Thermoelectric properties of Spark plasma sintered PbTe synthesized without any surfactant and organic solvent

Aqueous chemical route: Approximately 5 g NaOH powder was dissolved in double-distilled water at 323 K. When NaOH is completely dissolved, 2 g NaBH₄ and 0.01 mol Te metal powder were added to the aqueous NaOH solution with constant magnetic stirring. After the change in the color of the solution to dark purple, 0.01 mol lead acetate aqueous solution was added to the beaker drop-by-drop. Once the reaction is completed, the beaker was allowed to cool to room temperature, and the obtained black powder was filtered out and washed with distilled water, acetone, and ethanol. At last, the powder sample was dried in air ambient for 3 h at 323K.

Hydrothermal synthesis route: The stoichiometric amount of lead acetate and tellurium powder were added to a beaker containing an aqueous solution of NaBH₄ with constant magnetic stirring. After some time, 20 ml aqueous NaOH solution was added to the beaker, and the resulting solution was transferred to a Teflon-lined stainless-steel autoclave. Finally, the autoclave was filled to 80% of the total capacity with distilled water and maintained at 433 K for 14 h in a furnace. After the reaction was over, the autoclave was cooled to room temperatures. The obtained black powder was filtered out and washed with distilled water, acetone, and ethanol. Lastly, the powder sample was dried in air ambient for 3 h at 323 K, as in the former route.

The as-synthesized PbTe nanopowders were loaded in graphite die and consolidated using spark plasma sintering (SPS) using Syntex, Japan make furnace at a temperature of 773 K for 8 min under a pressure of 20 MPa.

A schematic of the synthesis process is shown in figure 1.

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The XRD measurements on as-synthesized Lead telluride samples were performed by employing a diffractometer (Philips Xpert Pro) using Cu-K_α radiation from 20° to 80° with a step size of 0.02°. Microstructure of both the as-synthesized samples and the SPS'ed (Spark Plasma Sintered) PbTe samples have been investigated by FESEM (field emission scanning electron microscope) (Jeol JSM-7800F) and (EDS) energy dispersive patterns. The morphology of nanostructured PbTe samples was determined by using TEM (Transmission electron microscopy) and HR-TEM (high-resolution transmission electron microscopy (JEM 2100F JEOL).

The thermoelectric properties of the SPS'ed lead telluride samples have been investigated from room temperature to 700 K. The thermal conductivity of the bulk nanostructured samples was calculated using the expression $\kappa =\rho {C}_{p}D,$ where ρ , C_p and D are the density, specific heat and thermal diffusivity of the sample, respectively. The density of the samples was measured using Archimedes method, and the thermal diffusivity (D) was determined by employing Laser Flash Apparatus ((LFA 1000), Linseis Messgeraete GmbH, Germany NPL). Specific heat capacity value has been adapted from data reported by Quin B et al, (2019) [19]. After the measurement of thermal diffusivity on the samples, rectangular pellets were cut out from the SPS'ed discs for the simultaneous measurements of Seebeck coefficient and electrical conductivity by employing ULVAC ZEM 3 in a Helium atmosphere.

The XRD diffractograms of the as-synthesized lead telluride samples from both the routes have been demonstrated in figure 1, along with the Rietveld refinement profiles.

As can be observed from figure 2, all the prominent diffraction peaks can be indexed to the FCC PbTe structure with space group Fm-3m (225) JCPDS #78-1905. There are no diffraction peaks from any impurity in the XRD diffractogram (See figure 2(b)), confirming that single phase PbTe has been successfully synthesized by using a simple hydrothermal reaction The intensity of the XRD peaks observed in the lead telluride sample synthesized by the hydrothermal method is relatively higher in contrast to the sample synthesized by aqueous chemical route (See figure 2). It indicates that the sample synthesized by the hydrothermal route is more crystalline in comparison to the sample synthesized by the aqueous chemical route. Further, the sharp peaks indicate that the crystallite size of the lead telluride sample synthesized by hydrothermal method is larger in comparison to the sample synthesized by aqueous chemical route which is also consistent with the HR- TEM observations (See figure 3).On the contrary, the sample synthesized by the aqueous chemical precipitation method results in PbTe product having relatively small-sized nanoparticles (broad peaks) with a few additional peaks corresponding to traces of unreacted Te in the sample (See figure 2(a)). Further, two peaks of very low intensity (marked by *) couldn't be identified as the Braggs position due to both the phase lies very close and hence peaks possibly. The calculated lattice constant using Rietveld refinement for the sample synthesized by hydrothermal route is 0.64556 nm and that synthesized using aqueous chemical route is 0.64590 nm; both values are in close agreement with the literature value (0.64540 nm) [20].

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The value of various parameters obtained by the Rietveld refinement has been shown in table 1.

We have calculated the crystallite size for the (200) peak using the Debye–Scherrer equation,

$$\begin{eqnarray}&&{D}_{XRD}=\displaystyle \frac{K\lambda }{\beta Cos\theta }\end{eqnarray} \tag{ 2 }$$

Where K, λ, β, and θ are Scherrer constant, wavelength of copper K_α radiation, FWHM (full width of the peak at half maximum) and Bragg's angle, respectively. The value of Scherrer constant lies between 0.89 to 0.94. The (200)-diffraction peak yields a crystallite size of 23.5 nm for the sample synthesized by the aqueous chemical route and 29.0 nm for the sample synthesized by the hydrothermal route. It is to be noted that crystallite size is not necessarily the same as that of particle size. Further, the broadening of the XRD peak can also be due to the internal stress and defects induced during the synthesis process. Hence, the crystallite size as estimated using Scherrer equation is expected to be smaller than the actual value [20]. The calculated crystallite size for both the synthesized samples is far lesser than Bohr's excitonic radius of lead telluride (∼152 nm) [21]. It is noteworthy that the Seebeck coefficient could be remarkably enhanced in the nanostructured systems due to carrier filtering and quantum confinement effects [22].

Figure 3(a) shows the TEM image of as-synthesized lead telluride nanoparticles synthesized by aqueous chemical route. It can be observed from the figure that the nanoparticles exhibit mixed morphology with particle size < 50 nm (See figure 3(a). The HR-TEM image of the lead telluride sample synthesized by aqueous chemical route has been depicted in figure 3(b). The clear lattice fringes as evident in HR-TEM image have the interplanar spacing of 0.323 nm and 0.228 nm, which corresponds to (200) and (220) plane of lead telluride. Figure 3(c) presents the TEM image of the lead telluride sample synthesized by hydrothermal route. It is evident from the figure that the as-synthesized nanoparticles have cubic morphology with edge length ∼50 nm. The inset in figure 3(c) shows the size distribution histogram for PbTe nanoparticles synthesized using the hydrothermal route. Figure 3(d) presents the HR-TEM image of the single particle of lead telluride sample synthesized by hydrothermal route. The lattice fringes observed in figure 3(d) have an interplanar spacing of 0.323 nm which corresponds to (200) plane of lead telluride crystal.

The clear lattice fringes observed in HR-TEM images (See figure 3(b) and d)), indicates that lead telluride nanoparticles synthesized from both the routes are perfectly crystalline. The FESEM-EDS images of SPS'ed PbTe samples are shown in figures 4(a)–(c) for aqueous chemical route and figures 4(d)–(f) for hydrothermal route. It was revealed from the elemental mapping images (b), (c), and (e), (f) that both the elements (Pb, Te) are uniformly distributed in the whole matrix, confirming that homogeneous composition has been achieved in both the samples.

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The FESEM images of the sintered pellets are shown in figure 5.

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The images reveal the porous nature of the samples synthesized by the aqueous chemical route, while the sample synthesized by the hydrothermal route appears relatively compact and distinctly dense. The existence of pores in the former is also consistent with the measurement of density in them. The density of the sample synthesized by the aqueous chemical route is close to 78%, and that of the sample synthesized by the hydrothermal route is 88% of the theoretical density. The relatively smaller density in our SPS'ed samples in comparison to the samples synthesized by the top-down approach (ball milling) [23] confirms the presence of porosity in the SPS'ed samples. In order to achieve higher densities, remarkably excessive sintering temperatures and prolonged durations are required, which may result in inevitable grain growth. The relationship between thermal conductivity and volume of pores inside a material is given by,

$$\begin{eqnarray*}&&\kappa ={\kappa }_{0}\left(1-{P}^{2/3}\right)\end{eqnarray*}$$

Where, κ is the coefficient of thermal conductivity with porosity, ${\kappa }_{0}$ is the coefficient of thermal conductivity with 100% total density, and P is the porosity. As per the equation, an increase in porosity has a detrimental effect on the thermal conductivity due to additional scattering of phonons across the pores in the porous material [24, 25].

It was reported that a nearly 100% enhancement in zT could be achieved by porous architecture engineering [26]. However, there will always be a tradeoff between the beneficial effect of reduction in thermal conductivity and the detrimental impact on the electrical conductivity.

In this section, the impact of microstructure on the transport properties of synthesized nanostructured lead telluride samples will be addressed. The temperature dependent Seebeck coefficient and electrical conductivity are presented in figures 6(a) and (c), respectively. Furthermore, the thermoelectric properties of raw ingot of the PbTe and also of the PbTe-nanowires synthesized by another chemical route retrieved from [27] and [28], respectively, will be used for the sake of comparison with the results of the present study.

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The SPS'ed discs of PbTe synthesized from both the aqueous chemical (aq) and hydrothermal (Hy) routes exhibited a p-type Seebeck coefficient from RT to ∼500 K as shown in figure 6(a), suggesting that the matrix is Te rich in both the samples. A conversion from p-type to n-type has been observed at ∼550 K for sample A (aq) and at ∼ 500 K for sample B (Hy). Such a behavior is also consistent with the PbTe raw ingot synthesized by conventional melting and quenching or chemical route [27, 28]. The total Seebeck coefficient of a semiconductor with bipolar conduction involving two different types charge carriers is given by,

$$\begin{eqnarray}&&S=\displaystyle \frac{{S}_{e}{\sigma }_{e}+{S}_{h}{\sigma }_{h}}{{\sigma }_{e}+{\sigma }_{h}}\end{eqnarray} \tag{ 3 }$$

Where S_e , and S_h are the Seebeck coefficients of the two types of carriers, and σ _e and σ _h are the contributions of the two charge carriers to electrical conductivity. At temperatures below 500 K, the second term in the numerator of equation (2), i.e., (S_h σ _h) dominates over the first term leading to a positive Seebeck coefficient in this temperature range. As PbTe is a narrow band gap semiconductor, the number of charge carriers in the conduction band increases progressively with the increase in temperature due to thermal excitation leading to an increase in S_e and the first term of equation (1). Consequently, the overall Seebeck coefficient decreases at T > 500 K. The dominance of intrinsic charge carriers at high temperatures is also consistent with the enhanced electrical conductivity at temperatures > 500 K, as evidenced in the lead telluride sample synthesized by hydrothermal method (See figure 6(c). The Seebeck coefficient exhibits a maximum value of 426 μV K⁻¹ at 375 K (Sample B) and 420 μV K⁻¹ at 425 K (Sample A). The values are comparable to the reported values in PbTe synthesized by chemical route (i.e., 322 μV K⁻¹ at ∼ 373 K [28]). It is to be noted that we obtained a significantly enhanced Seebeck coefficient in both types of samples (i.e., Aq & Hy types) compared to the bulk ingot 243 μV K⁻¹ at ∼ 320 K.

Figure 6(c) presents the temperature dependent electrical conductivity of both the samples in comparison with the literature. One can observe the temperature-activated electrical conductivity in both samples which is more intense in the lead telluride sample synthesized hydrothermal method. It was reported by Nolas et al, (2009) that the electrical transport properties of PbTe synthesized by wet chemical route depends upon the surface oxygen adsorption, stoichiometry, and density [29]. The density of our synthesized samples has been measured using the Archimedes method. We hypothesize that the lower density of sample synthesized by the aqueous chemical route is responsible for lower electrical conductivity relative to the sample synthesized by the hydrothermal route. The electrical conductivity of PbTe synthesized by hydrothermal route can be further improved by optimizing the sintering conditions (temperature, time, and pressure). Similar electrical behavior is observed in the nanostructured lead telluride samples synthesized from other wet chemical routes [29, 30]. Such temperature dependence of electrical conductivity can be ascribed to the charge carriers scattering from the potential barriers at crystal interfaces and grain boundaries. The charge carriers with low energies are trapped at grain boundaries, and those having sufficient energy are allowed to pass. The trapping of low-energy charge carriers by grain boundaries is also favorable for improving the Seebeck coefficient. The measurement of room temperature Seebeck coefficient is also in accordance with the hypothesis. The room temperature Seebeck coefficient of sample synthesized by aqueous chemical route is 347 μV K⁻¹, and that of sample synthesized by hydrothermal route is 380 μV K⁻¹, which are significantly enhanced as compared to Seebeck coefficient of bulk PbTe ingot 243 μV K⁻¹ at 320 K [22]. The remarkably enhanced Seebeck coefficient values observed in the present case could be a signature of the carrier filtering effect in our nanostructured samples [31]. The height of the potential energy barrier at crystal interfaces/grain boundaries can be estimated according to the following equation,

$$\begin{eqnarray}&&{\sigma }_{Eff}={T}^{-1/2}Exp\,\left(-\displaystyle \frac{{E}_{B}}{kT}\right)\end{eqnarray} \tag{ 4 }$$

Where σ _eff is the effective electrical conductivity, k is the Boltzmann constant, T is the absolute temperature, and E_B is the height of the grain boundary potential energy barrier. In order to justify this assumption, we have plotted ln( σ T^1/2) versus 1/kT in figure 6(b). The linear fit of the data sets confirms the applicability of the carrier filtering model. The height of the potential energy barrier comes out to be ∼ 210 meV in the sample synthesized by the aqueous chemical route and ∼201 meV in the sample synthesized by the hydrothermal route. The calculated height of the potential energy barrier is relatively large in contrast to the energy barrier reported by Scheele et al, E_B  = 140 meV [32]. These higher values of potential energy barrier are also consistent with the lower electrical conductivity observed in our synthesized samples.

Figure 6(d) presents the temperature-dependent power factor (=S² σ of the synthesized samples. It can be observed from the figure that the power factor of synthesized samples is significantly lower than the bulk ingot. The power factor of the sample synthesized by hydrothermal route reaches a value of 201 μW m⁻¹ K⁻² at a temperature ∼ 376 K, and 188 μW m⁻¹ K⁻² at 673 K. However, the power factor of sample synthesized by aqueous chemical route remains low ∼ 60 μW m⁻¹ K⁻² at room temperature. The lower power factor in our samples may be attributed to lower values of electrical conductivity due to the scattering of the charge carriers at numerous interfaces. We propose that the decreased electrical conductivity in our samples is not fully compensated by the enhanced Seebeck coefficient. The comparative values of thermoelectric parameters S, σ and κ at three different temperatures i.e., 303 K, 473K and 673K has been shown in table 2.

Table 2. The comparative values of thermoelectric parameters S, σ and κ at three different temperatures i.e., 303 K, 473K and 673K.

	Seebeck coefficient S (μV K−1)		Electrical conductivity σ (S m−1)		Power factor (S2 σ ) (μW m−1 K−2)		Thermal conductivity κ (W m−1 K−1)		Figure of merit (zT = $\tfrac{{S}^{2}\sigma }{\kappa }\,$T)
Temp. (K)	A (Aq)	B (Hy)	A (Aq)	B (Hy)	A (Aq)	B (Hy)	A (Aq)	B (Hy)	A (Aq)	B (Hy)
303 K	346.5	379.3	501	775	60	111	0.96	1.43	0.015	0.025
473 K	352.7	141.3	231	875	23	15	0.65	0.96	0.007	0.003
673 K	−135.7	−266.9	480	2800	9	188	0.66	0.76	0.003	0.176

The principal benefit of nanostructured materials in comparison to bulk materials is the significant decrease in lattice thermal conductivity. The temperature dependent total and lattice thermal conductivity has been presented in figures 7(a) and (b), respectively.

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It can be readily observed from figure 7, that the thermal conductivity of samples synthesized from both the routes is remarkably reduced in comparison to the thermal conductivity of bulk ingot [27] in the entire measurement temperature range. This result demonstrate successful reduction in the thermal conductivity to ∼38% in the sample synthesized by hydrothermal route and to ∼58% in the sample synthesized by aqueous chemical route when compared to the thermal conductivity of bulk ingot. Furthermore, figure 7 reveals that the total thermal conductivity is close to the lattice thermal conductivity, and it might be attributed to the inferior electrical conductivity of our synthesized samples.

The total thermal conductivity of a material is comprised of two components, i.e., the electronic component ( κ _El ) and the lattice component ( κ _L ). Further, the electronic component of thermal conductivity is directly proportional to electrical conductivity through Wiedemann–Franz law ( κ _El = LσT), where L,σ, and T are the Lorentz number, electrical conductivity, and absolute temperature, respectively. The value of Lorentz number can be estimated from the absolute values of Seebeck coefficient according to the following equation [33],

$$\begin{eqnarray}&&L=1.5+Exp\left(-\displaystyle \frac{S}{116}\right)\end{eqnarray} \tag{ 5 }$$

In the above equation, S is in the units of μV K⁻¹, and L is in 10⁻⁸ WΩK⁻².

The value of the electronic thermal conductivity is computed from the estimated values of Lorentz number and experimentally measured electrical conductivity values (figure 8(a)). It is directly observed from figure 8(a) that the electronic thermal conductivity increases with the increase in temperature and displays the same behavior as that of electrical conductivity.

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The electrical band gap of lead telluride is ∼0.3 eV at room temperature [34, 35]. In such a narrow band gap semiconductor, a notable contribution to thermal conductivity comes from bipolar diffusion of charge carriers. This contribution is termed as bipolar thermal conductivity and is found to increase with temperature. Besides an additional contribution to thermal conductivity, such as bipolar effects also lowers the Seebeck coefficient's absolute value and hence are responsible for inferior thermoelectric performance.

The total thermal conductivity is given by,

$$\begin{eqnarray}&&{\kappa }_{Tot}={\kappa }_{Latt}+{\kappa }_{Elec}+{\kappa }_{Bipolar}\end{eqnarray} \tag{ 6 }$$

To estimate the bipolar thermal conductivity, we have used a simple method proposed by Zhao et al, [9].

The lattice thermal conductivity can be approximated as

$$\begin{eqnarray}&&{\kappa }_{Latt}=3.5{\left(\displaystyle \frac{{k}_{B}}{h}\right)}^{3}\displaystyle \frac{M{V}^{1/3}{\theta }_{D}^{3}}{{\gamma }^{2}T}\end{eqnarray} \tag{ 7 }$$

Where M, V are the average mass and average atomic volume per atom, θ_D is the Debye temperature, and γ the Gruneisen parameter [10]. At low temperatures, we can approximate the acoustic phonon scattering as the governing mechanism and hence ${\kappa }_{Tot}\,-\,{\kappa }_{Elec},$ which is ${\kappa }_{Latt}.$ The latter is found to be inversely proportional to temperature (see figures 7(b), 8(c)). As the temperature is increased, ${\kappa }_{Latt}$ begins to diverge from the linear correspondence between ${\kappa }_{Latt}$ and inverse temperature. The value of bipolar thermal conductivity can be computed by extrapolating the linear correspondence between ${\kappa }_{Latt}$ and inverse temperature as specified by the solid line marked by blue color (Sample A) and green color (Sample B). The variation of bipolar thermal conductivity with temperature is presented in figure 8(b). It is evident from the figure that the bipolar thermal conductivity becomes considerable at temperatures more than 450 K.

The energy barrier for bipolar diffusion is computed using the relationship between ${\kappa }_{Bipolar}\,$and temperature,

$$\begin{eqnarray}&&{\kappa }_{Bipolar}=AExp\left(-\displaystyle \frac{{E}_{B}}{2kT}\right)\end{eqnarray} \tag{ 8 }$$

Where E_B is the energy barrier for bipolar diffusion and A is a pre-exponential coefficient. To estimate the energy barrier for bipolar diffusion, we have plotted ln( κ _Bipolar) versus 1/2kT for the synthesized samples (figure 8(d)). The linear fitting of the datasets yields an energy barrier value of 0.73 eV for sample B, and 0.53 eV for sample A.

The temperature dependent dimensionless figure of merit has been depicted in figure 9. The figure of merit of PbTe bulk ingot has been retrieved from [27]. Although the power factor of the sample synthesized by the hydrothermal route is relatively smaller than the bulk ingot, the figure of merit is distinctly on the higher side.

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The large value of figure of merit in case of PbTe synthesized using hydrothermal route is due to significantly smaller thermal conductivity in the nanostructured sample. The figure of merit reaches to a maximum value of 0.18 at 673 K in the sample synthesized by the hydrothermal route. On the other side, the dimensionless figure of merit of the sample synthesized by the aqueous chemical route remains low in the entire measurement temperature range owing to its poor electrical properties.