Phonon transport is an essential property of thermoelectric materials. Although the phonon carries heat, which reduces the thermoelectric efficiency, it contributes positively to the Seebeck coefficient S through the phonon-drag effect, as typified by the high-purity semiconductors, which show fairly large S at cryogenic temperatures. Although such a large S is attractive in terms of Peltier cooling, a clear guiding principle for designing thermoelectric materials enriched by the phonon-drag effect remains to be established. Here we demonstrate that a correlated semiconductor, FeSb2, is a promising thermoelectric material featuring quasi-ballistic phonons dragging d electrons with large effective mass. By changing the sample size within the sub-millimetre order for high-purity single crystals, we succeed in substantially increasing S to as much as −27 mV K−1 at low temperatures. Our results exemplify a strategy for exploring phonon-drag-based thermoelectric materials, the performance of which can be maximized by combining heavy electrons with ballistic phonons.
Recent progress in thermoelectric materials has been achieved primarily by understanding the phonon transport properties1,2,3,4. To improve the thermoelectric efficiency defined by the dimensionless figure of merit zT=S2Tρ−1κ−1, where S, ρ, κ and T are the Seebeck coefficient, electrical resistivity, thermal conductivity and absolute temperature, respectively5, reduction of the phonon thermal conductivity has been recognized as a key strategy. This concept is exemplified by nanostructuring techniques, which effectively reduces the mean free paths (MFPs) of the phonons6,7. On the other hand, phonons with long MFPs strongly affect S through the phonon-drag effect, which is a type of non-equilibrium phenomenon, that is, the phonon current drags the charge carriers, giving rise to the additional thermoelectric voltage8,9,10. The phonon-drag effect is proportional to the strength of the electron-phonon coupling and the relaxation time of the phonons coupled to electrons. To take advantage of the phonon-drag effect for thermoelectric applications, the magnitude of this effect has to be maximized, to overcome the disadvantage of κ enhancement caused by the long-MFP phonons. However, the development of phonon-drag-based thermoelectrics has been hampered by difficulty in describing the phonon-drag effect, as well as the absence of an ideal system for systematic investigations.
In the simple phenomenological model proposed by Herring9, the phonon-drag component of the Seebeck coefficient, Sph, is characterized by the relation, Sph∝lphμel−1, where lph and μel represent the phonon MFP and carrier(electron) mobility, respectively. This relation is consistent with the fact that a giant Seebeck effect has been reported for conventional high-purity semiconductors with ballistic phonons, the MFP of which is on the order of millimetres8,9,10. However, high thermoelectric efficiency has not been reported for conventional semiconductors because of the huge κ-values inherent in such long-MFP phonons11. Following the above discussion, high-purity correlated semiconductors promise to show an even higher thermoelectric efficiency, owing to the larger phonon-drag Seebeck effect, because the carrier mobility μ is inversely proportional to the effective mass m*, which has not been experimentally demonstrated.
To date, a colossal Seebeck coefficient for an intermetallic compound, FeSb2, (as large as |S|≃45 mV K−1) has been reported12,13. However, as far as we know, the reproducibility is questionable; thus, the origin of such a large Seebeck effect remains controversial14,15,16,17,18. There are two major explanations for its origin. One is based on the narrow gap structure formed by hybridization between localized Fe 3d electrons and itinerant Sb 5p electrons, as typified by Kondo insulators19,20,21,22,23. The other is based on the phonon-drag effect17,24,25. A recent first-principles study of FeSb2 predicts the presence of ballistic phonons with long MFPs on the order of 100 μm at low temperatures26. However, the former d–p hybridization gap scenario is not reproduced by band calculations considering the electron correlations24 and the latter phonon-drag scenario is inconsistent with the fact that the impurity concentration in the reported samples is higher than that of high-purity semiconductors15,18,27.
Here we show that FeSb2 is a canonical correlated semiconductor showing a colossal phonon-drag Seebeck effect, which features an exquisite combination of quasi-ballistic phonons and massive d electrons. Using transport measurements of high-purity single crystals with different dimensions, we identify the presence of nearly ballistic phonons that hugely enhance the Seebeck effect in FeSb2. In addition, the effective mass of charge carriers is found to be reasonably large by cyclotron resonance (CR) measurements at low temperatures.
Crystal size effect on the transport properties
Using high-purity single-crystalline samples of FeSb2 with different dimensions (80 × 160 μm2 ≤Sample cross-section ≤250 × 270 μm2), as shown in Fig. 1a,b, we studied the size effect on ρ, S and κ. Single crystals of FeSb2 reportedly exhibit two distinct types of transport properties, depending on the crystal growth conditions16: one shows semiconducting behaviour along all the axes, whereas the other shows semiconducting behaviour along the a- and b-axes but metallic behaviour along the c-axis above 50 K. Thus, to discuss the intrinsic size effect on the thermoelectric properties, we checked the crystallographic orientation dependence of ρ and found that our samples are nearly isotropic semiconductors (see Supplementary Figs 1 and 2). The crystallographic orientations of samples S1–S3 were characterized by Laue diffraction, as shown in Supplementary Fig. 3. Furthermore, to completely exclude the extrinsic effects arising from differences in the sample quality and crystallographic orientation, we prepared two samples, S4 and S5, which were obtained by cutting a single crystal into two pieces (see Supplementary Fig. 4). To determine S and κ, we used a steady-state technique, as shown in Fig. 1c, where two chromel-constantan thermocouples were mounted on the sample to measure the temperature difference and thermoelectric voltage. Figure 2a shows the temperature dependence of ρ; all values of ρ are almost the same, so it is essentially independent of the sample dimensions (the temperature dependence of ρ above 45 K is shown in Supplementary Fig. 5). From the magnetotransport properties, we estimated the carrier concentration n and mobility μ of all the samples from 8 to 30 K (see Supplementary Figs 6–8 and Supplementary Note 1). The obtained parameters are nearly invariant for all the samples with different dimensions, indicating that they share essentially the same electronic states with slightly different impurity levels.
In Fig. 2b,c, we plot S and κ as a function of temperature, revealing a significant sample-size effect. S4 has a considerably large |S| value of 27 mV K−1 at 10 K and the maximum value decreases to 10 mV K−1 (S3) with decreasing sample cross-section. As the electrical transport properties (ρ, n, and μ) are exactly the same in S4 and S5, it should be the sample size rather than the electronic conduction that plays a major role in the striking enhancement of S. Corresponding to the size dependence of S, the maximum value of κ also decreases from 770 W m−1 K−1 (S4) to 230 W m−1 K−1 (S3) at 15 K. The electronic component of κ, evaluated using ρ and the Wiedemann–Franz law, was found to be negligible for each sample, indicating that the size effect on κ stems from phonon transport, which is, therefore, likely to be related to the Seebeck effect. This idea is further supported by the fact that the |S| value for a low-purity (99.9% purity) polycrystalline sample (∼300 μV K−1) is two orders of magnitude smaller than that for the high-purity (99.999% purity) single crystals (Fig. 2b, inset), whereas the ρ-values of both samples at low temperatures are comparable (Fig. 2a). These results indicate that phonon scattering by crystal/grain boundaries and impurities significantly affects on the Seebeck effect27.
Surface scattering effect on the phonon MFP
Next, we evaluated the phonon MFPs using Fourier’s law (κ=1/3Cvlκ) with the Debye model (Debye temperature=340 K), where C, v and lκ are the specific heat, phonon velocity (v=3,100 m s−1) and MFP of the phonon involved in the thermal conductivity, respectively12. Here we assume that the phonon transport is isotropic, as it has been reported that the phonon velocity and thermal conductivity are almost independent of the crystal orientations12,28. As shown in Fig. 3a, lκ increases to on the order of 100 μm with decreasing temperature, suggesting that lκ is dominated by crystal-boundary scattering at low temperature. Here we define lb as the MFP dominated by such crystal boundary scattering. Provided that the sample surface with an appropriate roughness acts as diffuse scatterers of phonons, lb for the rectangular sample with side dimensions D and nD (Fig. 3a) can be evaluated as29
By using this equation, the lb value for samples S1, S2, S3, S4 and S5 were estimated to be 280, 230, 120, 290 and 150 μm, respectively. This estimation method loses its validity when the length scale of the rectangular sample, L, is comparable to the phonon MFP30, whereas this method is commonly used when L is much larger than the phonon MFP, which is in the order of D (ref. 29). This situation (L>>D∼lb) is satisfied in our experiment and, remarkably, the experimental data approach the evaluated values with equation (1) (dotted lines in Fig. 3a) at low temperature, indicating that boundary scattering dominates the thermal conductivity κ.
To clarify the relation between the phonon transport and the Seebeck effect, we quantitatively analyse S assuming the phonon-drag effect. In the approximate expression of the theory proposed by Herring, the phonon-drag component of S equals βvlsμ−1T−1, where ls is the MFP of the phonon involved in the phonon-drag effect and β is a parameter between 0 and 1 characterizing the relative strength of electron–phonon interactions9. If electrons are scattered only by phonons, β is 1, but additional scattering processes, such as impurity scattering, can bring it to 0 (ref. 9). From 30 to 15 K, μ increases with decreasing the temperature with the slope of T−3/2 (see Supplementary Fig. 8b), which is typically observed in the semiconductors, when the electrons mainly scattered by phonon. In addition, the increment of |S| is also observed in the same temperature range. These results imply that the carriers are primarily scattered by phonons. Therefore, we adopt the value β=1 for the evaluation of ls at 20 and 15 K. Here the electronic part of S is removed to evaluate ls at 15 and 20 K, which contributes <20% of the total values (see Supplementary Fig. 9 and Supplementary Note 2). Below 10 K, ls cannot be precisely evaluated, because |S(T)| is diminished, suggesting that β diverges from unity.
Here we compare the length scales of the thermal (lκ) and phonon-drag (ls) phonon MFPs, as shown in Fig. 3b. The values of lκ and ls have the same order of magnitude and show a clear proportional relationship. Although the phonon MFPs are distributed in a certain range at finite temperature, the linear relationship between ls and lκ strongly suggests that it is the crystal-boundary scattering that primarily determines the lengths of ls and lκ with a small wave vector (the phonon MFP is order of 100 μm). In other words, the similar size effect observed for lκ and ls indicates that the phonons, of which MFP is shortened with decreasing sample dimensions, contribute significantly to the colossal S value as well as κ. This result is the first direct evidence that the colossal S value in FeSb2 stems from the large enhancement of the phonon-drag effect by the presence of the nearly ballistic phonons.
According to the phonon-drag picture (S∝lsμ−1), the origin of the colossal S value of FeSb2 is ascribable not only to the nearly ballistic phonons but to the reduced electron mobility μ. As the ls value in FeSb2 is two orders of magnitude shorter than that in Si (∼1 mm), the μ-value of FeSb2 should be small enough that the S value is comparable to or even larger than that of Si (ref. 10). Indeed, the μ-value of FeSb2 is reported to be much smaller (∼5 × 103 cm2 V−1 s−1 at 15 K) than that of high-purity Si (∼2 × 105 cm2 V−1 s−1 at 15 K)15,31. In the Drude model, μ is expressed as eτem*−1, where τe is the relaxation time of carriers. To see whether the μ-value of FeSb2 is reduced by the enhanced electron mass, we measured the CR for a high-purity single crystal of FeSb2 at 5 K. As shown in Fig. 4a, the resonance peak shifts to higher fields as ν increases. The linear fit of the ν dependence of the resonant frequency gives m*=5.4m0 (Fig. 4b), which is 20 times larger than that of Si (∼0.3m0)31.
The significantly large effective mass of FeSb2 indicates that the low mobility μ (∼τem*−1) reflects the large effective mass, as τe of FeSb2 is comparable to that of Si. Therefore, we attribute the origin of the colossal Seebeck effect in FeSb2 to the huge enhancement of the phonon-drag effect by the combination of the fairly large ls (although it is not as large as that of high-purity Si) and the small μ inherent in the large m*. The large ls is anticipated in the first principles study by Liao et al.26, which found that the phonon MFPs at 20 K are concentrated in the range of 1–200 μm when the phonons are not scattered by the impurities and crystal boundaries. In our study, we have successfully prepared the high-purity single crystals with the large dimensions by a flux method using the high-purity metal powders (see Methods), resulting in the fairly large ls. On the other hand, the large value of m* presumably reflects the characteristic band structure as predicted by thermodynamic measurements and band calculations14,24. It has been reported that there exists a high and steep density of states just above the Fermi level in the conduction band, which mainly consists of the Fe 3d orbitals (Fig. 4c).
Our result demonstrates a simple strategy to design phonon-drag-based thermoelectric materials. We depict the thermoelectric efficiency associated with the phonon-drag effect in Fig. 3c. The linear relation between ls and lκ (ls=γlκ) yields a simple expression for the phonon drag’s dimensionless figure of merit zpT=Sph2σTκ−1=Alsnμ−1T−1 (σ=neμ, A=3β2evC−1γ−1), which can be improved by increasing ls and m*. Whereas zpT is restricted to a low value (<1/4) for conventional semiconductors11, strongly correlated semiconductors such as Kondo insulators have the potential to greatly improve their efficiency because of their fairly large m* values. Therefore, provided that m* is as large as 1,000m0, as observed in heavy fermion systems32, and the phonon MFP is on the order of sub-millimetre, zpT is expected to reach 1, even at low temperatures. In such strongly correlated systems, the strong electron–electron scattering creates the quasi-particle with the larger effective mass compared with the band mass. Thus, the interaction between these quasi-particles and the phonons may cause a significant phonon-drag effect beyond the result of FeSb2.
Single crystals of FeSb2 were grown by a self-flux method using metal powders of 99.999% (5N) pure Fe and 99.9999% (6N) pure Sb in an evacuated silica tube, as described in ref. 27. The single-crystalline nature was verified by Laue-pattern analysis (see Supplementary Figs 3 and 4). We measured the powder X-ray diffraction pattern of our samples and found no impurity peaks (see Supplementary Fig. 10). All samples were prepared simultaneously and then rubbed with sandpaper to yield five samples with different dimensions as shown in Fig. 1a. The prepared samples each have cross-section F and length L as shown in Table 1. S4 and S5 were obtained by cutting a single crystal into two pieces, meaning that their quality and crystallographic orientation should be identical. As all samples were ground by the same sandpaper, their surface conditions are expected to be almost the same. A polycrystalline sample was prepared by solid-state reaction using metal powders of 99.9% (3N) pure Fe and 99.9% (3N) pure Sb in an evacuated silica tube, as described in ref. 14.
The electrical resistivity ρ was measured as a function of temperature by a four-probe method.
The CR was measured at 5 K from 28.2 to 59.5 GHz. The CR apparatus consists of a 16 T superconducting magnet (14 T at 4.2 K and 16 T at 2.2 K, Oxford Instruments, UK), a vector network analyser (AB Millimeter Co. Ltd, France) and a home-built transmission-type CR cryostat (usually used for electron spin resonance). With this vector network analyser, we can detect not only the amplitude of the CR signal but also its phase. A variable temperature insert with an inner diameter of 37 mm is plugged into the bore of the magnet; thus, the temperature can be varied from 1.5 to 200 K.
The data that support the findings of this study are available from the corresponding author upon request.
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We thank M. Sato for the initial motivation of this work. We also appreciate H. Sakai, K. Akiba and M. Tokunaga for their helpful suggestion. This work was partially supported by Grant-in-Aid for Scientific Research, Japan Society for the Promotion of Science, Japan (Kakenhi No. 26287064), and the Strategic Japanese-Finland Cooperative Program on ‘Functional Materials,’ JST, Japan. H.T. acknowledges support from the JSPS Research Fellowship for Young Scientist.
The authors declare no competing financial interests.
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Takahashi, H., Okazaki, R., Ishiwata, S. et al. Colossal Seebeck effect enhanced by quasi-ballistic phonons dragging massive electrons in FeSb2. Nat Commun 7, 12732 (2016). https://doi.org/10.1038/ncomms12732
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