ZT p ) = C 1 log(A) + C 2 , where C 1 and C 2 are fitted as 0.69 and −0.039, respectively. Details of the data used in this fitting can be found in table S1. It should be noted that the experimentally obtained relationship between log(ZT n /ZT p ) and log(A) is in a reasonably good agreement with the theoretical calculation (section S1 and fig. S1). Specifically, when the weighted mobility ratio is larger than unity, the n-type material will outperform the p-type counterpart (ZT n /ZT p > 1). Conversely, when the weighted mobility ratio is smaller than unity, the p-type compound will demonstrate better thermoelectric performance (ZT n /ZT p < 1). When the thermoelectric performance for one type of a material has been experimentally studied, while the other type has not yet been investigated, it is possible to predict the ZT by using the identified relationship between the asymmetry and weighted mobility ratio. In this case, because the carrier mobility for the unreported type cannot be obtained, Eq. 3 can be adopted to estimate the weighted mobility ratio as it only involves the material’s basic parameters that can be determined by calculation. However, it should be noted that Eq. 3 is simplified by assuming that the carrier scattering is similar for electrons (n-type specimen) and holes (p-type specimen). Therefore, the estimated weighted mobility ratio by Eq. 3 could be different from the value experimentally determined by Eq. 2. Here, we have predicted the thermoelectric performance for several materials as shown in Fig. 2B. Our predictions indicate that reasonably high ZT can possibly be achieved in n-type SnTe, n-type GeTe, and n-type ZrCoBi. In addition, n-type TaFeSb and p-type NbCoSn are possible candidates for very promising thermoelectric performance. However, SnTe and GeTe have been proven as very persistent p-type materials, so we limited our investigation to the thermoelectric properties of the half-Heusler materials. Currently, we have not yet identified efficient dopants for p-type NbCoSn or n-type TaFeSb (fig. S2). Therefore, we will mainly discuss the thermoelectric properties of n-type ZrCoBi-based materials in this work. Fig. 2 Relationship between the asymmetrical thermoelectric performance and weighted mobility ratio. (A) Relationship between the ZT ratio (n-type versus p-type) and weighted mobility ratio. (B) Predicted thermoelectric performance for a number of materials.

Electronic structure of ZrCoBi Theoretical calculations of the electronic structure of ZrCoBi were first conducted to provide preliminary insight into its thermoelectric performance. The calculated band structure of ZrCoBi is shown in Fig. 3A. The valence bands show practically degenerate maxima at Γ and L, which is a previously discussed band convergence (42). It should be noted that there are strong splittings in the bands as one moves away from Γ. These are the splitting into a heavy- and light-hole band along Γ-X and an additional splitting of the upper heavy-hole band along Γ-L. This latter splitting would be forbidden without the combination of spin-orbit and non-centrosymmetry. These effects are strong due to the large difference between Co, Zr and the heavy p-electron element Bi, which leads to strong breaking parity symmetry, P. Although Γ has the full cubic symmetry, and the inverse effective mass tensor of a cubic material is isotropic, the bands at Γ are evidently highly anisotropic. This anisotropy with different curvatures in different bands and directions at Γ provides a mechanism for decoupling the electrical conductivity and the Seebeck coefficient, in particular with the high curvature areas, leading to high electrical conductivity. Even more notably, at the other pocket L, there is a pair of two-fold degenerate bands separated by 0.06 eV. Both are relevant for the thermoelectric transport due to this small splitting. The upper band is seen to be very strongly spin-split from L to W and somewhat more weakly so along L-Γ. This splitting would be forbidden by the combination of parity and time reversal in a centrosymmetric case. This is important because, in addition to the anisotropy allowed in a parabolic system around the L point [i.e., a transverse and a longitudinal mass, which can be different, as in GeTe (43)], there is an additional separation into a light and heavy band due to spin-orbit. This is highly favorable as it again decouples Seebeck coefficient and electrical conductivity. In contrast, the conduction band structure is apparently simpler, although it also shows strong influence of spin-orbit. This band shows a substantial spin-splitting along X-W but not X-Γ. This Dresselhaus spin-splitting shifts the conduction band minimum away from X perpendicular to the X-Γ line and can also decouple the Seebeck coefficient and electrical conductivity, although the decoupling is less pronounced than in the valence bands. Fig. 3 Calculated electronic structure of ZrCoBi. (A) Band structure. (B) Calculated inertial effective mass for n- and p-type ZrCoBi at 1000 K. (C) Valence band carrier pockets. (D) Conduction band carrier pockets. Visualization of the carrier pockets 0.1 eV from the band edges for the p- and n-type ZrCoBi are shown in Fig. 3 (C and D). The band degeneracy (N) is 10 for the p-type ZrCoBi but only 3 for its n-type counterpart. In addition, the calculated inertial effective mass is shown as a function of carrier concentration for n- and p-type ZrCoBi in Fig. 3B. It should be noted that the inertial effective mass (m I ) is very similar for n- and p-type ZrCoBi. As a result, the value of N/m I is ~5.3 m 0 −1 for p-type ZrCoBi and only ~2.2 m 0 −1 for its n-type counterpart, which corresponds to an n-type to p-type weighted mobility ratio of ~0.42. By using the experimentally determined thermoelectric performance of p-type ZrCoBi-based materials (i.e., peak ZT of ~1.4) and the weighted mobility ratio of ~0.42, the predicted ZT for n-type ZrCoBi is about ~0.71 (Fig. 2B). That is, despite the asymmetrical thermoelectric performance of ZrCoBi (i.e., the p-type outperforming the n-type), reasonably high thermoelectric performance could possibly be achieved in the n-type ZrCoBi.

Electronic properties of n-type ZrCoBi The n-type Ni-doped ZrCoBi-based materials were prepared by the ball-milling and hot-pressing method. All the samples are single phase (fig. S3) and highly dense (fig. S4). The measured electronic properties of the n-type ZrCoBi-based half-Heuslers are shown in Fig. 4. The undoped ZrCoBi shows a low electrical conductivity, and after Ni doping at the Co site, the electrical conductivity is substantially improved (Fig. 4A). Such an enhancement should be mainly attributed to the increased Hall carrier concentration. As shown in fig. S5, the carrier concentration increases monotonically with the Ni concentration. This indicates that Ni is an efficient dopant for supplying electrons to ZrCoBi, similar to the case of ZrCoSb (44). Accordingly, the Seebeck coefficient of ZrCo 1–x Ni x Bi decreases with the increase of Ni concentration (Fig. 4B), with the exception of the undoped ZrCoBi. By optimizing the carrier concentration, a reasonably high peak power factor (S2σ) of ~27 μW cm−1 K−2 can be realized by ZrCo 0.9 Ni 0.1 Bi (Fig. 4C). The power factor of p-type ZrCoBi 0.8 Sn 0.2 is also plotted for comparison (black solid line), and it is noticeably higher than that of the n-type specimens over the whole temperature range. The average power factor between 300 and 973 K can be calculated by the integration method, and it is ~20.8 μW cm−1 K−2 for n-type ZrCo 0.9 Ni 0.1 Bi and as high as ~33.6 μW cm−1 K−2 for p-type ZrCoBi 0.8 Sn 0.2 . Such a marked difference in the power factor between the n- and p-type ZrCoBi-based materials should be mainly ascribed to the difference in band degeneracy and pocket shape between the conduction and valence bands. Therefore, despite the similar inertial effective mass between the n- and p-type ZrCoBi, the density-of-state effective masses of p-type ZrCoBi-based materials are notably higher than those of their n-type counterparts (fig. S6A). Because the inertial effective masses are comparable between the n- and p-type ZrCoBi (Fig. 3B), the carrier mobility of the p-type ZrCoBi-based materials remains similar to that of their n-type counterparts (fig. S6B). To further reveal the effect of electronic structure on the thermoelectric properties, we calculated the product of carrier concentration and the square of Seebeck coefficient (S2n) for ZrCoBi-based materials, as shown in Fig. 4D. It can be observed that the S2n value of the p-type ZrCoBi 0.8 Sn 0.2 is substantially higher than that of its n-type counterparts over the whole temperature range. Fig. 4 Electronic properties for the n-type ZrCoBi-based half-Heuslers. (A) Electrical conductivity, (B) Seebeck coefficient, (C) power factor (PF), and (D) S2n of ZrCo 1–x Ni x Bi.

Thermal conductivity of the n-type ZrCoBi Thermal conductivity of the n-type ZrCoBi-based half-Heuslers is shown in Fig. 5. The room temperature thermal conductivity of ZrCo 1–x Ni x Bi is reduced considerably with the increase of Ni concentration (Fig. 5A). The room temperature thermal conductivity of the undoped ZrCoBi is ~8.9 W m−1 K−1, and it is only ~4.3 W m−1 K−1 for ZrCo 0.75 Ni 0.25 Bi, a reduction of ~52%. This can be attributed to the high concentration of substitutional point defects (i.e., Ni at the Co site) that disrupt the phonon propagation. To reveal the phonon scattering effect by Ni doping, we further calculated the lattice thermal conductivity, as shown in Fig. 5B. The room temperature lattice thermal conductivity is ~8.9 W m−1 K−1 for the pristine ZrCoBi, and it is ~3.4 W m−1 K−1 for ZrCo 0.75 Ni 0.25 Bi, a reduction of ~62%. In addition to the Ni doping, thermal conductivity can also be largely reduced by alloying ZrCoBi with ZrCoSb (fig. S7). Compared to the heavy Ni doping that leads to the inferior power factor (Fig. 4C), alloying ZrCoBi with ZrCoSb can effectively reduce the thermal conductivity while maintaining a relatively high power factor (fig. S7C). Fig. 5 Thermal conductivity of n-type ZrCoBi-based half-Heuslers. (A) Total thermal conductivity and (B) lattice thermal conductivity of ZrCo 1–x Ni x Bi.