Abstract
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 phonondrag effect, as typified by the highpurity 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 phonondrag effect remains to be established. Here we demonstrate that a correlated semiconductor, FeSb_{2}, is a promising thermoelectric material featuring quasiballistic phonons dragging d electrons with large effective mass. By changing the sample size within the submillimetre order for highpurity 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 phonondragbased thermoelectric materials, the performance of which can be maximized by combining heavy electrons with ballistic phonons.
Introduction
Recent progress in thermoelectric materials has been achieved primarily by understanding the phonon transport properties^{1,2,3,4}. To improve the thermoelectric efficiency defined by the dimensionless figure of merit zT=S^{2}Tρ^{−1}κ^{−1}, where S, ρ, κ and T are the Seebeck coefficient, electrical resistivity, thermal conductivity and absolute temperature, respectively^{5}, 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 phonons^{6,7}. On the other hand, phonons with long MFPs strongly affect S through the phonondrag effect, which is a type of nonequilibrium phenomenon, that is, the phonon current drags the charge carriers, giving rise to the additional thermoelectric voltage^{8,9,10}. The phonondrag effect is proportional to the strength of the electronphonon coupling and the relaxation time of the phonons coupled to electrons. To take advantage of the phonondrag effect for thermoelectric applications, the magnitude of this effect has to be maximized, to overcome the disadvantage of κ enhancement caused by the longMFP phonons. However, the development of phonondragbased thermoelectrics has been hampered by difficulty in describing the phonondrag effect, as well as the absence of an ideal system for systematic investigations.
In the simple phenomenological model proposed by Herring^{9}, the phonondrag component of the Seebeck coefficient, S_{ph}, is characterized by the relation, S_{ph}∝l_{ph}μ_{el}^{−1}, where l_{ph} and μ_{el} 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 highpurity semiconductors with ballistic phonons, the MFP of which is on the order of millimetres^{8,9,10}. However, high thermoelectric efficiency has not been reported for conventional semiconductors because of the huge κvalues inherent in such longMFP phonons^{11}. Following the above discussion, highpurity correlated semiconductors promise to show an even higher thermoelectric efficiency, owing to the larger phonondrag 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, FeSb_{2}, (as large as S≃45 mV K^{−1}) has been reported^{12,13}. However, as far as we know, the reproducibility is questionable; thus, the origin of such a large Seebeck effect remains controversial^{14,15,16,17,18}. 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 insulators^{19,20,21,22,23}. The other is based on the phonondrag effect^{17,24,25}. A recent firstprinciples study of FeSb_{2} predicts the presence of ballistic phonons with long MFPs on the order of 100 μm at low temperatures^{26}. However, the former d–p hybridization gap scenario is not reproduced by band calculations considering the electron correlations^{24} and the latter phonondrag scenario is inconsistent with the fact that the impurity concentration in the reported samples is higher than that of highpurity semiconductors^{15,18,27}.
Here we show that FeSb_{2} is a canonical correlated semiconductor showing a colossal phonondrag Seebeck effect, which features an exquisite combination of quasiballistic phonons and massive d electrons. Using transport measurements of highpurity single crystals with different dimensions, we identify the presence of nearly ballistic phonons that hugely enhance the Seebeck effect in FeSb_{2}. In addition, the effective mass of charge carriers is found to be reasonably large by cyclotron resonance (CR) measurements at low temperatures.
Results
Crystal size effect on the transport properties
Using highpurity singlecrystalline samples of FeSb_{2} with different dimensions (80 × 160 μm^{2} ≤Sample crosssection ≤250 × 270 μm^{2}), as shown in Fig. 1a,b, we studied the size effect on ρ, S and κ. Single crystals of FeSb_{2} reportedly exhibit two distinct types of transport properties, depending on the crystal growth conditions^{16}: one shows semiconducting behaviour along all the axes, whereas the other shows semiconducting behaviour along the a and baxes but metallic behaviour along the caxis 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 steadystate technique, as shown in Fig. 1c, where two chromelconstantan 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 samplesize 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 crosssection. 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 lowpurity (99.9% purity) polycrystalline sample (∼300 μV K^{−1}) is two orders of magnitude smaller than that for the highpurity (99.999% purity) single crystals (Fig. 2b, inset), whereas the ρvalues 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 effect^{27}.
Surface scattering effect on the phonon MFP
Next, we evaluated the phonon MFPs using Fourier’s law (κ=1/3Cvl_{κ}) with the Debye model (Debye temperature=340 K), where C, v and l_{κ} are the specific heat, phonon velocity (v=3,100 m s^{−1}) and MFP of the phonon involved in the thermal conductivity, respectively^{12}. 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 orientations^{12,28}. As shown in Fig. 3a, l_{κ} increases to on the order of 100 μm with decreasing temperature, suggesting that l_{κ} is dominated by crystalboundary scattering at low temperature. Here we define l_{b} as the MFP dominated by such crystal boundary scattering. Provided that the sample surface with an appropriate roughness acts as diffuse scatterers of phonons, l_{b} for the rectangular sample with side dimensions D and nD (Fig. 3a) can be evaluated as^{29}
By using this equation, the l_{b} 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 MFP^{30}, 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∼l_{b}) 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.
Phonondrag effect
To clarify the relation between the phonon transport and the Seebeck effect, we quantitatively analyse S assuming the phonondrag effect. In the approximate expression of the theory proposed by Herring, the phonondrag component of S equals βvl_{s}μ^{−1}T^{−1}, where l_{s} is the MFP of the phonon involved in the phonondrag effect and β is a parameter between 0 and 1 characterizing the relative strength of electron–phonon interactions^{9}. 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 l_{s} at 20 and 15 K. Here the electronic part of S is removed to evaluate l_{s} at 15 and 20 K, which contributes <20% of the total values (see Supplementary Fig. 9 and Supplementary Note 2). Below 10 K, l_{s} cannot be precisely evaluated, because S(T) is diminished, suggesting that β diverges from unity.
Here we compare the length scales of the thermal (l_{κ}) and phonondrag (l_{s}) phonon MFPs, as shown in Fig. 3b. The values of l_{κ} and l_{s} 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 l_{s} and l_{κ} strongly suggests that it is the crystalboundary scattering that primarily determines the lengths of l_{s} and l_{κ} with a small wave vector (the phonon MFP is order of 100 μm). In other words, the similar size effect observed for l_{κ} and l_{s} 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 FeSb_{2} stems from the large enhancement of the phonondrag effect by the presence of the nearly ballistic phonons.
CR measurements
According to the phonondrag picture (S∝l_{s}μ^{−1}), the origin of the colossal S value of FeSb_{2} is ascribable not only to the nearly ballistic phonons but to the reduced electron mobility μ. As the l_{s} value in FeSb_{2} is two orders of magnitude shorter than that in Si (∼1 mm), the μvalue of FeSb_{2} should be small enough that the S value is comparable to or even larger than that of Si (ref. 10). Indeed, the μvalue of FeSb_{2} is reported to be much smaller (∼5 × 10^{3} cm^{2} V^{−1} s^{−1} at 15 K) than that of highpurity Si (∼2 × 10^{5} cm^{2} V^{−1} s^{−1} at 15 K)^{15,31}. In the Drude model, μ is expressed as eτ_{e}m^{*−1}, where τ_{e} is the relaxation time of carriers. To see whether the μvalue of FeSb_{2} is reduced by the enhanced electron mass, we measured the CR for a highpurity single crystal of FeSb_{2} 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*=5.4m_{0} (Fig. 4b), which is 20 times larger than that of Si (∼0.3m_{0})^{31}.
Discussion
The significantly large effective mass of FeSb_{2} indicates that the low mobility μ (∼τ_{e}m^{*−1}) reflects the large effective mass, as τ_{e} of FeSb_{2} is comparable to that of Si. Therefore, we attribute the origin of the colossal Seebeck effect in FeSb_{2} to the huge enhancement of the phonondrag effect by the combination of the fairly large l_{s} (although it is not as large as that of highpurity Si) and the small μ inherent in the large m*. The large l_{s} 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 highpurity single crystals with the large dimensions by a flux method using the highpurity metal powders (see Methods), resulting in the fairly large l_{s}. On the other hand, the large value of m* presumably reflects the characteristic band structure as predicted by thermodynamic measurements and band calculations^{14,24}. 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 phonondragbased thermoelectric materials. We depict the thermoelectric efficiency associated with the phonondrag effect in Fig. 3c. The linear relation between l_{s} and l_{κ} (l_{s}=γl_{κ}) yields a simple expression for the phonon drag’s dimensionless figure of merit z_{p}T=S_{ph}^{2}σTκ^{−1}=Al_{s}nμ^{−1}T^{−1} (σ=neμ, A=3β^{2}evC^{−1}γ^{−1}), which can be improved by increasing l_{s} and m*. Whereas z_{p}T is restricted to a low value (<1/4) for conventional semiconductors^{11}, 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 1,000m_{0}, as observed in heavy fermion systems^{32}, and the phonon MFP is on the order of submillimetre, z_{p}T is expected to reach 1, even at low temperatures. In such strongly correlated systems, the strong electron–electron scattering creates the quasiparticle with the larger effective mass compared with the band mass. Thus, the interaction between these quasiparticles and the phonons may cause a significant phonondrag effect beyond the result of FeSb_{2}.
Methods
Sample preparation
Single crystals of FeSb_{2} were grown by a selfflux 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 singlecrystalline nature was verified by Lauepattern analysis (see Supplementary Figs 3 and 4). We measured the powder Xray 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 crosssection 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 solidstate 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.
Transport measurement
The electrical resistivity ρ was measured as a function of temperature by a fourprobe method.
CR measurements
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 homebuilt transmissiontype 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.
Data availability
The data that support the findings of this study are available from the corresponding author upon request.
Additional information
How to cite this article: Takahashi, H. et al. Colossal Seebeck effect enhanced by quasiballistic phonons dragging massive electrons in FeSb_{2}. Nat. Commun. 7:12732 doi: 10.1038/ncomms12732 (2016).
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Acknowledgements
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 GrantinAid for Scientific Research, Japan Society for the Promotion of Science, Japan (Kakenhi No. 26287064), and the Strategic JapaneseFinland Cooperative Program on ‘Functional Materials,’ JST, Japan. H.T. acknowledges support from the JSPS Research Fellowship for Young Scientist.
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H. Takahashi and I.T. designed the study; H. Takahashi performed the experiments and analysed the data. A.O. and M.H. carried out the CR experiment. R.O. provided technical assistance. R.O., S.I., H. Taniguchi and I.T. provided critical review. H. Takahashi, S.I. and I.T. wrote the manuscript. I.T. supervised the study.
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Correspondence to H. Takahashi or I. Terasaki.
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Supplementary Information
Supplementary Figures 110, Supplementary Notes 12 and Supplementary References (PDF 1934 kb)
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Takahashi, H., Okazaki, R., Ishiwata, S. et al. Colossal Seebeck effect enhanced by quasiballistic phonons dragging massive electrons in FeSb_{2}. Nat Commun 7, 12732 (2016). https://doi.org/10.1038/ncomms12732
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