Achieving high power factor and output power density in p-type half-Heuslers Nb1-xTixFeSb

Ran He; Daniel Kraemer; Jun Mao; Lingping Zeng; Qing Jie; Yucheng Lan; Chunhua Li; Jing Shuai; Hee Seok Kim; Yuan Liu; David Broido; Ching-Wu Chu; Gang Chen; Zhifeng Ren

doi:10.1073/pnas.1617663113

Contributed by Ching-Wu Chu, October 24, 2016 (sent for review September 7, 2016; reviewed by Jing-Feng Li and Silke Paschen)

Significance

Thermoelectric technology can boost energy consumption efficiency by converting some of the waste heat into useful electricity. Heat-to-power conversion efficiency optimization is mainly achieved by decreasing the thermal conductivity in many materials. In comparison, there has been much less success in increasing the power factor. We report successful power factor enhancement by improving the carrier mobility. Our successful approach could suggest methods to improve the power factor in other materials. Using our approach, the highest power factor reaches ∼106 μW⋅cm⁻¹⋅K⁻² at room temperature. Such a high power factor further yields a record output power density in a single-leg device tested between 293 K and 868 K, thus demonstrating the importance of high power factor for power generation applications.

Abstract

Improvements in thermoelectric material performance over the past two decades have largely been based on decreasing the phonon thermal conductivity. Enhancing the power factor has been less successful in comparison. In this work, a peak power factor of ∼106 μW⋅cm⁻¹⋅K⁻² is achieved by increasing the hot pressing temperature up to 1,373 K in the p-type half-Heusler Nb_0.95Ti_0.05FeSb. The high power factor subsequently yields a record output power density of ∼22 W⋅cm⁻² based on a single-leg device operating at between 293 K and 868 K. Such a high-output power density can be beneficial for large-scale power generation applications.

The majority of industrial energy input is lost as waste heat. Converting some of the waste heat into useful electrical power will lead to the reduction of fossil fuel consumption and CO₂ emission. Thermoelectric (TE) technologies are unique in converting heat into electricity due to their solid-state nature. The ideal device conversion efficiency of TE materials is usually characterized by (1)η=TH−TCTH⋅1+ZT¯−11+ZT¯+TCTH,[1]where ZT¯ is the average thermoelectric figure of merit (ZT) between the hot side temperature (T_H) and the cold side temperature (T_C) of a TE material and is defined asZT=PFκtotT[2]PF=S2σ[3]κtot=κL+κe+κbip,[4]where PF, T, κ_tot, S, σ, κ_L, κ_e, and κ_bip are the power factor, absolute temperature, total thermal conductivity, Seebeck coefficient, electrical conductivity, lattice thermal conductivity, electronic thermal conductivity, and bipolar thermal conductivity, respectively. Higher ZT corresponds to higher conversion efficiency.

One effective approach to enhance ZT is through nanostructuring that can significantly enhance phonon scattering and consequently result in a much lower lattice thermal conductivity compared with that of the unmodified bulk counterpart (2). This approach works well for many inorganic TE materials, such as Bi₂Te₃ (2), IV–VI semiconductor compounds (3, 4), lead–antimony–silver–tellurium (LAST) (5), skutterudites (6), clathrates (7), CuSe₂ (8), Zintl phases (9), half-Heuslers (10 ⇓–12), MgAgSb (13, 14), Mg₂(Si, Ge, Sn) (15, 16), and others.

However, nanostructuring is effective only when the grain size is comparable to or smaller than the phonon mean free path (MFP). In compounds with a phonon MFP shorter than the nanosized grain diameters, nanostructuring might impair the electron transport more than the phonon transport, thus potentially decreasing the power factor and ZT. In contrast, improving ZT by boosting the power factor has not yet been widely studied (17 ⇓⇓–20). To the best of our knowledge, there is no theoretical upper limit applied to the power factor. Additionally, the output power density ω of a device with hot side at T_H and cold side at T_C is directly related to the power factor by (21)ω=(TH−TC)4L2PF¯,[5]where L is the leg length of the TE material and PF¯ is the averaged power factor over the leg. As contact resistance limits the reduction of length L, higher power factor favors higher power density when heat can be efficiently supplied and removed.

One group of thermoelectric materials that may have high power factor is half-Heusler (HH) compounds. Among the various HH compounds, ZrNiSn-based n-type and ZrCoSb-based p-type materials have been widely studied due to their satisfactory ZT ∼ 1 at 873–1,073 K (10), low cost (11), excellent mechanical properties (22), and nontoxicity. Recently, the NbFeSb-based p-type materials were found to possess good TE properties. Joshi et al. (23) and Fu et al. (24) reported ZT ∼ 1 with Ti substitution. Moreover, a record-high ZT ∼ 1.5 at 1,200 K was reported with Hf substitution (12). These works mark HH compounds among the most promising candidates for thermoelectric conversion in the mid-to-high temperature range.

As reported by Joshi et al. (23), a high power factor of ∼38 μW⋅cm⁻¹⋅K⁻² was realized in the p-type half-Heusler Nb_0.6Ti_0.4FeSb_0.95Sn_0.05 at 973 K. However, the composition with 40% Ti substitution strongly scatters the electrons as well. A subsequent work by Fu et al. (24) reported a higher power factor of ∼62 μW⋅cm⁻¹⋅K⁻² at 400 K in Nb_0.92Ti_0.08FeSb that was attributed to less electron scattering. However, they studied only one sintering temperature at 1,123 K (12, 24). Because high-temperature heat treatment for TE materials can be beneficial to TE performance (25), further optimization may be achieved in the NbFeSb system. Here we report the thermoelectric properties of the Nb_1-xTi_xFeSb system with Ti substitution up to x = 0.3 prepared by using arc melting, ball milling, and hot pressing (HP) at 1,123 K, 1,173 K, 1,273 K, and 1,373 K. We find that higher HP temperature enhances the carrier mobility, leading to a high power factor of ∼106 μW⋅cm⁻¹⋅K⁻² at 300 K in Nb_0.95Ti_0.05FeSb. Such an unusually high power factor has previously been observed only in metallic systems such as YbAl₃ and constantan (26, 27). Furthermore, a record output power density of ∼22 W⋅cm⁻² with a leg length ∼2 mm is experimentally obtained with T_C = 293 K and T_H = 868 K. We also observe that the lattice thermal conductivity hardly changes within the range of grain sizes studied in this work. Thus, by using a higher HP temperature of 1,373 K and changing the Ti concentration, we have achieved higher power factor and ZT than the previously reported results for the same compositions (24).

Materials and Methods

Synthesis.

Fifteen grams of raw elements (Nb pieces, 99.9%, and Sb broken rods, 99.9%, Atlantic Metals & Alloy; Fe granules, 99.98%, and Ti foams, 99.9%, Alfa Aesar) are weighed according to stoichiometry. The elements are first arc melted multiple times to form uniform ingots. The ingots are ball milled (SPEX 8000M Mixer/Mill) for 3 h under Ar protection to produce nanopowders. The powders are then consolidated into disks via hot pressing at 80 MPa for 2 min at 1,123 K, 1,173 K, 1,273 K, or 1,373 K with an increasing temperature rate of ∼100 K/min.

Characterization.

An X-ray diffraction machine (PANalytical X’Pert Pro) is used to characterize the sample phases. Morphology and elemental ratios of the samples are characterized by a scanning electron microscope (SEM) (LEO 1525) and electron probe microanalysis (EPMA) (JXA-8600), respectively. A transmission electron microscope (TEM) (JEOL 2100F) is used to observe the detailed microstructures.

Measurement.

The thermal conductivity is calculated as a product of the thermal diffusivity, specific heat, and mass density that are measured by laser flash (LFA457; Netzsch), a differential scanning calorimeter (DSC 404 C; Netzsch), and an Archimedes’ kit, respectively. Bar-shaped samples with sizes around 2 mm × 2 mm × 10 mm are used for measuring the electrical conductivity and the Seebeck coefficient in a ZEM-3 (ULVAC). Hall concentrations (n_H) are measured using the Van der Pauw method in a physical properties measurement system (PPMS) (Quantum Design) under ±3 Tesla magnetic induction. The uncertainties for electrical conductivity, Seebeck coefficient, and thermal conductivity are 4%, 5%, and 12%, respectively. In particular, the thermal conductivity uncertainty comprises 4% for thermal diffusivity, 6% for specific heat, and 2% for mass density. As a result, the combined uncertainties for power factor and ZT are 10% and 20%, respectively. To increase the readability of the figures presented, we add error bars only to some of the curves (Figs. 3 and 5B).

Results and Discussion

Enhanced Power Factor with Higher Hot Pressing Temperature.

All of the compositions in this work possess pure half-Heusler phases (Fig. S1). Fig. 1 A–F shows the thermoelectric properties of Nb_0.95Ti_0.05FeSb with hot pressing temperatures of 1,123 K, 1,173 K, 1,273 K, and 1,373 K. As shown in Fig. 1A, the power factor (PF) improves significantly at below 573 K. The peak value reaches ∼106 μW⋅cm⁻¹⋅K⁻² at 300 K and is the highest measurement value in half-Heusler compounds. When the temperature is higher than 873 K, the power factor values converge.

Fig. 1.

Thermoelectric property dependence on temperature for the half-Heusler Nb_0.95Ti_0.05FeSb hot pressed at 1,123 K, 1,173 K, 1,273 K, and 1,373 K. (A) Power factor, (B) electrical conductivity, (C) Seebeck coefficient, (D) total thermal conductivity, (E) lattice plus bipolar thermal conductivity, (F) bipolar thermal conductivity, (G) ZT, and (H) ZT with T from 300 K to 573 K. The green dashed lines in B and E represent the T^−3/2 and T⁻¹ relations, respectively. The magenta dashed line in C shows the calculated Seebeck coefficient using the SPB model.

Fig. S1.

Sample characterizations. (A) Selected area electron diffraction pattern. (B) High-resolution TEM image of Nb_0.95Ti_0.05FeSb. (C) X-ray diffraction (XRD) patterns of Nb_1-xTi_xFeSb. All of the compositions are pressed at 1,373 K and possess pure half-Heusler phase.

Fig. 1B shows that the high power factor is mainly due to the improved electrical conductivity. Above 673 K, the electrical conductivity values converge and follow the T^−3/2 law, suggesting that acoustic phonons dominate carrier scattering (28). In contrast, Fig. 1C shows that the Seebeck coefficient changes little regardless of the hot pressing temperature. The dashed line in Fig. 1C is the calculated Seebeck coefficient using the single parabolic band (SPB) model (29),S=+(kBe)[2F1(η)F0(η)−η][6]Fn(η)=∫0∞χn1+eχ−ηdχ,[7]where η is related to n_H throughnH=4π(2mh∗kBTh2)3/24F02(η)3F−1/2(η),[8]where η, n_H, mh∗, and k_B are the reduced Fermi energy, the Hall carrier concentration, the density of states (DOS) effective mass of holes, and the Boltzmann constant, respectively. The n_H values are obtained through the Hall measurement and are presented in Table 1. The DOS hole effective mass (mh∗) is obtained by fitting the Pisarenko relation (as is shown in TE Properties of p-Type Nb_1-xTi_xFeSb). F_n(η) is the Fermi integral of order n. The good agreement between the calculated result and experimental data shows the adequacy of the SPB model in describing the hole transport.

View this table:

Table 1.

RT Hall carrier concentration (n_H), Hall mobility (μ_H), deformation potential (E_def), relative density, and EPMA composition of Nb_1-xTi_xFeSb at different Ti concentrations and hot pressing temperatures

The total thermal conductivity is a multiplication of bulk density, thermal diffusivity, and specific heat (Fig. S2). The total thermal conductivity is slightly lower for samples hot pressed at lower temperature (Fig. 1D). This is mainly due to the difference in the electronic thermal conductivity originating from the difference in the electrical conductivityκe=LσT,[9]where L is the Lorenz number evaluated using the SPB model (29). By subtracting the κ_e from κ_tot, we plot the sum κ_L + κ_bip in Fig. 1E. The sums of κ_L + κ_bip are barely affected by the hot pressing temperature. Furthermore, at temperatures above 773 K, the thermal conductivity trend deviates slightly from the T⁻¹ behavior, indicating some minor bipolar effects even though the Seebeck coefficient seems not to show such an effect. The κ_bip is calculated using a three-band model (Fig. 1F, Fig. S3, and Table S1). The peak κ_bip reaches ∼0.4 W⋅m⁻¹⋅K⁻¹ at 973 K, a small value compared with the κ_L.

Fig. S2.

Temperature dependence of (A) thermal diffusivity and (B) specific heat of Nb_1-xTi_xFeSb pressed at 1,373 K with x = 0, 0.04, 0.05, 0.06, 0.07, 0.1, 0.2, and 0.3.

Fig. S3.

(A) DFT band structures calculated by QE (gray, squares) and elk (red, solid line) with the full-electron approach. (B) Estimated band gap using the conductivity of intrinsic (undoped) NbFeSb. (C) Estimated band offset between the two valence bands. (D) Bipolar thermal conductivity among different bands. κ_C-VL, κ_C-VH, and κ_VL-VH are the bipolar thermal conductivity values between the conduction-V_L band, the conduction-V_H band, and the V_L band-V_H band, respectively.

View this table:

Table S1.

n_H and μ_H of NbFe_1-yCo_ySb at room temperature

Because of the enhanced power factor and the almost unaffected thermal conductivity, the ZT improves with elevated hot pressing temperatures (Fig. 1G). This is especially obvious at temperatures below 573 K, as shown in Fig. 1H, where the power factor shows larger differences (Fig. 1A).

SEM images show significant enlargement of the grains with higher hot pressing temperatures (Fig. 2 A–D). The average grain sizes are found to be ∼0.3 μm, ∼0.5 μm, ∼3.0 μm, and ∼4.5 μm for samples pressed at 1,123 K, 1,173 K, 1,273 K, and 1,373 K, respectively (Fig. S4). Meanwhile, the room temperature (RT) Hall measurement (Table 1) shows an ∼73% enhancement of Hall mobility (μ_H) of samples pressed at 1,373 K over those pressed at 1,123 K. Because the lattice thermal conductivity changes little with the grain size (Fig. 1E), it is very interesting to investigate why the enlarged grain size affects the electron transport so differently.

Fig. 2.

SEM images of Nb_0.95Ti_0.05FeSb hot pressed at (A) 1,123 K, (B) 1,173 K, (C) 1,273 K, and (D) 1,373 K.

Fig. S4.

Normalized grain size distribution of Nb_0.95Ti_0.05FeSb pressed at (A) 1,123 K, (B) 1,173 K, (C) 1,273 K, and (D) 1,373 K.

Effect of Grain Size on Lattice Thermal Conductivity and Carrier Mobility.

The lattice thermal conductivity (κ_L) of sample pressed at 1,373 K is obtained by subtracting κ_e and κ_bip from κ_tot. To describe the lattice thermal conductivity, we use the Klemens (30) model and split the phonon scattering into four different sources: three-phonon (3P) processes, grain boundary (GB) scattering, point defects (PD) scattering, and electron–phonon (EP) interaction. The calculation details and the fitting parameters are given in Supporting Information, Klemens Model. The complete results combining 3P, GB, PD, and EP are shown in TE Properties of p-Type Nb_1-xTi_xFeSb. Here, in Fig. 3A, we show only the effect of GB scattering. Clearly, the calculated reduction in the lattice thermal conductivity is small with decreasing grain size, only ∼9% when the grain size decreases by more than one order of magnitude from 4.5 μm to 0.3 μm. The insensitivity of the phonon transport to the grain size may indicate that the dominant thermal MFPs that contribute to the thermal conductivity of Nb_0.95Ti_0.05FeSb may be less than 0.3 μm. We are currently in the process of measuring the phonon MFP (31 ⇓⇓⇓⇓–36) distributions for Nb_0.95Ti_0.05FeSb and some preliminary measurement data at room temperature are included in Supporting Information (Fig. S5). The results suggest that the dominant phonon MFPs of Nb_0.95Ti_0.05FeSb are in the range of a few tens to a few hundreds of nanometers. This supports our experimental observation of the insensitivity of the thermal conductivity to the grain size because the grain sizes in our measurement range are significantly larger than the dominant phonon MFPs.

Fig. 3.

Effect of grain size on (A) κ_L and (B) μ_H (=σR_H,300K). The symbols are obtained from the measurements and the lines are fitted using the transport models. The error bars in A and B show 12% and 4% relative error, respectively.

Fig. S5.

(A) A representative SEM image of the microfabricated metallic grating on the Nb_0.95Ti_0.05FeSb substrate. (B) TDTR measured size-dependent Nb_0.95Ti_0.05FeSb thermal conductivities at room temperature. The error bars represent the SD of the measurement uncertainty. (C) Reconstructed phonon MFP distribution of Nb_0.95Ti_0.05FeSb at room temperature.

To analyze the measured carrier mobility, we assume a weak temperature dependence of n_H from room temperature through 573 K. This assumption is reasonable within the temperature region where the bipolar effect is small and weakly temperature dependent (Fig. 1F). This effect was experimentally observed in another half-Heusler system (29). Thus, the carrier mobility, μ_H = σR_H from room temperature up to 573 K can be approximated by the measured quantity σR_H,300K,μH=σRH,300K=σ(qnH,300K)−1,[10]where q is the carrier charge, and R_H,300K and n_H,300K are the Hall coefficient and carrier concentration at 300 K, respectively.

At high temperatures, the dominant electron scattering is from acoustic phonons (AP), and thus the mobility is expressed asμAP=F0(η)2F1/2(η)22eπℏ43(kBT)3/2vl2d(Nv)5/3Edef2(mh∗)5/2,[11]where v_l is the longitudinal phonon velocity and calculated from the elastic constants (37), d is the mass density, and N_v is the valley degeneracy. E_def is the deformation potential that is obtained by extrapolating the high-temperature mobility back to room temperature, using the T^−3/2 law. E_def is found to be ∼12 eV for Nb_0.95Ti_0.05FeSb, a relatively small value compared with other TE systems like InSb (∼33 eV) (38), PbTe (∼22.5 eV) (39), and Bi₂Te₃ (∼20 eV) (40), suggesting a weaker EP in the HH systems (29) that can benefit the electron transport and the power factor.

The mobility with the GB scattering is written as (41, 42)μGB=De(12πmb∗kBT)1/2exp(−EBkBT),[12]where D is the grain size and E_B is a commonly fitted barrier energy of the GB. Combining the two scattering mechanisms according to Matthiessen’s rule yields the expression for Hall mobility:μH−1=μAP−1+μGB−1.[13]The calculated mobility (μ_H) with E_B ∼ 0.1 eV and the measured quantity σR_H,300K are shown in Fig. 3B. The good fitting demonstrates the importance of the grain boundaries in scattering carriers below 573 K. At temperatures higher than 573 K the charge carriers are mainly scattered by the acoustic phonons, and thus the mobility curves converge. In addition, Fig. 3B shows that the GB scattering becomes stronger when the grain size is smaller: For decreasing size from 0.5 μm to 0.3 μm, the mobility decreases by ∼30%, but for decreasing size from 4.5 μm to 3.0 μm, the mobility drops by only 5%.

TE Properties of p-Type Nb_1-xTi_xFeSb.

Fig. 4 shows the TE properties of Nb_1-xTi_xFeSb pressed at 1,373 K with x = 0, 0.04, 0.05, 0.06, 0.07, 0.1, 0.2, and 0.3 as a function of temperature. The electrical conductivity (σ) of undoped NbFeSb (i.e., x = 0) increases with temperature, consistent with typical semiconductor behavior (Fig. 4A, Inset). For the highly doped samples, σ obeys the T^−3/2 law, suggesting the dominant acoustic phonon scattering of charge carriers (28). Meanwhile, σ increases with increasing x up to 0.2 because of the increased n_H; upon further increase of x to 0.3, σ decreases due to the decreased μ_H because of the stronger alloy scattering (Table 1). The Seebeck coefficient varies with n_H in good agreement with the Pisarenko relation using a DOS effective mass mh∗=7.5 m0 at 300 K (Fig. 5A) (43),S=8πkB23eh2 mh∗T(π3nH)2/3.[14]

Fig. 4.

Thermoelectric property dependence on temperature for Nb_1-xTi_xFeSb with x = 0, 0.04, 0.05, 0.06, 0.07, 0.1, 0.2, and 0.3. (A) Electrical conductivity, (B) Seebeck coefficient, (C) power factor, (D) total thermal conductivity, (E) lattice and bipolar thermal conductivity, and (F) ZT. In A, the purple dashed line and Inset show the ∼T^−3/2 relation and the measured conductivity of undoped NbFeSb, respectively.

Fig. 5.

(A) Pisarenko plot at 300 K, with DOS effective mass mh∗=7.5m0 for holes, showing the validity of the SPB model in describing hole transport. (B) The contribution of different phonon scattering mechanisms to the lattice thermal conductivity of Nb_1-xTi_xFeSb at RT. The 3P, GB, PD, and EP processes are shown. For GB, the grain size is set at 4.5 μm. The error bars show 12% relative error.

The power factors of the p-type Nb_1-xTi_xFeSb are very high at a wide temperature range (Fig. 4C). These values are much higher than those reported by Fu et al. (24), where a temperature of 1,123 K was used for sintering. As shown in Fig. 6A, higher pressing temperature accounts for the higher power factor. Moreover, even when comparing the samples pressed at 1,123 K, our work also shows higher power factor, with the probable reason being the higher sample densities (∼99%) in the current work compared with those reported by Fu et al. (24) (∼95%).

Fig. 6.

TE property comparison. (A) Power factor and (B) ZT of the Nb_1-xTi_xFeSb systems from different reports (23, 24). (C) Power factor and (D) ZT among Nb_0.95Ti_0.05FeSb, constantan (27), and YbAl₃ (26), with peak power factor exceeding 100 μW⋅cm⁻¹⋅K⁻².

The total thermal conductivities (κtot) are shown in Fig. 4D. With increased Ti doping, the lattice thermal conductivity decreases significantly (Fig. 4E). As a result, the peak ZT reaches 1.1 for Nb_0.8Ti_0.2FeSb (Fig. 4F) at 973 K with a linear upward trend suggesting even higher ZT at higher operating temperatures (Figs. 4F and 6B).

To analyze the RT lattice thermal conductivity, we use the Klemens model incorporating the 3P, GB, PD, and EP processes, as mentioned in the previous section (30). The fitting parameters are listed in Supporting Information (Table S2). As shown in Fig. 5B, the GB scattering is much weaker compared with the other scattering processes. Note that the EP interaction is quite important in this system. Similar results were also reported by Fu et al. (12) with Zr and Hf doping.

View this table:

Table S2.

Important parameters in Klemens model for Nb_1-xTi_xFeSb

Fig. 6 C and D compares the power factor and ZT, respectively, of a few materials with very high power factor, including YbAl₃ single crystal (26), Nb_0.95Ti_0.05FeSb (this work), and constantan (27). These materials possess peak power factors of at least 100 μW⋅cm⁻¹⋅K⁻². It should be noted that YbAl₃ and constantan are essentially metals where the anomalous Seebeck coefficients are due to Kondo resonance (44) and virtual bound states (45), respectively. To the best of our knowledge, such high power factor above RT has never been realized before in semiconductor-based TE materials. Despite similar power factors, the other two materials possess very high thermal conductivity because they are essentially metals, and thus their ZT are much lower than Nb_0.95Ti_0.05FeSb.

Power Output.

Due to the higher power factor, higher power output is expected. Following the approach of Kim et al. (46), the output power density (ω) and efficiency (η_max) under a large temperature gradient are calculated. With T_C = 293 K and a leg length L = 2 mm, the calculated ω and η_max are ∼28 W⋅cm⁻² and 8.8%, respectively, for Nb_0.95Ti_0.05FeSb when T_H is 868 K (Fig. 7 A and B). The corresponding experimental measured values are ∼22 W⋅cm⁻² and 5.6%, respectively, which are lower than the calculated results, probably due to imperfect device fabrication and possible parasitic heat losses during device measurement (see Supporting Information for details, Fig. S6). To the best of our knowledge, this work obtains the highest power density for bulk thermoelectric materials, which can be important for power generation applications (12, 23, 47, 48).

Fig. 7.

Calculated (dotted lines) and measured (symbols) (A) output power density and (B) conversion efficiency of Nb_1-xTi_xFeSb (x = 0.05 and 0.2) samples with the cold side temperature at ∼293 K and the leg length ∼2 mm, calculated (C) output power density and (D) conversation efficiency of Nb_1-xTi_xFeSb (x = 0.05 and 0.2) samples with hot side temperature at ∼873 K and the leg length ∼2 mm but varying the cold side temperature, and comparison of (E) power factor and (F) ZT of Nb_0.95Ti_0.05FeSb and Nb_0.8Ti_0.2FeSb.

Fig. S6.

(A) A TE sample brazed to the copper hot-junction heater assembly. (B) TE single leg device mounted to the test rig and surrounded by a guard heater to minimize parasitic heat losses from the hot-junction heater. The cold-junction temperature is maintained with a TEC mounted onto a liquid-cooled cold stage. (C) TE voltage (V_TE) and output power density as functions of the TE current (I_TE) at constant hot and cold side temperatures of 773 K and 293 K, respectively.

For power generation applications, the hot side temperature could be easily maintained at about 873 K as long as enough heat is supplied. However, maintaining the cold side temperature at 293 K could be challenging, and it would be more practical to maintain the cold side temperature at around ∼320–340 K by convection heat removal. Thus, we calculate the cold side temperature (T_C)-dependent output power density (ω) and efficiency (η_max) with fixed T_H at 873 K, as shown in Fig. 7 C and D, respectively. Clearly, the cold side temperature is very important to both the output power density and conversion efficiency. However, the calculated output power density for Nb_0.95Ti_0.05FeSb remains quite high (∼25 W⋅cm⁻²) under more attainable temperature boundaries (T_C = 330 K and T_H = 873 K).

Fig. 7 E and F shows the power factor and ZT, respectively, of Nb_0.95Ti_0.05FeSb and Nb_0.8Ti_0.2FeSb. Clearly, from Fig. 7 A and E, higher power factor leads to higher output power density with the same leg length. Similarly, higher ZT results in higher efficiency, as shown in Fig. 7 B and F.

Conclusions

A higher hot pressing temperature up to 1,373 K is found to be beneficial for higher carrier mobility due to larger grain size. The resulting increase in electrical conductivity leads to a much higher power factor of ∼106 μW⋅cm⁻¹⋅K⁻² in the p-type half-Heusler Nb_0.95Ti_0.05FeSb. With the high power factor, a record output power density of ∼22 W⋅cm⁻² is experimentally achieved, which can be important for power generation applications.

Sample Characterization

Fig. S1 shows sample characterization.

Thermal Diffusivity and Specific Heat

Fig. S2 shows thermal diffusivity and specific heat.

Band Structure and Bipolar Thermal Conductivity

The density functional theory (DFT) band structure of the base composition NbFeSb is shown in Fig. S3A. Two software packages were used, Quantum Espresso (QE) (gray squares) and elk (red curves); the latter uses the full electron approach and gives almost identical results to those of QE. It shows an indirect gap of 0.53 eV between the conduction band minima (CBM) X point (with valley degeneracy N_c = 3) and valance band maxima (VBM) L point (valley degeneracy N_v = 4). In addition, there are a light band (V_L) and a heavy band (V_H) at the VBM, and therefore the total N_v = 8.

The Fermi–Dirac integral is defined asFn(η)=∫0∞χn1+eχ−ηdχ,[S1]where η is the reduced Fermi energy.

The calculated carrier concentration for each band isnc=4π(2mC∗kBTh2)3/2F1/2(ηC)rH(ηC)[S2]pVL=4π(2mVL∗kBTh2)3/2F1/2(ηVL)rH(ηVL)[S3]pVH=4π(2mVH∗kBTh2)3/2F1/2(ηVH)rH(ηVH),[S4]where k_B is the Boltzmann constant, r_H is the Hall factor, and h is the Planck constant.

The carrier concentration of each band obeys the neutrality equation as follows:Nexp=pVL+pVH−nC.[S5]In this work, we assumed that N_exp remains constant throughout the entire temperature range.

The relationship between the reduced Fermi energies of each band isηVH=−ηC−E/kBT[S6]ηVL=−ηC−(E+H)/kBT,[S7]where η_C, η_VH, and η_VL are the reduced Fermi energy for the conduction band, heavy valence band, and light valence band, respectively. E is the bandgap and H is the band offset between the two valence bands.

Eqs. S1–S7 are used to solve for the reduced Fermi energy for each band.

Bandgap.

E₀ is the bandgap at 0 K, in our calculation; the E₀ of NbFeSb is taken from the DFT calculation, ∼0.53 eV.

Dependence of E on temperature can be expressed asdE/dT=a1×10−4 eV/K,[S8]where a₁ could be estimated in the following way.

For undoped NbFeSb, the conductivity at high temperature satisfiesln(σ)∝E×−12kBT[S9]and E is ∼0.51 eV using the data between 673 K and 973 K (Fig. S3B). Therefore, a₁ varies between 0.2 and 0.4. In the calculation, a value of 0.3 was used.

The bandgap, E, can be expressed as follows:E=E0+(dE/dT)×T.[S10]

Band Offset Between the Valence Bands.

H₀ is the band offset at 0 K. In our case, the H₀ of NbFeSb was estimated to be ∼0.003 eV (Fig. S3C) by low-temperature Hall measurement, as (49)β∝exp(−H0kBT)[S11]|log(β)|∝H0kBT,[S12]whereβ=RT−R0R0.[S13]R_T is the Hall coefficient at temperature T.

The dependence of the band offset H on temperature can be expressed asdH/dT=a2×10−4 eV/K,[S14]where a₂ needs to be fitted. The fitted value of a₂ equals −1.

The calculated carrier mobility of each band includes scattering by APs within the deformation potential approximation.

The mobility for AP scattering μ_ph can be expressed asμAP=2eπℏ43(kBT)3/2vl2dEdef2(mb∗)5/2F0(η)F1/2(η),[S15]where v_l is the longitudinal sound wave, ρ is the density, and E_ph is the deformation potential due to the phonon wave.

The electrical conductivity for each band can be calculated as follows:σC=enCμC[S16]σVH=enVHμVH[S17]σVL=enVLμVL.[S18]Total electrical conductivity can be calculated as follows:σtot=σC+σVL+σVH.[S19]The Lorenz number for each band can be calculated as follows. Assuming that the AP scattering is the dominant scattering mechanism for electrons in each band, the scattering parameter is therefore r = −1/2:LC=(kBe)2{(r+7/2)Fr+5/2(ηC)(r+3/2)Fr+1/2(ηC)−[(r+5/2)Fr+3/2(ηC)(r+3/2)Fr+1/2(ηC)]2}[S20]LVH=(kBe)2{(r+7/2)Fr+5/2(ηVH)(r+3/2)Fr+1/2(ηVH)−[(r+5/2)Fr+3/2(ηVH)(r+3/2)Fr+1/2(ηVH)]2}[S21]LVL=(kBe)2{(r+7/2)Fr+5/2(ηVL)(r+3/2)Fr+1/2(ηVL)−[(r+5/2)Fr+3/2(ηVL)(r+3/2)Fr+1/2(ηVL)]2}.[S22]The total Lorenz number L_tot is the weighted value of the Lorenz number for each band (50),Ltot=LCσC+LVHσVH+LVLσVLσC+σVH+σVL.[S23]The Seebeck coefficient can also be calculated for each band as follows:SC=−kBe[(r+5/2)Fr+3/2(ηC)(r+3/2)Fr+1/2(ηC)−ηC][S24]SVH=+kBe[(r+5/2)Fr+3/2(ηVH)(r+3/2)Fr+1/2(ηVH)−ηVH][S25]SVL=+kBe[(r+5/2)Fr+3/2(ηVL)(r+3/2)Fr+1/2(ηVL)−ηVL].[S26]Note that the Seebeck coefficient for the conduction band is negative, whereas it is positive for the valence band.

The total Seebeck coefficient is the weighted value of the Seebeck coefficient of each band (51):Stot=SCσC+SVHσVH+SVLσVLσC+σVH+σVL.[S27]The measured Seebeck coefficient in conjunction with electrical resistivity could be used to fit the value of a₂.

To complete the analysis, we need to know the electron’s mobility in n-type NbFeSb. With the same sample process routine, we found Co doping effectively makes n-type materials. The required carrier mobility values are listed in Table S1.

The bipolar thermal conductivity of each band can be calculated as follows:κC_VL=σCσVLσtot(SC−SVL)2T[S28]κC_VH=σCσVHσtot(SC−SVH)2T[S29]κVL_VH=σVLσVHσtot(SVL−SVH)2T.[S30]Total bipolar thermal conductivity is the sum of the abovementioned three components:κbip=κC_VL+κC_VH+κVL_VH.[S31]The calculated results are shown in Fig. S3D.

Grain Size Distribution

We etched the fine-polished sample surfaces to analyze the grain size distribution. The sample surfaces were first mechanically polished by alumina sandpaper with particle size of ∼1 μm, followed by diamond suspension polishing with particle size ∼0.25 μm, and finishing with ∼50 nm SiO₂ suspension on a VibroMet machine, usually resulting in a surface roughness on the order of ∼10 nm (22). The smooth surfaces were then chemically etched for 10–15 s, using etchant containing 50 mL distilled water, 1 mL HF acid (38%), and 2 mL H₂O₂ (35%). The results of normalized grain size distribution are shown in Fig. S4.

Measuring Phonon MFP Distribution in Nb_0.95Ti_0.05FeSb

We use a recently developed phonon MFP spectroscopy technique (33 ⇓⇓–36, 52, 53) to approximately measure the RT distribution of phonon MFPs that contribute to thermal transport in Nb_0.95Ti_0.05FeSb hot pressed at 1,273 K. Briefly, the spectroscopy technique is based on observing nondiffusive thermal transport at length scales comparable to or smaller than the dominant thermal phonon MFPs (54). To access the nondiffusive transport regime, nanoscale metallic gratings of variable line widths are microfabricated on top of the Nb_0.95Ti_0.05FeSb samples that have been fine polished to achieve a root-mean-square roughness ∼5 nm before microfabrication. Fig. S5A shows a representative SEM image of the fabricated metallic grating on top of the sample. The metallic gratings serve as both heaters and thermometers in the time-domain thermoreflectance (TDTR) measurements that are used to measure the size-dependent thermal conductivities. When the heater size (i.e., grating line width) is much larger than the phonon MFPs, phonons experience sufficient scattering to establish a local thermodynamic equilibrium after they traverse the metal–substrate interface, resulting in a diffusive transport regime that can be accurately described by Fourier’s law. However, when the heater size becomes comparable with the dominant thermal phonon MFPs, some long-MFP phonons do not scatter as inherently assumed by the heat diffusion theory, leading to a nondiffusive transport regime where the measured thermal conductivity becomes size dependent (54).

An in-house two-tint TDTR setup is used to probe the size-dependent thermal transport in the fabricated material system (34, 55, 56). Details of the measurement technique are described in ref. 34. Briefly, the two-tint TDTR uses an ∼791-nm pump beam to heat up the sample and uses another time-delayed ∼780-nm probe beam to monitor the reflectance change at the sample surface. The delay time is regulated by a mechanical delay line on the laser table. Because the system is in the linear response regime, measuring the reflectance change is equivalent to measuring the temperature change at the sample surface. To avoid carrier excitation in the substrate, the microfabricated metallic gratings are designed with subwavelength gaps between neighboring metal lines that prevent the pump and probe light from transmitting through the grating when the laser beams are polarized parallel to the grating (33). We use a combination of a quarter waveplate and a linear polarizer to align the laser beams with the metallic gratings during the measurement. The size-dependent effective thermal conductivities are extracted by matching the measured reflectance signals with the prediction from the heat diffusion theory.

Fig. S5B shows the measured size-dependent thermal conductivities of Nb_0.95Ti_0.05FeSb (hot pressed at 1,273 K) at room temperature. As shown in Fig. S5B, the transport transitions from the near-diffusive regime to the nondiffusive regime when the heater size is systematically reduced from ∼500 nm to ∼100 nm, consistent with our understanding of nondiffusive thermal transport. The size-dependent thermal conductivities are used to approximately extract the phonon MFP distribution in Nb_0.95Ti_0.05FeSb via a MFP reconstruction algorithm that is described in detail in ref. 34. Fig. S5C shows the reconstructed phonon MFP spectra of Nb_0.95Ti_0.05FeSb at room temperature. The dominant thermal phonon MFPs at room temperature are in the range of a few tens to a few hundreds of nanometers. In particular, phonons with MFPs shorter than 300 nm contribute ∼70% to the total thermal conductivity. Due to increasing 3P scattering processes with increasing temperatures, the dominant thermal phonon MFPs are typically further suppressed at elevated temperatures. Consequently, the grain size dependence of the thermal conductivity of Nb_0.95Ti_0.05FeSb at elevated temperatures is expected to be much weaker than that at room temperature when the studied grain sizes are larger than 0.3 µm, consistent with our experimental observation shown in Fig. 1 D and E in the main text.

Klemens Model

In the Klemens model (30) the lattice thermal conductivity isκL=kB2π2v(kBℏ)3T3∫0θD/Tτ(x)x4ex(ex−1)2dx,[S32]where kB, v, ℏ, T, θD, and τ are the Boltzmann constant, the phonon velocity, the Planck constant, the absolute temperature, the Debye temperature, and the phonon relaxation time, respectively. x is defined as x=ℏω/kBT (ω is the phonon frequency). It is important to evaluate the phonon relaxation time as a function of frequency, i.e., τ(x).

The Debye temperature and phonon velocity are related to each other bykBθD=ℏv(6π2n)1/3,[S33]where n is the number of atoms per unit volume.

The evaluation of phonon relaxation time follows Matthiessen’s rule (57),τ−1=τPD−1+τ3P−1+τEP−1+τGB−1,[S34]where the subscripts PD, 3P, EP, and GB are as described previously.

Point Defect Scattering.

The relaxation time for PDs is (58)τPD−1=Ax4, A=(kBTℏ)4VatomΓ4πv3.[S35]Vatom is the volume occupied per atom in solid (not the volume of an atom), which is inversely proportional to the density ρ,ρVatom=M¯.[S36]M¯ is the average atomic mass.

Γ is an important parameter that is called the scattering parameter. It could be written as (59, 60)Γ=13(m¯M¯)2[∑ifi(1−mim¯)2+ε∑ifi(1−rir¯)2].[S37]m¯ and r¯ are the average atomic mass and average radius of the substituted sites, respectively. fi, mi, and ri are the fractional concentration, atomic mass, and atomic radius of the ith substitution atom, respectively, and ε is a phenomenological parameter for fitting.

Phonon–Phonon Interaction.

Roufosse and Klemens (61) studied the relaxation time of 3P processing of monatomic cubic crystals. Due to its higher complexity, the phonon relaxation time of HHs should be lower than that following the routine by Roufosse and Klemens. As pointed out by Geng et al. (62), by introducing a reduced anharmonicity constant γ1, the phonon relaxation time of HHs due to 3P processing could be written asτ3P−1=Bx2, B=4π2 kBγ12Vatom1/3mM¯v3T(kBTℏ)2.[S38]m is the unit atomic mass and γ1 is taken as the fitting parameter. Here, we use linear interpolation when changing the compositions; thus B is not a constant value.

EP Interaction.

Phonon relaxation time under EP interaction satisfiesτEP−1=Cx2.[S39]As pointed out by Shi et al. (63), C is proportional to the four-thirds power of carrier concentration (C ∼ n_H^4/3).

GB Scattering.

τGB−1=v/D.[S40]D is the grain size, which is set as 4.5 μm. v is the averaged phonon velocity of one longitudinal branch (vL) and two transverse branches (vT1 and vT2),1v3=13(1vL3+1vT13+1vT23).[S41]The three speed branches are estimated through the elastic constants (C₁₁, C₁₂, and C₄₄). For Nb_1-xTi_xFeSb, we assume their phonon velocities can be linearly interpolated between NbFeSb and TiFeSb. Therefore, we need to know only the elastic constants of the compositions at the two ends. The elastic constants of NbFeSb are calculated by Hong et al. (37). However, we cannot find any previous report on the elastic properties of ternary TiFeSb, and thus the values of TiCoSb are used because the elastic constants barely change in the TiCo_1-δFe_δSb systems (64). Table S2 lists the fitting parameters.

Calculation of Efficiency and Power Density

(PF)eng=(∫TCTHS(T)dT)2∫TCTHρ(T)dT[S42](ZT)eng=(PF)eng∫TCTHκ(T)dT(TH−TC)[S43]ω=(PF)eng(TH−TC)Lmopt(1+mopt)2[S44]ηmax=ηC1+(ZT)eng(α1/ηC)−1α01+(ZT)eng(α1/ηC)+α2[S45]mopt=1+(ZT)engα1ηc−1[S46]ηC=TH−TCTH[S47]αi=STH(TH−TC)∫TCTHS(T)dT−∫TCTHτ(T)dT∫TCTHS(T)dTWTηC−iWJηC (i=0,1,2)[S48]τ=TdS(T)dT[S49]WT=∫TCTH∫TTHτ(T)dTdTΔT∫TCTHτ(T)dT[S50]WJ=∫TCTH∫TTHρ(T)dTdTΔT∫TCTHρ(T)dT.[S51]

Measurement of Efficiency and Power Density

The efficiency and power output measurement on a single leg are performed using an established approach (14). In this work the Nb_0.95Ti_0.05FeSb and Nb_0.8Ti_0.2FeSb are cut and polished to size ∼1.3 × 1.3 mm² in cross-section and ∼2 mm in height. The TE leg is directly brazed (Ag_0.45Cu_0.15Zn₁₆Cd₂₄, liquidus 618 °C) to the copper enclosure of the hot-junction heater assembly (Fig. S6A). The cold side of the TE leg is first electroplated with copper, nickel, and gold and then soldered (In₅₂Sn₄₈, melting point 118 °C) onto the cold-junction copper electrode (Fig. S6B). The hot- and cold-junction braze/solder joints are mechanically strong and show insignificant electrical contact resistance at RT. However, due to the excellent braze wettability of the samples, it was difficult to prevent sidewall wetting completely. Due to the nature of the single TE leg experiment, the electrical current needs to be supplied at the hot junction of the TE leg. Thus, the hot-junction heater assembly also functions as the electrical hot junction for the TE leg. The large TE current due to the low electrical conductance of the samples can be a significant challenge due to joule heating in the hot-junction current leads. The Joule heating is minimized by choosing a large-diameter copper current wire. The single TE leg device is surrounded by an electrically heated copper guard heater to which the leads of the heater assembly (electrical current, voltage, and thermocouple leads) are thermally grounded to minimize parasitic heat losses. The experiments are performed under high vacuum (below 10⁻⁴ mbar) to eliminate convection and air conduction losses. The copper cold-junction electrode is soldered onto a TE cooler (TEC) that is soldered onto a liquid-cooled cold plate to enable accurate cold-junction temperature control. The cold-junction temperature is measured with a T-type thermocouple that is embedded in the cold-junction copper electrode. The hot-junction temperature is measured with a K-type thermocouple embedded inside the copper enclosure of the heater assembly. During the experiments the guard heater temperature (measured with a K-type thermocouple) is maintained close to the temperature of the hot-junction heater to minimize parasitic heat losses. The single TE leg power output (density) and efficiency experiment is performed by measuring the TE voltage (V_TE) as a function of the TE current (I_TE) and recording the electrical input power to the hot-junction heater at various hot-junction temperatures (T_H) while the cold-junction temperature (T_C) is maintained at 293 K. Fig. S6C shows an example current-voltage and current-power curve for the single TE leg device operating with a hot-junction temperature of 773 K. Due to the large conductance of the measured TE samples, the radiative thermal shunting between the hot and cold sides is conservatively estimated to be below 0.5% in the measured temperature range. The effect of radiative heat transfer between the TE leg and its surroundings on the measurement is similarly insignificant.

Acknowledgments

This work is funded in part by the US Department of Energy under Contract DE-SC0010831 (materials synthesis and characterizations) and in part by the “Solid State Solar Thermal Energy Conversion Center,” an Energy Frontier Research Center funded by the US Department of Energy, Office of Science, Office of Basic Energy Science under Award DE-SC0001299 (output power density measurement), as well as by US Air Force Office of Scientific Research Grant FA9550-15-1-0236, T. L. L. Temple Foundation, John J. and Rebecca Moores Endowment, and the State of Texas through the Texas Center for Superconductivity at the University of Houston.

Footnotes

↵¹To whom correspondence may be addressed. Email: cwchu{at}uh.edu, gchen2{at}mit.edu, or zren{at}uh.edu.

Author contributions: R.H. and Z.R. designed research; R.H. performed research; D.K., L.Z., Y. Lan, C.L., J.S., H.S.K., Y. Liu, and D.B. contributed new reagents/analytic tools; R.H., J.M., Q.J., G.C., and Z.R. analyzed data; and R.H., D.K., L.Z., D.B., C.-W.C., G.C., and Z.R. wrote the paper.
Reviewers: J.-F.L., Tsinghua University; and S.P., Vienna University of Technology.
The authors declare no conflict of interest.
This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10.1073/pnas.1617663113/-/DCSupplemental.

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Achieving high power factor and output power density in p-type half-Heuslers Nb_1-xTi_xFeSb

Significance

Abstract

Materials and Methods

Synthesis.

Characterization.

Measurement.

Results and Discussion

Enhanced Power Factor with Higher Hot Pressing Temperature.

Effect of Grain Size on Lattice Thermal Conductivity and Carrier Mobility.

TE Properties of p-Type Nb_1-xTi_xFeSb.

Power Output.

Conclusions

Sample Characterization

Thermal Diffusivity and Specific Heat

Band Structure and Bipolar Thermal Conductivity

Bandgap.

Band Offset Between the Valence Bands.

Grain Size Distribution

Measuring Phonon MFP Distribution in Nb_0.95Ti_0.05FeSb

Klemens Model

Point Defect Scattering.

Phonon–Phonon Interaction.

EP Interaction.

GB Scattering.

Calculation of Efficiency and Power Density

Measurement of Efficiency and Power Density

Acknowledgments

Footnotes

References

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Main menu

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Search

Significance

Abstract

Materials and Methods

Synthesis.

Characterization.

Measurement.

Results and Discussion

Enhanced Power Factor with Higher Hot Pressing Temperature.

Effect of Grain Size on Lattice Thermal Conductivity and Carrier Mobility.

TE Properties of p-Type Nb1-xTixFeSb.

Power Output.

Conclusions

Sample Characterization

Thermal Diffusivity and Specific Heat

Band Structure and Bipolar Thermal Conductivity

Bandgap.

Band Offset Between the Valence Bands.

Grain Size Distribution

Measuring Phonon MFP Distribution in Nb0.95Ti0.05FeSb

Klemens Model

Point Defect Scattering.

Phonon–Phonon Interaction.

EP Interaction.

GB Scattering.

Calculation of Efficiency and Power Density

Measurement of Efficiency and Power Density

Acknowledgments

Footnotes

References

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TE Properties of p-Type Nb_1-xTi_xFeSb.

Measuring Phonon MFP Distribution in Nb_0.95Ti_0.05FeSb