Enhancement of thermoelectric properties by effective K-doping and nano precipitation in quaternary compounds of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05

Abstract

We investigated thermoelectric properties in K-doped quaternary compounds of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 (x ≤ 0.03). In terms of two valence bands model, we argue that the L-band approaches to the Σ-band with increasing the K-doping concentration resulting in the increase of carrier concentration and effective mass of carrier due to the increase of band degeneracy. The effective K-doping by x = 0.02 and PbS substitution causes high power factor and low thermal conductivity, resulting in the comparatively high ZT value of 1.72 at 800 K. The low thermal conductivity for (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05 compound is attributed from the lattice distortion and line dislocation in a phase of nano precipitation. By optimizing K-doping and PbS substitution, we achieved the enhancement of practical thermoelectric performance such as average ZT_avg = 1.08, engineering (ZT)_eng = 0.81, maximum efficiency η_max = 11.63%, and output power density P_d = 6.3 W cm⁻², with temperature difference ΔT = 500 K, which has practical applicability in waste heat power generation technologies.

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Introduction

Thermoelectric materials (TE) have been investigated for several decades due to possible applications in environmentally friendly energy harvesting technologies. The efficiency of thermoelectric materials is determined by dimensionless figure of merit, ZT = S²σT/κ, where S, T, σ, and κ are the Seebeck coefficient, absolute temperature, electrical conductivity, and thermal conductivity, respectively. Recently, several effective strategies have been successfully employed to enhance thermoelectric performance such as band engineering,¹ band distortion by resonance balance band,² nano particle dispersion and/or separation,^3–5 bulk nano composites,^6–9 and panascopic approach.^10–12

Lead tellurides (PbTe) and their alloys have a simple face-centered-cubic (FFC) rock salt structure and have been studied over several decades for thermoelectric applications at the intermediate temperature range (600–800 K).¹³ Nanostructuring by phase separation and precipitation in PbTe matrix causes the reduction of lattice thermal conductivity, resulting in the enhancement of thermoelectric performance.^14,15 In addition, high thermoelectric performance of PbTe can be achieved through the precise control of hole concentration in p-type PbTe (from 10¹⁸ to 10²⁰) by substitution of monovalent atoms such as thallium (Tl),² sodium (Na)¹⁶ on the lead sublattice. The Tl- or Na-doping in PbTe increases the density of states (DOS) by controlling the Fermi level close to the heavy hole band¹⁷ which attributes to the enhancement of the Seebeck coefficient.^18–20

Combining the nano structuring with the Fermi level tuning by Na-doping, high thermoelectric performance was revealed in Na-doped binary composites, for example, PbTe–PbS (ZT = 1.8 at 800 K) and Na-doped PbTe–PbSe (ZT = 1.8 at 850 K) as result of nano structuring with sulfur and band engineering by alloying selenide, respectively.^21,22 In order to maximize the performance, complex quaternary composites of Na-doped PbTe–PbSe–PbS were investigated. Quaternary compounds of (PbTe)_1−2x(PbSe)_x(PbS)_x (x = 0.07) exhibited ZT ≈ 2.0 at 800 K due to the valence band modification and phonon scattering from alloy scattering and point defects.²³ The Na-doped PbTe_0.9−xSe_0.1S_x compounds had a high ZT ≈ 1.6 near 750 K.²⁴

Theoretical investigations suggested that power factor of PbTe based compounds can be enhanced by resonant like DOS distortions in p-type Pb_1−xA_xTe (A = K, Rb, and Cs but not Na).²⁵ However, the other theoretical study showed that K- and Tl-doping do not form the resonance state but can control the energy difference of the maxima of two primary valance sub-bands in PbTe.^17,26 The K-doped PbTe compounds showed relatively high ZT values reaching 1.3 at 673 K (ref. 27) which is comparable with those of Na-doped PbTe at the same temperature range.¹⁸ Furthermore, a high thermoelectric performance was obtained in binary compounds of K-doped PbTe_1−xSe_x (ZT ≈ 1.6 at 773 K) due to the increase of DOS near the Fermi level, resulting in the enhancement of Seebeck coefficient for band convergence of two valance bands in PbTe_1−xSe_x.²⁷

From the valance band converge of PbTe–PbSe^22,27 and the nano precipitation in PbTe–PbS,²¹ here we examine the effect of K-doping in quaternary system of PbTe–PbSe–PbS composites in order to maximize ZT value by combining of the band converge and nano precipitation. We found that the K-doping enhances the Seebeck coefficient via heavy doping, which increases the DOS near the Fermi level. Combined effect of low lattice thermal conductivity due to nanostructure and dislocation with high power factor, we obtained a high ZT ≈ 1.73 at 750 K for (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05, which is an 32% improved value comparing with the one of K-doped PbTe and K-doped PbTe_1−xSe_x compounds.²⁷ In addition, we analysed the thermoelectric performance in terms of newly established parameters, such as the engineering ZT_eng, device efficiency η, and so on.

Experimental section

Sample synthesis

We firstly synthesized the precursor compounds of PbSe and PbS by melting and hot press sintering. The high purity elements of Pb (99.999%), dried S (99.999%), and Se (99.999%) were sealed in evacuated quart ampoules at residual pressure of 10⁻⁴ Torr and melted at 1150 °C to produce high purity starting materials of PbSe and PbS. The polycrystalline samples of K-doped Pb_1−xK_xTe_0.7Se_0.25S_0.05, which will be denoted hereafter as (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 (x = 0.01, 0.015, 0.02, 0.025, and 0.03), were synthesized by melting and hot press sintering using the following process.

We loaded the synthesized precursor powders of PbSe and PbS and the elements Pb (99.999%), Te (99.999%), and K (99.99%) in carbon-coated fused silica tubes and sealed them under vacuum at 10⁻⁴ Torr. The elements and powders sealed in the quartz ampoules were heated to 1100 °C, kept at the temperature for 10 hours, and were then quenched in ice cold water. The ice water quenched ingots were annealed at 500 °C for 72 hours. The ingots were hand ground and hot pressed in a graphite mold having a 12.7 mm diameter using a hot press sintering machine under a uniaxial pressure of 40 Mpa for 1 hour at 500 °C under vacuum environment.

Characterization and measurement of thermoelectric properties

Phase identification and lattice parameters were obtained by powder X-ray diffraction (XRD) measurements with Cu Kα radiation (D8 Advance, Bruker, Germany).

The Seebeck coefficient S and electrical conductivity σ were measured by the four-point probe method using a thermoelectric measurement system (ZEM-3 ULVAC, Japan). The isothermal Hall resistivity ρ_xy was measured by five-point contact method with sweeping magnetic fields from −5 T to 5 T using a physical property measurement system (PPMS Dynacool-14T, Quantum Design, USA). The Hall coefficient R_H, Hall carrier density n_H, and Hall mobility μ_H were obtained using the relations of R_H = ρ_xy/H, n_H = −1/(R_He), and μ_H = 1/(ρn_He), respectively.

Thermal conductivity κ was obtained by the relation κ = ρ_sλC_p where ρ_s, λ, and C_p are sample density, thermal diffusivity, and specific heat, respectively. The ρ_s, λ, and C_p were measured by Archimedes method, laser flash method (LFA-447, NETZSCH, Germany), and high temperature fitting of the Dulong–Petit law by measuring specific heat using a physical property measurement system (PPMS Dynacool-14T, Quantum Design, USA), respectively. The combined uncertainty for all measurements involved in ZT determination was around 20%.

TEM were prepared by Nova 600 nanolab Focused Ion Beam (FEI) using 5–30 kV. TEM images (High Resolution images/STEM/ED pattern) were collected using a JEOL 2100F at 200 kV. Energy dispersive X-ray spectrometer (EDS) analysis were obtain using Oxford Instruments (INCA platform) detector equipped on JEOM 2100F.

Results and discussion

The powder XRD diffractions of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 (x = 0.01, 0.015, 0.02, 0.025, and 0.03) are presented in Fig. 1 which shows the single phases of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 for all K-doping in the compounds. All peaks of XRD diffraction pattern represent to the cubic structure. The lattice parameter is almost not changed with respect to K-doping as show in Table 1 indicating that small amount of K-doping does not change the crystal structure of matrix (PbTe)_0.70(PbSe)_0.25(PbS)_0.05.

Fig. 1 Powder X-ray diffraction of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 compounds (x = 0.010, 0.015, 0.020, 0.025, and 0.030).

Table 1 Relative atomic concentrations of the Pb_1−xK_xTe_0.7Se_0.25S_0.05 compounds from the XPS measurements

x	Pb (at%)	K (at%)	Pb	K
0.01	42.17	2.69	0.94	0.06
0.015	47.48	7.58	0.86	0.14
0.02	47.81	6.83	0.87	0.13
0.025	38.2	2.39	0.94	0.06
0.03	49.5	9.31	0.84	0.16

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In order to identify the K-doping concentration in the compounds, we measured X-ray photoelectron spectroscopy (XPS) for the whole series compounds of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 as shown in Fig. 2. Fig. 2(a) represent full energy scan of the compounds. The peaks represent the chemical elements of Pb 4f and 4d_5/2, Te 3d_3/2, S 3p_3/2, Se 3d_5/2, K 2p_3/2, and other impurities such as O s- and C 1s-orbitals. The oxygen and carbon impurities come from the surface oxidation and contamination which is widely observed in XPS measurement.

Fig. 2 Full scan of the XPS spectra (a) and expanded region near the K 2P_3/2 core level (b) for (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 compounds (x = 0.01, 0.015, 0.02, 0.025, and 0.03).

Fig. 2(b) is the expanded scale near the K 2P_3/2 core level for the compounds. From the integration of the peaks for each elements, we quantified the atomic concentration of the elements. Table 1 listed the atomic (at%) and relative concentrations of Pb and K elements for the initial stoichiometric compounds of Pb_1−xK_xTe_0.7Se_0.25S_0.05 (x = 0.01, 0.015, 0.02, 0.025, and 0.03). It shows the non-systematic concentration of K with respect to initial compositions. It is not surprising because XPS is a surface sensitive measurement tool and the K-doping concentration is relatively small. In spite of that, it seems that the potassium does not be distributed homogeneously.

From the room temperature Hall resistivity measurement, we obtained the Hall carrier density n_H and Hall mobility μ_H of p-type (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 system. Fig. 3 shows the Hall carrier density (left axis, black closed square) and Hall mobility (right axis, blue open square) at room temperature of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 system as function of K-doping concentration x. The Hall carrier concentration is abruptly increased at K-doping range x ≥ 0.02 from 4.33 × 10¹⁹ cm⁻³ (x = 0.02) to 7.08 × 10¹⁹ cm⁻³ (x = 0.025), which indicates that the K-doping tunes the carrier concentration in (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 system, similarly observed in (PbTe)_1−x(PbSe)_x (x = 0, 0.1, 0.25 and 0.75) compounds.²⁸ The maximum Hall mobility is found μ_H = 303.2 cm² V⁻¹ S⁻¹ at x = 0.1 and is decreased with increasing K-doping concentration due to the increase of interfacial and defect scattering of carriers.²⁹

Fig. 3 Hall carrier mobility μ_H (blue open square, right axis) and Hall carrier contraction n_H (black closed square, left axis) of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 compounds with respect to K-doping concentration x.

In order to understand the K-doping effect on thermoelectric performance, we measured temperature-dependent Seebeck coefficient S(T) and electrical resistivity ρ(T) from 300 K to 800 K of quaternary (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 compounds and K-doped Pb_0.98K_0.02Te (black square) for comparison as shown in Fig. 4(a) and (b), respectively. The compounds show a typical behaviour of degenerated semiconductor which S(T) and ρ(T) are increased with increasing temperature. The Seebeck coefficients show a broad maximum at high temperature range (T ≥ 700 K) depending upon the K-doping concentration as shown in Fig. 4(a). The S(T) is not changed significantly with respect to K-doping concentration except the x = 0.02 K-doped one and Pb_0.98K_0.02Te.

Fig. 4 Temperature dependence of (a) Seebeck coefficient S, (b) electrical resistivity ρ, (c) power factor S²/ρ of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 and Pb_0.98K_0.02Te compounds. (d) Room temperature Pisaranko plot of Seebeck coefficient S versus Hall carrier concentration n_H for (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 compounds as well as p-type PbTe,^18,33 Pb_1−xK_xTe,²⁷ and Na-doped PbTe¹⁸ compounds based on the two band model.

From the Seebeck coefficient, we can obtain a rough estimation of the effective mass of carrier based on a single band model:³⁰

where η is the reduced Fermi energy, r_H is the Hall factor taking into account band anisotropy and relaxation time of carriers which is about unity for parabolic bands and a spherical Fermi surface, and k_B and h are the Boltzmann and Plank constants, respectively. The calculate effective masses of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 compounds are presented in Table 2. The effective mass is increased with the increase of K-doping except x = 0.15 case. The estimated values of effective masses at room temperature for x = 0.01 (m* = 0.309 m_e), 0.020 (m* = 0.310 m_e), 0.025 (m* = 0.365 m_e) and 0.030 (m* = 0.339 m_e) are higher than 0.25 m_e or 0.27 m_e for PbTe.^29,31 The large effective masses are related to the two valance bands of L- and Σ-bands.^22,27 In the two valance bands case, the total Seebeck coefficient is given by:

S_total = (σ_LS_L + σ_ΣS_Σ)/(σ_L + σ_Σ)

here, S_L and S_Σ denote the Seebeck coefficients of the light hole L-band and heavy hole Σ-band, while σ_L and σ_Σ refer to the electrical conductivity from light hole and heavy hole bands, respectively. The L-and Σ-bands have temperature dependent behaviour.^22,32 The energy band gap ΔE_g between the conduction band and L-valence band increases while the extremum valence band difference ΔE_v between the L- and Σ-bands decreases with increasing temperature as following relation.

ΔE_C–Σ = 0.36 + 0.10x

Table 2 Lattice parameter a, Hall carrier concentration n_H, Hall mobility of carrier μ_H, Seebeck coefficient S, and effective mass m* for (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 compounds

x	a (Å)	nH (×10^19 cm^−3)	μH (cm^−2 V^−1 S^−1)	S (μV K^−1)	m* (me)
0.01	6.355	4.932	303.487	46.997	0.309
0.015	6.353	4.875	268.894	33.492	0.220
0.02	6.353	4.336	224.972	51.401	0.310
0.025	6.356	7.720	166.440	41.106	0.365
0.03	6.355	7.085	123.253	40.436	0.339

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The energy band gap ΔE_g between the conduction band and L-valence band increases while the extremum valence band difference ΔE_v between the L- and Σ-bands decreases with increasing temperature.^22,32 Because the L- and Σ-bands come closer to each other with increasing temperature, holes transfer from the L-band to the Σ-band, which is the cause of the broad maximum near 700 K. The electrical resistivity shows the similar trend with Seebeck coefficient as shown in Fig. 4(b), which means the metallic behaviour with temperature. In terms of two valance bands model, the L-band approaches to the Σ-band with increasing K-doping concentration. The two bands overlap increases carrier concentration as well as effective mass of carrier due to the high density of states. Table 2 shows the distinguishably increase of carrier concentration and effective mass at x ≥ 0.02, implying the band degeneracy between two bands.

Fig. 4(c) presents the power factor S²/ρ of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 system and Pb_0.98K_0.02Te for comparison. The maximum power factor is 27.78 μV cm⁻¹ K⁻² at 600 K for x = 0.01. The high power factor for the compound is due to the significant reduction in electrical resistivity. Fig. 4(d) presents room temperature Pisarenko plot (Seebeck coefficient vs. Hall carrier concentration) based on two band model for (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 compounds as well as p-type PbTe,^18,33 Pb_1−xK_xTe,²⁷ and Na-doped PbTe¹⁸ compounds. Two band model for p-type PbTe was calculated based on m^*_h = 1.2 m_o and m^*₁ = 0.36 m_o and includes band gap ΔE ≈ 0.12 eV at room temperature. The (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 system follows the Pisarenko plot. The flattening of the Seebeck coefficient with increasing carrier concentration indicates the presence of the second valance band.^23,33 The experimental data points are below the dashed line of two band model, which is due to lower effective mass on the compounds than p-type PbTe. Based on the Pisaranko plot in Fig. 4(d), we do not find any indication of the existence of resonance scattering.

Temperature-dependent total thermal conductivity κ(T) of K-doped compounds decreases with increasing temperatures as shown in Fig. 5(a) which is the typical behavior of acoustic phonon scattering. Even though the K-doping effect does not show a systematic behavior, the lowest thermal conductivity is obtained for x = 0.2 K-doped compound (1.9 W m⁻¹ K⁻¹ at 300 K). The significant decrease of thermal conductivity of the compound than those of Pb_0.98K_0.02Te (3.4 W m⁻¹ K⁻¹ at 300 K) may come from the additional multiple elements doping by Se and S which will be discussed in detail later. The κ(T) for the samples of x ≤ 0.02 weakly increases with increasing temperature at the high temperature region (T ≥ 700 K). The small upturn of total thermal conductivity is also found in Na-doped (PbTe)_1−x(PbS)_x compound indicating small bipolar contribution of carriers.³⁴

Fig. 5 Total thermal conductivity κ (a) and lattice thermal conductivity κ_L (b) of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 and Pb_0.98K_0.02Te compounds.

The lattice thermal conductivity κ_L(T) of the compounds (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 is shown in Fig. 5(b) by subtracting the electronic thermal conductivity κ_e from the Wiedermann–Franz law: κ_e = LT/ρ, where ρ, T, and L are the electrical resistivity, absolute temperature, and the Lorenz number. The temperature-dependent Lorenz number is calculated within an assumption of a parabolic band model with acoustic phonon scattering.²⁹ L is expressed by the Fermi integral formalism as following relations:³⁵

where F_j, η, f, and ε are the Fermi integral, reduced electrochemical potential, Fermi function, and energy, respectively. The Seebeck coefficient S and Lorenz number L are given by:

As a result, the lattice thermal conductivity of the (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 compounds are shown in Fig. 5(b). The lowest lattice thermal conductivity is found in (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05 with 0.91 W m⁻¹ K⁻¹ at room temperature and 0.69 W m⁻¹ K⁻¹ at 800 K. The lattice thermal conductivity of the compound is strongly reduced as much as 63.4% and 40.64% at room temperature comparing with the previous report of Pb_0.98K_0.02Te and (Pb_0.98K_0.02Te)_0.75(PbSe)_0.25.²⁷ The compounds showing low lattice thermal conductivity (x = 0.01, 0.02, and 0.025) do not follow the T⁻¹ dependence indicating that the additional contribution of phonon scattering attributes to the thermal conductivity as well as acoustic phonon contribution.

In order to clarify the phonon scattering mechanism, we calculate the thermal conductivity based on the PbTe_1−xSe_x alloy model. Theoretical model of lattice thermal conductivity of PbTe_1−xSe_x alloy can be calculated as follows which is given by Klemens:^23,36

where θ_D is the Debye temperature, Ω is the molar volume, ν is the velocity of sound, and Γ is a disorder scaling parameter that depends on mass and strain field fluctuations (Δm/m and Δα/α).

When we plot the lattice thermal conductivity for various PbTe based compounds including the (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05, Pb_0.98K_0.02Te (in this work) and (Pb_0.98K_0.02Te)_1−x(PbSe)_x²⁷ with the theoretical estimation of lattice thermal conductivity in the alloy model as shown in Fig. S2 of ESI,† the experimental lattice thermal conductivity of (Pb_0.98K_0.02Te)_1−x(PbSe)_x follows the trend of theoretical of lattice thermal conductivity for (PbTe)_1−x(PbSe)_x system. Therefore, the strong phonon scattering in (Pb_0.98K_0.02Te)_1−x(PbSe)_x comes from the point defect scattering created by Te/Se mixed occupation in rock salt structure. On other hand, the lowest lattice thermal conductivity of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 compounds lie below the theoretical lattice thermal conductivity of (PbTe)_1−x(PbSe)_x. It implies that the low lattice thermal conductivity of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 has another scattering sources as well as the alloy scattering and/or point defect scattering by Te/Se mixed occupation.

One possible phonon scattering mechanism is the nano/micro-grain boundary scattering. We examine the microstructural analysis of (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05 by high resolution transmission electron microscopy (HR-TEM) as shown in Fig. 6. Fig. 6(a) and (b) show mid-magnified images of (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05. It displays numerous nano-precipitates with dark contrast for size distribution of 5–15 nm. The electron diffraction pattern, in the inset of Fig. 6(a), of (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05 sample shows cubic structure. The lattice parameters of PbTe, PbSe, and PbS are 3.22 Å (PbTe), 3.07 Å (PbSe), and 2.965 Å (PbS), respectively, along the (200) plane.^34,37 In order to study the nano-precipitates of the compound, we measure high-resolution transmission electron microscope (HR-TEM) images as shown in Fig. 6(c). We can identify the several homogeneous nano-precipitates (dark contrast) in (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05 with sizes of 2–5 nm in the matrix. From the electron diffraction, we identified the nano-precipitates are composed with (200) and {111} plane as shown in the inset of Fig. 6(c). Fig. 6(d) is the inverse fast Fourier transform (IFFT) image of the Fig. 6(c). The IFFT images along the (200) plane show that there are many line dislocations at the coherent interfaces between the nano-precipitates and PbTe matrix which is marked by red circles.

Fig. 6 Transmission electron microscope (TEM) images of (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05 for low magnified image showing numerous nano-precipitation and spinodal decomposition region (a) and its enlarged image (b). Inset of (a) is the electron diffraction pattern. High resolution TEM (HR-TEM) image of nano-precipitation with semi-coherent interface (c) and electron diffraction (inset of (c)). Inverse fast Fourier transform IFFT image of nano-precipitates along (200) plane (d).

The nano-precipitates have coherent or semi-coherent interface depending on each of the crystal plane as shown in Fig. 7. Previous reports on the PbTe/PbS system showed that the nano-precipitation was formed by spinodal decomposition or nucleation, depending on composition and temperature conditions, using a miscibility gap in the PbTe/PbS system.^38–40 Fig. 7(a)–(c) present high resolution TEM images of [110] plane at different regions. The electron diffraction peaks (inset of Fig. 7(b) and (c)) and HR-TEM images show the identical crystal symmetry. On the other hand, the inverse fast Fourier transform (IFFT) images in Fig. 7(d) and (e) present the line dislocations and small lattice distortion. The line dislocations and lattice distortions may come from the phase separation due to the compositional complexity by multiple elements doping or substitutions.^23,41

Fig. 7 HR-TEM image of (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05 for selected area (region A and region B) (a), HR-TEM image of precipitate image of region A (b) and region B (c), inverse-FFT analysis for precipitation area for region A (d) and region B (e).

Although it is difficult to quantitatively determine the composition of precipitates due to overlap with matrix, we investigate the elemental analysis of the matrix and the precipitates by energy dispersive X-ray spectroscopy (EDS) for (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05. Fig. 8(a) and (b) are the low magnified images of the compound, which clearly show the nano precipitates and spinodal decompositional phase separation in dark regions. By comparing the EDS spectrum 1 (matrix), spectrum 2 (nano-dot), and spectrum 3 (spinodal) regions as shown in the marked regions of Fig. 8(b), we identify the elemental compositions of the compound. We found that the precipitates and spinodal decompositional regions corresponds to the PbS rich phases while the region 1 corresponds to the PbTe matrix as shown in Fig. 8(d).

Fig. 8 (a) Low magnified image of bright-field HR-TEM micrograph for (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05 compound. (b) Enlarged view of HR-TEM image of selected area of (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05: indicating (1) matrix, (2) nano-dot and (3) spinodal decomposition region. (c) Energy dispersive X-ray spectroscopy images at each different regions of matrix (region 1), nano-dot (region 2), and spinodal decomposition (region 3). (d) Expected compositional concentration of PbTe and PbS for different regions.

We analyze the lattice thermal conductivity of (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05 in terms of various scattering mechanisms based on the Callway model for PbTe and (PbTe)_0.70(PbSe)_0.25(PbS)_0.05. The lattice thermal conductivity can be expressed by the Callaway's model:^37,42

where k_B, ħ, T, ν, θ, and x are the Boltzmann's constant, Plank's constant, absolute temperature, sound velocity, Debye temperature, and x = ħω/k_BT, respectively. τ_N and τ_c are the relaxation times due to normal phonon–phonon scattering and the combined relaxation time. From the Matthiessen's rule, the relaxation time is obtained by integrating the relaxation times from various scattering processes.

Based on TEM studies, for the same frequency, the relaxation time depends mainly on scattering from the nanostructuring precipitates, dislocation boundaries and phonon–phonon interactions. The combined relaxation time is given as:

τ⁻¹_c = τ⁻¹_U + τ⁻¹_N + τ⁻¹_B + τ⁻¹_S + τ⁻¹_D + τ⁻¹_P

where τ_U, τ_N, τ_B, τ_S, τ_D, and τ_P are the relaxation time corresponding to scattering from Umklapp process, normal process, boundaries, strain, dislocations, and precipitates.^43–48 In this work, we consider four different scattering mechanisms such that the point-defect scattering, phonon–phonon scattering, grain boundary scattering, and nanoprecipitated (with radius r and volume fraction n) scattering. Therefore the combine phonon relaxation time can be expressed as
where L is the average grain size, and the coefficient A and C are fitting parameters. The value of C depends only on the crystal structure of (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05 thus, we obtain the value of C by fitting the above equation to the lattice thermal conductivity of un-doped (PbTe)_0.70(PbSe)_0.25(PbS)_0.05 one. Because we cannot distinguish grain boundary scattering, it is convenient to define an effective mean free path:⁴⁹

The parameters can be obtained from TEM investigations and some from ref. 49.

Fig. 9 shows the theoretical lattice thermal conductivity of Pb_0.98K_0.02Te (black square) and (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05 (magenta inverse triangle) based on the Callaway's model (open symbols) with experimental data (closed symbols). The experimental data of Pb_0.98K_0.02Te (black closed square) fit very well with the theoretical lattice thermal conductivity model of Pb_0.98K_0.02Te (black open square). However, the experimental lattice thermal conductivity (closed magenta inverse triangle) of (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05 does not fit to the theoretical values (open magenta inverse triangle). The discrepancy of the lattice thermal conductivity of the compound can be considered by several reasons: (1) the Lorenz number calculation is not exact in nano-structured composites,^5,50 (2) an anharmonic phonon excitation (3) partial local collapse of nano-structure at high temperature.^40,51 This phenomena was also found in (PbTe)_0.88(PbS)_0.12 by Bi doping⁵¹ and (Pb_0.95Sn_0.05Te)_1−x(PbS)_x⁴⁰ alloys. We surmise that the nano precipitation gives rise to significant phonon scattering resulting in the low lattice thermal conductivity. It needs further investigation to clarify the cause of thermal transport on the compounds.

Fig. 9 Experimental (closed symbols) and theoretical (open symbols) lattice thermal conductivities of PbTe (black square) and (PbTe)_0.75(PbSe)_0.20(PbS)_0.05 (magenta inverse triangle). Theoretical calculation is based on the phenomenological effective medium theory and Callaway's model (described in the text).

The apparently low thermal conductivity and high power factor of the nanostructured compounds can have a direct impact on thermoelectric performances. Fig. 10(a) displays dimensionless thermoelectric figure-of-merit ZT of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_x compounds. The compound of x = 0.02 shows the highest ZT value of 1.72 at 750 K, which higher than those of Pb_0.98K_0.02Te (ZT of 1.11 at 800 K) and (Pb_0.98K_0.02Te)_0.75(PbSe)_0.25 (ZT of 1.60 at 773 K)²⁷ as much as 35% and 7.0%, respectively (S3 of ESI†). We also calculated the average ZT_avg values of (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05, Pb_0.98Na_0.02Te, Pb_0.98K_0.02Te, (Pb_0.98Na_0.02Te)_0.88(PbS)_0.12,²¹ (Pb_0.98Na_0.02Te)_0.75(PbSe)_0.25,²² (Pb_0.98K_0.02Te)_0.75(PbSe)_0.25 (ref. 27) and (Pb_0.98Na_0.02Te)_0.84(PbSe)_0.07(PbS)_0.07.²³ The average ZT_avg is given by:⁵²

where Z is the figure-of-merit, T_c is the cold side temperature, T_h is the hot side temperature, T_avg is the average temperature (T_h + T_c)/2, and ΔT is the temperature difference between hot and cold sides (T_h − T_c). Fig. 10(b) shows that the average ZT_avg of (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05 has the highest average ZT_avg values (1.08) comparing to the other compounds in the previous report.^21–23,27

Fig. 10 (a) Temperature dependent ZT values for (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 and Pb_0.98K_0.02Te compounds and (b) average ZT value ZT_avg, which is defined in the text, for various Pb-based chalcogenides in a condition of T_h = 800 K and T_c = 300 K comparing with references.

For more practical point of view, the engineering ZT_eng predicts more reliable and practical conversion efficiency of thermoelectric.⁵² The engineering ZT_eng is expressed as:⁵²

where S(T), κ(T), and ρ(T) are the temperature dependent Seebeck coefficient, total thermal conductivity, and electrical resistivity, respectively. The (PF)_eng is called the engineering power factor having a unit of W m⁻¹ K⁻¹ not the conventional unit of W m⁻¹ K⁻² since its effect associated with the temperature gradient. Fig. 11(a) and (b) show (ZT)_eng and (PF)_eng with respect to temperature gradient for T_c = 300 K of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 compounds. It show the maximum (ZT)_eng is 0.81 at ΔT = 500 K with x = 0.02.

Fig. 11 Temperature-dependent engineering dimensionless figure-of-merit ZT_eng (a), engineering power factor PF_eng (b), thermoelectric efficiency η (c), and output power density P_d (d) with respect to temperature difference at T_c = 300 K for series compounds of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05.

We compare the (ZT)_eng of x = 0.02 doped compound with the K- and Na-doped PbTe alloys^21–23,27 in S4(a) of ESI.† It clearly shows that (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25(PbS)_0.05 has the highest (ZT)_eng comparing to the other compounds: such as Pb_0.98Na_0.02Te (ZT_eng ≈ 0.43 at ΔT = 500 K), Pb_0.98K_0.02Te (ZT_eng ≈ 0.50 at ΔT = 500 K), (Pb_0.98Na_0.02Te)_0.88(PbS)_0.12 (ZT_eng ≈ 0.52 at ΔT = 500 K),²¹ (Pb_0.98Na_0.02Te)_0.75(PbSe)_0.25 (ZT_eng ≈ 0.57 at ΔT = 500 K),²² (Pb_0.98K_0.02Te)_0.70(PbSe)_0.25 (ZT_eng ≈ 0.64 at ΔT = 500 K),²⁷ and (Pb_0.98Na_0.02Te)_0.84(PbSe)_0.07(PbS)_0.07 (ZT_eng ≈ 0.74 at ΔT = 500).²³

Using the (ZT)_eng, we calculated the individual legs' maximum efficiency (η_max) based on temperature independence. The calculated the maximum efficiency is defined by⁵²

where η_c is the Carnot efficiency (T_h − T_c)/T_h, and [small alpha, Greek, circumflex] is dimensionless intensity factor of the Thomson effect defined as . Fig. 11(c) displays engineering efficiency of the compounds (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05. An efficiency of 11.63% is obtained for the individual legs of x = 0.02 compound, when the cold and hot sides temperature are set to be 300 K and 800 K, respectively. This result is comparable or higher than the other PbTe alloy materials such as Pb_0.98Na_0.02Te (η_max ≈ 7.70%), Pb_0.98K_0.02Te (η_max ≈ 8.58%), (Pb_0.98Na_0.02Te)_0.88(PbS)_0.12 (η_max ≈ 8.92%), (Pb_0.98Na_0.02Te)_0.75(PbSe)_0.25 (η_max ≈ 9.26%), (Pb_0.98Na_0.02Te)_0.75(PbSe)_0.25 (η_max ≈ 9.88%), (Pb_0.98Na_0.02Te)_0.75(PbSe)_0.07(PbS)_0.07 (η_max ≈ 10.70%) as shown in S4(b) of ESI.† These result shows that K-doping improves the maximum efficiency η_max comparing with the Na-doped ones due to high Seebeck coefficient.

The engineering power factor (PF)_eng determines the output power density for given temperature gradient. Due to higher efficiency and engineering power factor, we expect a high output power density of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 system. The output power density can derived as:⁵²

where m_opt is the optimized ratio defined as and L is the length of the thermoelectric element. Fig. 11(d) shows output power density of the compounds with L = 1 mm and T_c = 300 K. The maximum output power density is 6.34 W cm⁻² for x = 0.01 compound due to high power factor.

Conclusions

In summary, we synthesized the compounds of (Pb_1−xK_xTe)_0.70(PbSe)_0.25(PbS)_0.05 (x ≤ 0.03) and investigated thermoelectric properties. The carrier concentration and effective mass of carriers are distinguishably enhanced near the x ≥ 0.02 K-doped region. In terms of two band model, we argue that the L-band become close to the Σ-band giving rise to the enhancement of band degeneracy, resulting in the increase of relative Seebeck coefficient due to the enhancement of effective mass as well as carrier concentration. Nano precipitation of PbS in the matrix causes the reduction of lattice thermal conductivity. The effective alloy scattering and phonon scattering by line dislocation and lattice distortion in nano-structured phase induce very low lattice thermal conductivity for x = 0.02 K-doped compound. The high ZT value of 1.72 at 800 K for (Pb_0.98K_0.020Te)_0.70(PbSe)_0.25(PbS)_0.05 compound is the comparatively larger than the previously reported ones as well as the enhancement of values in average ZT_avg = 1.08, engineering (ZT)_eng = 0.81, maximum efficiency η_max = 11.63%, and output power density P_d = 6.3 W cm⁻², with temperature difference ΔT = 500 K. The enhancement of thermoelectric performance of the K-doped Pb-based quaternary compounds indicates that the effective band convergence and homogeneous nano-structuring with line dislocation and lattice distortion are an effective way to increase thermoelectric properties for waste heat power generation.

Acknowledgements

This work was supported by the Samsung Research Funding Centre of Samsung Electronics under Project Number SRFC-TA1403-02.

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Footnote

† Electronic supplementary information (ESI) available. See DOI: 10.1039/c6ra11299d

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