Right sizes of nano- and microstructures for high-performance and rigid bulk thermoelectrics

Significance

PbTe is known to be a promising thermoelectric material for waste heat recovery, so it has been the subject of extensive research involving new approaches. It is important to note that the performances of these developed materials can depend on the material synthesis conditions. We investigated three different routes of synthesizing 2% Na-doped PbTe and found that its thermoelectric figure of merit, zT, can be enhanced to ∼2.0 at 773 K. Also, the mechanical hardness of the sample synthesized by this condition was nearly double than that of the other samples. Our study shows that the size of nano- and microstructures can vary significantly by the choice of synthesis methods, which can explain the variation in zTs and mechanical hardness.

Abstract

In this paper, we systematically investigate three different routes of synthesizing 2% Na-doped PbTe after melting the elements: (i) quenching followed by hot-pressing (QH), (ii) annealing followed by hot-pressing, and (iii) quenching and annealing followed by hot-pressing. We found that the thermoelectric figure of merit, zT, strongly depends on the synthesis condition and that its value can be enhanced to ∼2.0 at 773 K by optimizing the size distribution of the nanostructures in the material. Based on our theoretical analysis on both electron and thermal transport, this zT enhancement is attributed to the reduction of both the lattice and electronic thermal conductivities; the smallest sizes (2∼6 nm) of nanostructures in the QH sample are responsible for effectively scattering the wide range of phonon wavelengths to minimize the lattice thermal conductivity to ∼0.5 W/m K. The reduced electronic thermal conductivity associated with the suppressed electrical conductivity by nanostructures also helped reduce the total thermal conductivity. In addition to the high zT of the QH sample, the mechanical hardness is higher than the other samples by a factor of around 2 due to the smaller grain sizes. Overall, this paper suggests a guideline on how to achieve high zT and mechanical strength of a thermoelectric material by controlling nano- and microstructures of the material.

A thermoelectric (TE) device is a solid-state device that converts heat directly into electricity and vice versa (1–5). As there are no moving parts involved and the device configuration is simple, TE devices have demonstrated long-term reliability in various space missions, usually running for tens of years without maintenance (6). However, they are not yet widely used in many other energy conversion applications on earth mainly due to their low conversion efficiencies. The conversion efficiency of a TE device largely depends on the material properties, i.e., the figure of merit (1, 3), zT = [S²/ρ(κ_L + κ_e)]T, where T is the absolute temperature, S is the Seebeck coefficient, ρ is the electrical resistivity, and κ_L and κ_e are, respectively, the lattice (or phonon) and electronic thermal conductivities. Increasing the zT has proven challenging because the constituent TE properties are interdependent; for example, decreasing the electrical resistivity results in decreasing the Seebeck coefficient and increasing the electronic thermal conductivity.

Among the various TE materials, PbTe is known to be a very useful TE material for waste heat recovery in the temperature range of 600–900 K (1, 4, 6–8). This material has been used widely since the onset of the Space Age in the late 1950s. According to an excellent review of 20th century TE materials by Wood et al. (7) in 1988, PbTe was the core material used in the first radioisotope TE generator launched into space, called SNAP-3G. SNAP-3G had been operational for more than 26 y without any failures since its launch in 1961 (7). Interestingly, the zT of PbTe was believed to be around 0.7 at 600∼800 K (7). However, according to a later paper by Pei et al. (9), the zT of Na-doped PbTe was found to be as high as ∼1.4 at 750 K after more precise thermal conductivity measurements at high temperatures. Recently, PbTe has been the subject of extensive research involving various new approaches for enhancing the TE properties. These efforts include band convergence via alloys (10) and electronic density of states distortion by resonant levels (11, 12) to enhance the power factor (13, 14), as well as a variety of embedded nanostructures to reduce the thermal conductivity (15–19). Heremans et al. (11, 12) showed that the power factor of PbTe can be enhanced by doping with an appropriate impurity such as Tl, which creates resonant levels inside the electronic band of PbTe. They reported a zT value of 1.5 at 773 K for 2% Tl-doped PbTe. Band convergence has been suggested as another method of increasing the power factor; Pei et al. (10) reported an enhancement of the figure of merit to 1.8 at around 850 K due to an optimized power factor via the convergence of at least 12 valleys in Pb_0.98Na_0.02Te_1-xSe_x alloys. In addition, by tuning the energy separation between the three bands (C + L + Σ) to achieve an optimal carrier concentration in 2 mol % Na-doped Mg_xPb_1-xTe alloys (20), a significant enhancement of the zT over a wide temperature range was achieved, with a peak zT value of ∼1.7 at 725 K. It is now well known that nanostructures such as nanoparticles (15–17, 21) and nanosized grain boundaries (18) can scatter phonons effectively, thereby reducing the thermal conductivity. Girard et al. (22) reported that the Na-doped PbTe–PbS 12% formed PbS nanostructures, which reduced the lattice thermal conductivity significantly and thus achieved a maximum zT of 1.8 at 800 K. Recently, a significant enhancement of the figure of merit of 2.2 at 915 K in spark-plasma-sintered (SPS) 2% Na-doped PbTe–SrTe alloys was reported by Biswas et al. (15) These authors attributed the large increase in zT to the thermal conductivity reduction achieved by “all-scale hierarchical architectures,” which cause extensive phonon scattering over a broad range of phonon wavelengths with structures of various length scales, i.e., atomic-scale impurities for scattering short-wavelength phonons, nanoscale precipitates for midwavelength phonons, and mesoscale grains for long-wavelength phonons.

The performances of these recently developed nanostructured TE materials can depend heavily on the material synthesis method and conditions used; their enhanced TE properties largely depend on the micro- and nanostructures in the materials, which can be significantly altered by small changes in the synthesis conditions. Despite recent extensive research on PbTe and its alloys, however, the effects of different synthesis methods and conditions on their TE properties have been neglected in the research thus far. In this paper, we systematically investigate three different routes of synthesizing 2% Na-doped PbTe to study their effects on the TE properties. After melting the elements, three different material synthesis routes are used: (i) quenching followed by hot-pressing (QH), (ii) annealing followed by hot-pressing (AH), and (iii) quenching and annealing followed by hot-pressing (QAH). We compare the TE properties of the samples synthesized by these three methods and investigate the variation of the nanostructures and grain sizes in the samples using scanning electron microscopy (SEM) and transmission electron microscopy (TEM). It was found that the zT strongly depends on the synthesis condition and that its value can be enhanced from 1.5∼1.7 (AH, QAH) to 2.0 or higher (QH) at 773 K. Based on a theoretical analysis of both electron and thermal transport, we attribute this zT enhancement to the reduction of both the lattice and electronic thermal conductivities; we found that the sizes of the nanoscale precipitates and microscale grains vary significantly among the samples when different synthesis methods are used. Our theoretical analysis reveals that this clearly affects the TE properties of the samples. In particular, QH showed the smallest sizes of both the precipitates and the grains, thus exhibiting a zT of 2.0 at 773 K. We also tested the hardness of the samples, which is important for mechanical stability in practical applications. The hardness of the QH samples was the highest among all samples and was nearly double that of the QAH samples.

Results and Discussion

Based on the powder X-ray diffraction (XRD) pattern for the three Pb_0.98Na_0.02Te samples synthesized by the three different processes of QH, AH, and QAH (shown in Fig. S1), we confirmed that the XRD patterns of the three samples are fairly similar to one another in the measured angle range. All of the typical peaks of the three samples can be indexed to the face-centered-cubic rock-salt structure. We obtained SEM images to determine the morphological differences among the three different samples (Fig. 1 A–F). All of the figures are cross-sectional images. In the low-magnification images (Fig. 1 A–C), it is clearly observed that the grain sizes are homogeneous in the AH sample, with an average grain size of ∼100 μm (Fig. 1A). On the other hand, in the QH sample, although the grain sizes are generally homogeneous, their average size (around 5 μm) is much smaller than that of the AH samples (Fig. 1C). Interestingly, these two different length scales of grains coexist in the QAH sample (Fig. 1B). This can be observed more clearly in the high-magnification SEM images shown in Fig. 1 D–F. The grain boundaries are clearly shown in the figures, and the grains are closely packed with almost no voids between them. In general, the AH grain sizes are fairly large, such that whole grains cannot be captured in the low-magnification SEM image (Fig. 1D), although few smaller grains are observed in the grain boundaries. Smaller grains in the QAH (Fig. 1E) and grains in the QH (Fig. 1F) samples are clearly observable in the images. This result may be attributed to the insufficient time for the grain to grow larger in the QH sample due to the quenching process, whereas sufficient time was given, along with sufficient atomic kinetics, during the annealing process for small grains to grow into much larger grains in the AH and QAH samples. Therefore, small grains ∼5 μm in size on average could be formed by the quenching process, whereas larger grains ∼100 μm in size are grown during the annealing process. TEM images of the samples are shown in Fig. 1 G–L. The TEM sample was prepared by a focused ion beam, and the thickness variation should therefore be negligible. In the high-magnification TEM images (Fig. 1 G–I), nanostructures with different sizes depending on the synthesis method, i.e., QH, QAH, or AH, are clearly shown. This is consistent with recent reports by He et al. (23, 24) in which Na precipitates in 2% Na-doped PbTe were observed, as the sodium content exceeded the solubility limit (0.5 mol %) of the Na in the PbTe. The 0.5% sodium content, below the solubility limit, introduces point defects (solid solution formation) in PbTe, and the remainder would precipitate as nanostructures. This solubility limit of 0.5 mol % was later confirmed by Yamini et al. (25) in their study on the temperature-dependent solubility of Na in the PbTe. They also showed that the additional Na in PbTe beyond the solubility limit induces the formation of a Na-rich liquid phase at temperatures above 633 K at grain boundaries. Magnified images of the nanostructures are provided in Fig. 1 J–L. Although these nanostructures are coherent, the shapes are quite different; whereas a spherical shape with a diameter of a few nanometers is observed for QH; irregular shapes are shown in the QAH and AH results. The spherical nanostructures may have agglomerated during the annealing process. A similar finding was reported by Lensch-Falk et al. (26) in their study on Ag₂Te precipitates in a PbTe matrix; they found that Ag₂Te precipitates form as coherent spherical nanoparticles in the quenched sample, yet these evolve into flattened semicoherent disks in the annealed samples. The sizes of these nanostructures are quantified and shown in Fig. 1 M–O. The mean size of the AH sample is the largest, and that of the QH sample is the smallest. The mean size of QAH sample is between these values, but it is more toward the AH size. Also, the distribution of the sizes in the AH set is the largest. As expected, quenching is a very effective means of forming smaller nanostructures, whereas annealing makes them grow into larger structures.

Fig. 1.

SEM and TEM images of the Pb_0.98Na_0.02Te samples. Low- (A–C) and high- (D–F) magnification SEM images of the QH, AH, and QAH (27) samples showing microsized grains. Nanostructures of different sizes are observed in the low- (G–I) and high- (J–L) magnification TEM images. The sizes of these nanostructures are quantified and presented in M–O.

The temperature-dependent figures of merit for the QH, AH, and QAH samples are presented in Fig. 2. In addition, schematics of their nano- and microstructures are shown. The zT value for the QH sample at each temperature is the average value of 13 measurements. Error bars indicate deviations in the zT measurements. It was predicted in a previous report (10) that an optimized zT in heavily doped p-type PbTe could be as high as 1.7 at 750 K based on a model that takes multiple bands into account. However, the zT of our QH sample is even higher than that, reaching ∼2.0 at 773 K. On the other hand, the AH and QAH samples had zT values of ∼1.5 and 1.6, respectively, at 773 K, which is lower than that of the QH sample at the same temperature. Our theoretical analysis, explained later in this paper, revealed the following: At high temperature, the power factor of QH sample is around 4% higher than that of QAH sample, yet the thermal conductivity of QH is around 14% lower than that of the QAH. Also, even though the electrical resistivity of QH sample is lower than that of QAH, i.e., higher electronic thermal conductivity of QH over that of QAH, the total thermal conductivity of QH is lower than that of QAH. This clearly indicates the lattice thermal conductivity of QH is lower than that of QAH. The theoretical analysis reveals that the smaller sizes of nanostructures in QH samples led to reduced interparticle spacing in nanostructures which effectively scattered phonons. For comparison, we also plotted in Fig. 2 previously reported zT values of PbTe:Na (9, 15). Pei et al. (9) synthesized PbTe:Na by melting–quenching–annealing followed by a hot-pressing process similar to our QAH process. The zT value for this reference material is ∼1.4 at a temperature of ∼750 K, which is slightly lower than the zT of our QAH. This may have arisen because our QAH sample was synthesized based on an optimized hot-pressing condition (27). In addition, Biswas et al. (15) synthesized PbTe:Na via melting–quenching followed by an SPS process and reported a zT close to 1.4 at 800 K. We suspect that a different synthesis process might have caused this deviation. Furthermore, it is known that an alkali-doped (2P) material has poorer mechanical properties (28). However, in our mechanical hardness analysis, shown in Fig. 2 (Inset), the hardness of the QH sample is nearly double that of the QAH sample. The mechanical hardness was measured by a Zwick µ-S Vickers hardness tester. These values are based on the average of five measurements each. It is known (29) that the hardness increases as the grain size becomes smaller, given that a smaller grain restricts slip and dislocation (30).

Fig. 2.

TE figures of merits for the QH, AH, and QAH samples with other reference data (9, 15). The zT values of the QH sample are based on the average values of 13 measurements. The error bar indicates the highest and the lowest zT values in the measurements. The schematics of morphologies of the sample are also shown. (Inset) Normalized mechanical hardness of the QH, AH, and QAH samples.

The electrical resistivities, Seebeck coefficients, and power factors of the three samples are presented as a function of the temperature in Fig. 3. Along with the experimental data, the theoretical calculations for bulk p-type PbTe are plotted in the figure. Details regarding the theoretical transport modeling can be found in SI Text. The electrical resistivities for all three samples increase with an increase in the temperature, as similarly occurs in metals. The difference as regards metals is that the hole concentration increases with the temperature in all of the samples, as evidenced by the Hall effect measurement results shown in Fig. S2. According to earlier work (ref. 15), the sodium atoms trapped at the grain boundaries cannot donate carriers to the matrix. However, these sodium atoms diffuse into the matrix as the temperature rises, becoming electrically active to provide more carriers at a higher temperature. The AH sample had a lower hole concentration (3.5 × 10¹⁹ cm⁻³) compared with the QAH and QH samples (5.0 × 10¹⁹ cm⁻³ and 6.5 × 10¹⁹ cm⁻³_, respectively) at room temperature, but it increased more rapidly with the temperature than the others, particularly at high temperatures above 600 K, such that the AH sample showed a slight decrease in its resistivity when the temperature exceeded 650 K despite the fact that the mobility decreases with the temperature due to the increased acoustic phonon scattering. This is consistent with the theoretical calculations shown in Fig. 3A. The higher experimental resistivity compared with the calculation for all of the samples is believed to be due to additional carrier scattering by the grain boundaries and nanostructures in the material, although this was not included in the modeling.

Fig. 3.

TE properties of the QH, AH, and QAH samples: (A) electrical resistivities; (B) Seebeck coefficients; (C) the total electronic and bipolar thermal conductivities; and (D) the lattice thermal conductivities. The dots indicate the experimental data, and the line denotes the theoretical analysis data.

The Seebeck coefficient also increases with an increase in the temperature for all of the samples, but it becomes saturated at 250∼300 μV/K above 650 K. Typically, the Seebeck coefficient tends to increase steadily with the temperature at a constant carrier concentration. However, in these samples, the hole concentration rapidly increases with the temperature at high temperatures above 650 K, which prevents the Seebeck coefficient from increasing further. The QH sample had the highest hole concentration (∼1.0 × 10²⁰ cm⁻³ at 773 K) among the three samples over the entire temperature range, although it is not much different from that of the QAH sample. Therefore, the QH sample has the lowest Seebeck coefficient and lowest resistivity among them. The reason for the variation in hole concentration among the samples is not yet clearly understood, but it may be related to the densities of the defects and/or electrically active impurities such as Na in the matrix.

Fig. 3C shows the measured total thermal conductivity (κ) along with the calculated electronic and bipolar thermal conductivities for the three samples. The electronic thermal conductivity (κ_e) includes the bipolar thermal conductivity (κ_bi), and it is used to extract the lattice thermal conductivity via κ_l = κ−κ_e. More details about the bipolar thermal conductivity calculation are found in SI Text. The electronic thermal conductivity before the bipolar term is included is calculated by the Wiedemann–Franz law as κ_e−κ_bi = LT/ρ, where L is the Lorenz number. The Lorenz number is calculated from the multiband Boltzmann transport model described in SI Text. As the Lorenz number remains almost constant for all three samples at a given temperature, as shown in Fig. S3A, the electronic thermal conductivity without the bipolar term is largely determined by the resistivity. As shown in Fig. 3C, bipolar thermal conductivity starts to appear beyond 600 K in all three samples, becoming as large as 0.27 W/m K at 773 K for the AH sample. The QH and QAH samples have bipolar thermal conductivity of ∼0.15 W/m K at the same temperature, which is lower than that of the AH sample due to their relatively higher hole concentrations compared with the AH sample, which lowers the contribution from the electrons to the bipolar thermal conductivity. As a result, the AH sample has higher electronic thermal conductivity (∼0.46 W/m K at 773 K) than the QH (0.4 W/m K) and QAH (0.36 W/m K) samples, although it has the highest resistivity among the samples. The large bipolar thermal conductivity of the AH sample is responsible for the slight increase in the measured total thermal conductivity at high temperatures beyond 750 K for the AH sample, as shown in Fig. 3C.

The lattice thermal conductivity κ_L is then obtained by subtracting the electronic thermal conductivity from the total thermal conductivity, as shown in Fig. 3D. To understand why the QH exhibited the lowest thermal conductivities among the samples, we used the Callaway model (31). A detailed analysis is available in SI Text, and the results are presented as solid curves in Fig. 3D. We reflected in the simulation what we observed in the SEM and TEM images (Fig. 1). First, we set the size of the grain to be 100, 8, and 3.7 μm for the AH, QAH, and QH samples, respectively, based on the SEM images in Fig. 1. Also, based on findings by He et al. (23), the solubility limit of Na in PbTe is 0.5%. The remaining 1.5% should be precipitated, although the exact amount of this should be varied among samples due to the different synthesis condition, measurement history, sample variation, etc. Based on the SEM and TEM images, we assumed that the remaining 1.5% was precipitated in the form of Na₂Te, with a diameter of around 10, 7, and 2.4 nm for the AH, QAH, and QH samples, respectively. However, the solubility limit of Na atoms in PbTe can increase as temperature increases (25), which can be the reason that the carrier concentration increases with temperature due to the further diffusion of Na atoms into the matrix at higher temperatures. This may affect the size distribution of the nanoprecipitates at high temperatures, which is subject to future study. As shown in Fig. 3D, incorporation of these values indeed explains why the QH sample has the lowest thermal conductivity. In the simulation, lattice thermal conductivities of QAH and AH are similar despite the fact that they have different grain sizes and diameters of their nanostructures. It is known that phonon grain boundary scattering is usually dominant at temperatures lower than the Debye temperature (32) unless the sizes of the grains are very small, e.g., on the order of a few tens of nanometers (16). Considering that the Debye temperature of PbTe (33) is 136 K, the differences in the grains did not affect the lattice thermal conductivities significantly in the measured temperature range. Thus, scattering due to the nanostructures indeed causes this discrepancy. In fact, given that we fixed the concentration of the nanostructures at 1.5%, the differences in the diameters of the nanostructures, which effectively scatter mid- to long-wavelength phonons (17), affected the lattice thermal conductivities. It is known (34) that, given the same nanostructure concentration, the mean size of the nanostructure is directly related to the interparticle spacing assuming homogeneous distribution of nanostructures; the smaller the size of the nanostructure, the closer the interparticle spacing. The spacings between the nanostructures could be estimated as 22.7, 15.8, and 5.4 nm for the AH, QAH, and QH samples, respectively. This suggests that the phonon mean-free path of scattering processes other than scattering due to the nanostructures should be shorter than 15.8 nm because the lattice thermal conductivities of QAH and AH do not appreciably depend on the sizes of the nanostructures. Lattice thermal conductivities of the QH, AH, and QAH samples with and without nanoparticles (NP) were calculated as shown in Fig. S4. It is clear in the figure that effects of NPs on the lattice thermal conductivities are evident in the QH but are negligible in the others. This clearly suggests that, given the same nanostructure concentration, the sizes of the nanostructures are closely related to the interparticle spacing which affects the phonon mean-free path (35).

Conclusions

We demonstrated a zT of ∼2.0 at 773 K in Na-doped PbTe in one of the synthesis methods, (QH), used in this paper. Based on our theoretical analysis, the main reason for such a high zT in QH is the reduction of the thermal conductivity. Interestingly, although the microsized grains and nanostructures occur in all three samples, QH, AH, and QAH, nanostructures only a few nanometers in diameter present in QH effectively reduced the lattice thermal conductivity. Based on this finding, we should emphasize that nanostructures are not always effective in reducing the lattice thermal conductivity. They are the most effective when the spacing between them is smaller than the phonon mean-free paths of other scattering processes. Therefore, optimal synthesis condition should be sought to find the right sizes of nanostructures for this material. Our study shows that the size of nanostructures can vary significantly by the choice of synthesis methods and parameters, which can explain the variation in zTs among the samples from different synthesis methods. In addition to the high zT, the QH sample possesses mechanical hardness superior to that of the other samples by a factor of 2. This is due to the relatively small grain sizes compared with the others (29, 30). Overall, this QH sample demonstrates great potential for vehicle waste heat recovery not only due to its high TE figure of merit but also due to its high mechanical hardness for practical use.

Materials and Methods

Materials.

Pb_0.98Na_0.02Te was synthesized by the three different synthesis processes of QH, AH, and QAH. Elemental lead, tellurium, and sodium were used as the starting materials. They were weighted and mixed in a carbon-coated quartz tube under a N₂-filled glove box. The tube was then evacuated and sealed. The elements were heated at 1,073 K for 2 h followed by melting at 1,273 K for 6 h. The ingots were then prepared by the following three different processes: (i) The QH sample was obtained by quenching in cold water followed by hot-pressing. (ii) For the AH sample, after the melting process, the sample was annealed at 973 K for 2 d in a furnace. After this process, the sample was cooled naturally to room temperature in the furnace, after which it was hot-pressed. (iii) For the QAH sample, the melted sample was quenched into cold water. The ingot formed in this way was placed into another carbon-coated quartz tube and was annealed at 973 K for 2 d. The sample was then cooled to room temperature. The QAH sample was obtained after hot-pressing this ingot. In all three processes, the resulting ingot was then ground into powder and subsequently hot-pressed in a uniaxial hot-pressing machine at 100 MPa at 773 K for 1 h, which was found to be the optimal condition for hot-pressing in our previous work (27). Electrical heaters supplied heat in the hot-pressing machine. Details regarding material synthesis can be found in Supporting Information.

Thermoelectric Properties.

The electrical resistivity and Seebeck coefficient were measured simultaneously over the temperature range between 300 and 800 K using a commercial ULVAC ZEM-3 system. The thermal diffusivity was measured using a laser flash apparatus (Netzsch LFA 457; see Fig. S5). The heat capacity was estimated by the relationship of C_p (k_B per atom) = 3.07+4.7 × 10⁻⁴ × (T/K − 300), which was obtained by fitting the data to the experimental data reported by Blachnik and Igel (36). The thermal conductivity (κ) was extracted from the thermal diffusivity (λ), the specific heat capacity (C_p), and the density (d) based on the relationship κ = λC_pd. Here, d was measured by the Archimedes method. The figure of merit was then obtained from the transport property measurements above. We only measured samples with a relative density (see Table S1). Details of other characterization tools used here, such as the mechanical hardness, XRD, and Hall measurements, the theoretical transport modeling, etc. are available in SI Text. Specially, sensitivity analysis on lattice thermal conductivity regarding the variation of sizes of nanoparticles and microsized grains is provided in Fig. S6, and the repeated thermoelectric properties of the QH samples are shown in Fig. S7. Additionally, Fig. S8 shows the thermoelectric properties, measured a few months after the synthesis, for QH while heating and cooling the sample.