Experimental

Nanostructured FeSb₂ samples were synthesized by first forming an ingot through melting and solidification. The ingot was ball milled and hot pressed at different temperatures to obtain the nanocomposite samples with different grain sizes ranging from nanometers to micrometers. The labels given to the different samples and the corresponding processing parameters are shown in Table I. The Seebeck coefficient (S), electrical resistivity (ρ), thermal conductivity (κ), and Hall coefficient (R _H) were measured on a Physical Property Measurement System (PPMS) from Quantum Design Inc. (San Diego, CA). Sample preparation and measurements were performed in the manner described in Ref. Reference Zhao, Pokharel, Zhu, Chen, Lukas, Qing, Opeil, Chen and Ren6.

Table 1.

Assigned IDs, processing temperature, and average grain sizes estimated from the SEM images for the four nanocomposite FeSb₂ samples.

Results and discussions

Figure 1 shows the temperature dependence of the Seebeck coefficient for our four nanocomposite samples. Sample S-600 was measured both in magnetic fields of 0 and 9 T. The Seebeck coefficients of all samples are significantly smaller than that of a single crystal at low temperature, whereas at high temperature the values are comparable. A room temperature value of S ~ 26 µV/K was observed for all our samples. This value is comparable to S ~ 31 µV/K reported for the single crystal.^[Reference Abrikosov and Petrova^15] A decrease in the Seebeck coefficient at low temperature in polycrystal,^[Reference Bentien, Madsen, Johnsen and Iversen^11] thin films^[Reference Sun, Johnsen, Eklund, Sillassen, Bøttiger, Oeschler, Sun, Steglich and Iversen^16] and arsenic-substituted FeSb₂ single crystals^[Reference Sun, Oeschler, Johnsen, Iversen and Steglich^9] was reported earlier. As shown in the inset of Fig. 1, the peak value of the Seebeck coefficient (S _max) decreases with decreasing grain size. This is expected for the phonon-drag effect, because the non-electronic scattering (grain boundary scattering in the case of nanocomposites) reduces the phonon mean free path, which in turn decreases the phonon-drag contribution. A similar conclusion was drawn in Ref. Reference Tomczak, Haule, Miyake, Georges and Kotliar14. Yet another example of the reduction of phonon mean free path decreasing phonon drag contribution was shown by Weber et al.^[Reference Weber, Lehr and Gmelin^17] It was demonstrated, using point contacts, that when the contact size becomes comparable to the mean free path of the relevant phonons, the phonon-drag part of the thermopower is suppressed by the boundary scattering. Here, we also note that the temperature profile of the Seebeck coefficient for the samples with larger grains follows the typical behavior of phonon-drag system as suggested by Blatt.^[Reference Blatt^18]

Figure 1.

Temperature dependence of the Seebeck coefficient for the four samples. Sample S-600 was measured both at 0 and 9 T magnetic fields. Inset: The grain size dependence of the peak value (S _max) and the peak position (T _max) of the Seebeck coefficient.

The Seebeck peaks shift to the higher temperature when the grain size decreases (inset of Fig. 1). This type of size-dependent shift in the Seebeck peak is one of the characteristics of the phonon-drag-dominated systems, as pointed out previously.^[Reference Geballe and Hull^8, Reference Hou, Gu, Chen and He^19, Reference Issi and Boxus^20]

Usually, a small magnetothermopower is expected for the phonon-drag dominated system. For sample S-600 at 25 K, we observed a relative change in the Seebeck coefficient, (S(9T) − S(0T))/S(0T) = 0.059. This small value supports the non-electronic origin of the large Seebeck coefficient in this sample.

Figure 2 shows the temperature dependence of the thermal conductivity (κ) for the four samples. Calculations based on the Wiedemann–Franz law show that below 50 K, more than 97% of the total thermal conductivity comes from the lattice contribution. For large-grained samples, a temperature dependence of the form κ ~ T ² is nearly obeyed at temperatures below 50 K. Weber et al.^[Reference Weber and Gmelin^21] reported a similar T ² behavior in silicon below 20 K that they attribute to strong electron-phonon scattering. As the grain size decreases, we find a gradual deviation from the T ² law. This can be understood as follows: for a large-grained sample, the number of available phonons interacting with the carriers is large which causes significant phonon-drag. As the grain size decreases the number of relevant phonons decreases and as a result the preferential scattering of electrons by phonons becomes weaker. Evidence for electron–phonon coupling in FeSb₂ single crystals was also reported by Perucchi et al.^[Reference Perucchi, Degiorgi, Hu, Petrovic and Mitrović^22] who found a large change in phonon lifetimes using optical spectroscopy. Lazarevic et al.^[Reference Lazarević, Popović, Hu and Petrovic^23] later suggested that the electron–phonon coupling is temperature dependent below 40 K.

Figure 2.

Temperature dependence of the thermal conductivity for the four samples. Fitting to the power law was done for all the samples below 50 K. Inset: Seebeck coefficient as a function of thermal conductivity at 25 and 50 K.

The inset of Fig. 2 shows the change in the Seebeck coefficient with thermal conductivity for 25 and 50 K. At 25 K, the Seebeck coefficient increases linearly with the thermal conductivity. The linear relationship is expected because in a phonon-drag dominated system S _p is expected to follow the same temperature dependence as the phonon heat capacity so that S _p and the lattice thermal conductivity (κ _l) are linearly related. This indicates that phonons play a significant role in determining the Seebeck coefficient values of our samples at around 25 K. At 50 K, however, the Seebeck coefficient decreases with increasing thermal conductivity without any obvious trend. This is an indication that the phonon-mean free path decreases and the phonon-drag effect becomes weak at higher temperature.

Significant phonon drag effects are expected to occur when the dominant phonons acquire sufficient momentum to scatter carriers across the Fermi surface.^[Reference Morelli^24] In a rough approximation,$\; T_{{\rm max}} \approx {\textstyle{1 \over {10}}}\theta _D $, where T _max is the temperature at which the phonon drag peak occurs and θ _D is the Debye temperature. For rutile TiO₂,^[Reference Tang, Wang, Zhao and Li^25]θ _D is 450–780 K and the Seebeck peak occurs at 10–30 K. For bismuth,^[Reference Issi and Boxus^20] another well-known phonon drag system, θ _D is 119 K, whereas the Seebeck peak occurs at 2–3 K. For the FeSb₂ polycrystal, θ _D ≈ 330 − 350K was reported in Ref. Reference Bentien, Madsen, Johnsen and Iversen11 and 256 K in Ref. Reference Petrovic, Lee, Vogt, Lazarov, Bud'ko and Canfield26. The Seebeck peak in single crystal occurs at around 10 K. For our sample S-600, the peak occurs at 25 K. Hence, comparing with the other phonon drag systems, the scaling between T _max and θ _D in FeSb₂ is consistent with the phonon-drag picture.

Following Herring,^[Reference Herring^7] the electron-diffusion part of the Seebeck coefficient, in μV/K, for a semiconductor is given by

(2)$$S_{\rm d}=\mp \; 86.2\left[{{\rm ln}\displaystyle{{4.7 \times 10^{15} } \over n}+\displaystyle{3 \over 2}{\rm ln}\displaystyle{{m^{\ast} } \over m}+\displaystyle{{\vert \Delta E\vert } \over {kT}}+\displaystyle{3 \over 2}{\rm ln}T} \right]\comma \;$$

where n is the charge carrier density expressed in cm⁻³, m and m* are the bare and effective masses of the electron, respectively, and ΔE is the average energy of the transported electrons relative to the band edge. In Eq. (2), the upper and lower signs are for n-type and p-type materials, respectively.

For the case of lattice scattering by long wavelength phonons, |ΔE| /kT can be approximated by^[Reference Herring^7]

(3)$$\left\vert {\displaystyle{{\Delta E} \over {kT}}} \right\vert =\displaystyle{5 \over 2}+r\comma \;$$

where the scattering parameter r is taken to be −1/2. S _d was calculated taking m* = m and using the charge carrier density (n) calculated from the Hall-coefficient measurements. The diffusion part was then subtracted from the total measured Seebeck coefficient to obtain the drag contribution shown in Fig. 3. For S-600 at 25 K, we find the diffusion value, S _d = −110 µV/K. Using Eq. (1), S _p was calculated to be −238 µV/K. However, for S-300 the calculated diffusion contribution turned out to be slightly larger than the measured values. Hence, no significant phonon drag contribution is expected for this sample. This is understandable, because the phonon mean free path for sample S-300 is drastically reduced so that the dominant phonons do not carry sufficient momentum to scatter carriers. Moreover, the phonon drag is expected to weaken with increasing carrier concentration and, in fact, it has been proposed that a saturation effect occurs at high concentration level. The overestimation of the diffusion part of the Seebeck coefficient in S-300 is probably due, in part, to the approximations made in the above calculations.

Figure 3.

Temperature dependence of the calculated value of diffusion and drag part for the sample S-600 based on Eqs. (1) and (2). The carrier concentration obtained from the Hall coefficient data was used in the calculation. The measured total Seebeck coefficient is also plotted for comparison.

The phonon-drag thermopower for semiconductors, in the first-order approximation, can be written as^[Reference Herring^7]

(4)$$\; S_{\rm p}=\displaystyle{{\beta v_{\rm s} \lambda _{\rm p} ne} \over {\sigma T}}\comma \;$$

where v _s is the velocity of sound, λ_p the mean free path of the interacting phonons, n the charge carrier density, σ the electrical conductivity, T is the absolute temperature, and β is the dimensionless parameter with its value ranging from 0 to 1 depending upon the strength of the interaction. Equation (4) can be used to find approximate values of the mean free path of t phonons interacting with the electrons. The mean free path of an average phonon can also be estimated from the lattice thermal conductivity based on kinetic theory^[Reference Ziman^27]

(5)$$K_{\rm l}=\displaystyle{1 \over 3}c_V \widetilde{\lambda _{\rm P} }v_{\rm S}\comma \;$$

where K _l is the lattice thermal conductivity and c _V the phonon contribution to the total specific heat capacity. The lattice contribution of the total thermal conductivity was calculated assuming κ_total = κ_l + κ_e and κ_e = LσT, where κ_e is the electronic contribution to the total thermal conductivity. The Lorenz number in the free electron model, L = 2.45 × 10⁻⁸ W/K²/Ω was used, c _V was calculated using C _P − γT = βT ³ with γ = 3.98 × 10⁻³/JK²/mol, as reported in Ref. Reference Bentien, Madsen, Johnsen and Iversen11, and the C _P values were obtained from the same reference. Figure 4 shows the temperature dependence of the two length scales for samples S-600. It turns out that with ʋ _s = 3116 m/s and β = 0.5, the two length scales, which come from two independent calculations, are of the same order of magnitude and more importantly they behave roughly the same way as a function of temperature. This suggests that similar wavelength phonons are playing a role in both the thermal conductivity and the phonon-drag effect. A similar analysis was performed in Ref. Reference Thurber and Mante28 to describe the unusually large Seebeck coefficient in rutile TiO₂ at low temperature.

Figure 4.

Temperature dependence of the phonon mean free paths calculated from two independent calculations using Eqs. (4) and (5).

According to Keyes et al.^[Reference Keyes, Heikes and Ure^29] the value of the figure of merit (Z _p) that can be reached using phonon drag is rather low. Based on their argument, Goldsmid^[Reference Goldsmid^30] shows that Z _pT is less than ¼ for a bulk thermoelectric material. Ivanov et al.^[Reference Ivanov and Rowe^31] reported recently a similar conclusion for low-dimensional structures. These conclusions are consistent with our earlier work^[Reference Zhao, Pokharel, Zhu, Chen, Lukas, Qing, Opeil, Chen and Ren^6] where, despite the dramatic reduction in the thermal conductivity by three orders of magnitude, ZT values were not improved significantly Fig. 3(d) in Ref. [Reference Zhao, Pokharel, Zhu, Chen, Lukas, Qing, Opeil, Chen and Ren6].

Finally, our nanocomposite data and analysis of phonon drag do not support other mechanisms that might explain the large Seebeck effects reported by Bentien et al.^[Reference Bentien, Johnsen, Madsen, Iversen and Steglich^5] Although one cannot preclude the presence of electron–electron correlation effects, their role in this phenomenon may be a minor one. The recent analysis of electron correlations using a hybrid functional approach of Becke^[Reference Becke^32] and Hegin's GW functional approach^[Reference Hedin^33] by Tomczak et al.^[Reference Tomczak, Haule, Miyake, Georges and Kotliar^14] suggest that the high thermopower in FeSb₂ should not be understood in the context of local correlations, but rather by utilizing vertex corrections to the transport coefficients. Such vertex corrections describe the phonon-drag effect. The phonon-drag effects in FeSb₂ are similar to those described in p-type Ge by Geballe and Hull.^[Reference Geballe and Hull^8] In a similar vein the study of magnetoresistance and Hall effect by Takahashi et al.^[Reference Takahashi, Okazaki, Yasui and Terasaki^13] concludes that the large Seebeck coefficient in FeSb₂ is unlikely to originate from electron–electron correlations because they have an insignificant effect on the Seebeck coefficient in the low-temperature insulating regime. Our data on FeSb₂ nanocomposite supports their conjecture that the phonon-drag effect plays an essential role in enhancing the Seebeck coefficient in the low-temperature regime, as shown in other semiconductor materials e.g., InSb^[Reference Frederikse and Mielczarek^34] and weakly P doped Si.^[Reference Weber and Gmelin^21]