Contrasting Thermoelectric Transport Behaviors of p-Type PbS Caused by Doping Alkali Metals (Li and Na)

PbS is a latent substitute of PbTe thermoelectric materials, which is on account of its superiority in low cost and earth abundance. Here, the thermoelectric transport properties of p-type PbS by doping alkali metals (Na and Li) are investigated and it is verified that Li is a more effective dopant than Na. By introducing Li, the electrical and thermal transport properties were optimized collectively. The electrical transport properties were boosted remarkably via adjusting carrier concentration, and the maximum power factor (PFmax) of ~11.5 μW/cmK2 and average power factor (PFave) ~9.9 μW/cmK2 between 423 and 730 K in Pb0.99Li0.01S were achieved, which are much higher than those (~9.5 and ~7.7 μW/cmK2) of Pb0.99Na0.01S. Doping Li and Na can weaken the lattice thermal conductivity effectively. Combining the enlarged PF with suppressed total thermal conductivity, a maximum ZT ~0.5 at 730 K and a large average ZT ~0.4 at 423-730 K were obtained in p-type Pb0.99Li0.01S, which are higher than ~0.4 and ~0.3 in p-type Pb0.99Na0.01S, respectively.


Introduction
The search for reliable and environmentally friendly new energy has attracted worldwide attention because of the shortage of fossil energy. A thermoelectric device is capable of transforming heat into electric energy immediately, which has shown great prospect in clean energy field [1][2][3][4][5]. The thermoelectric device efficiency is positively associated with the dimensionless figure of merit [6][7][8][9], ZT = S 2 σT/κ tot , where T represents absolute temperature, S expresses the Seebeck coefficient, σ denotes electrical conductivity, and κ tot represents total thermal conductivity comprising electronic (κ ele ) and lattice (κ lat ) contributions [2,10,11].
Lead telluride-(PbTe-) based materials, as a kind of medium-temperature operating materials, have attracted extensive research interests on account of outstanding thermoelectric performance [12][13][14]. However, considering the high costs and low earth abundance of the Te element, the thermoelectric materials with rich resources should be developed. To date, one research hotpot in this field is to find an alternative material to substitute PbTe which possesses prominent thermoelectric properties [14][15][16]. As a similar alternative of PbTe, PbS possesses a NaCl structure and an alike band structure. Nevertheless, the poor electrical properties and large lattice thermal conductivity historically make PbS become an inferior thermoelectric material [11]. Aimed at solving the shortcomings of PbS, the approaches including carrier concentration optimization [17], band manipulation [18], and microstructure engineering [19][20][21][22] have been proved as effective strategies to manipulate electrical properties and thermal conductivity; the achievements realized through the above strategies well elucidate the potential performance of PbS.
Usually, doping is a powerful tactic to enhance ZT. Doping is essential in adjusting carrier concentration, and it is the prerequisite to gain a high ZT as all of those thermoelectric properties are interlinked by carrier concentration. On the assumption that the scattering or band structure is not modified obviously by a dopant, the Seebeck coefficient and electrical conductivity of degenerated semiconductor with a single parabolic band can be given using Equations (1) and (2) [23,24].
where k B displays the Boltzmann constant, e shows the electron charge, h denotes the Planck constant, m * expresses effective mass, n is the carrier concentration, and μ n represents the carrier mobility. Apparently, S and σ are in an inversely proportional relationship. Therefore, adjusting carrier concentration through balancing the relationship between S and σ is an important method to boost power factors. Doping with different elements may induce diverse impact on carrier concentration optimization and band structure manipulation. For example, p-type Na-doped PbS with CdS as second phases attains a large ZT~1.3 at 923 K owing to extensive phonon scattering by nanophase precipitates and better electrical transport [25]; the ZT of p-type Tl-doped PbTe reaches~1.5 at 773 K due to greatly enhanced Seebeck coefficients by deformation of electronic density of states [17]. It is meaningful to speculate the impacts by doping other alkali metals on thermoelectric performance in PbS.
In this article, we focused on PbS, which contains highly earth-abundant elements and owns higher melting point compared to PbTe. The thermoelectric properties in PbS doped by Li and Na were investigated systematically. The consequences reveal that the electrical and thermal properties were optimized synchronously through alkali metal doping. The electrical properties were improved through adjusting carrier concentration, and the PF max of Pb 0.99 Li 0.01 S reached~11.5 μW/cmK 2 , which is far greater than~9.5 μW/cmK 2 in Pb 0.99 Na 0.01 S. Both Li and Na can bring down the κ lat because of point defects in PbS matrix. Li was more effective than Na in reducing κ lat on account of larger mass and strain field fluctuations. Integrating enhanced PF and reduced κ tot , a higher ZT value~0.5 at 730 K and average ZT~0.4 at 423-730 K can be reached in Pb 0.99 Li 0.01 S, which are higher than~0.4 and~0.3 in ptype Pb 0.99 Na 0.01 S, respectively, indicating that Li doping can improve thermoelectric performance of PbS more effectively than Na doping.

Preparation Method.
High-purity chemicals of Pb particle (99.99%), S (99.99%), Na (99.99%), and Li (99.99%) were weighed and loaded into carbon-coated quartz ampules under a N 2 -filled glove box. The ampules of the chemicals were evacuated under vacuum and flame-sealed. The pure chemicals were gradually warmed up to 723 K in 12 h, elevated to 1423 K in 7 h before keeping stable at 1423 K for 6 h, and finally naturally cooled to indoor temperature. The prepared specimens were pulverized and filtered with 400mesh sieves for sintering through spark plasma sintering (SPS-211LX) using a pressure of 50 MPa at 923 K for 10 min.

Thermoelectric
Properties. The acquired cylinder bulk materials were incised for measuring relevant thermoelectric properties. The CTA system was applied to measure electrical parameters (S and σ) at 300-730 K under He gas, and the samples were polished in a rectangular shape of 10 mm × 3 mm × 3 mm. The surfaces of measured samples were sprayed with thin-layer BN, which can inhibit volatilization and protect instrument [26]. The cylindrical disks with thickness of 1 mm and diameter of 6 mm were used to measure the thermal diffusivity (D). The thermal conductivity is computed through κ tot = DC p ρ, and the thermal diffusivity was characterized using a Netzsch LFA457 instrument with a laser flash method [27]. A thin graphite film on the surface of samples was utilized to cut down errors of emissivity for testing D. The density (ρ) was obtained based on mass and volume. All the densities of samples are around 7.2 g/cm 3 . The heat capacity (C p ) was computed using the Debye model [28].
2.3. X-Ray Diffraction. The phase structure was investigated using an X-ray diffraction technique with D/MAX2200pc system with CuKα at 2θ = 20 -80°(Rigaku, Japan, 40 kV, 40 mA). The scanning speed and step size of the XRD measurement are 6°min -1 and 0.02°, respectively.

Theoretical
Calculations. The density functional theory (DFT) calculations were acquired through a projectoraugmented wave (PAW) strategy [29,30] with the Vienna Ab initio Simulation Package (VASP) [31]. The Perdew-Burke-Ernzerhof (PBE) exchange-correlation functional was used to model crystal and electronic structure. The used kinetic cutoff energy of plane waves is 500 eV. A 3 × 3 × 3 supercell (Pb 27 S 27 ) was constructed to evaluate the defect formation energy of Li-(Pb 26 LiS 27 ) and Na-(Pb 26 NaS 27 ) doped systems. The internal coordinates of all atoms are entirely relaxed while the maximum residual ionic force is lower than 0.01 eV Å -1 , and the total energy difference approaches 10 -6 eV.
The formation energy of the defect Q (Q = Na, Li) in charge q is calculated by [32,33] where EðPbS, Q q Þ and EðPbSÞ refer to the total energy of defect α contained and undoped PbS in supercells with the same dimension, separately. μ i , E i , and n i are the chemical potential, average energy of element in its most stable crystal structure, and the number of the atom i added to (n i < 0) or taken from (n i > 0) the host, respectively. E F presents the Fermi level relative to energy location of valence band maximum (E v ), which changes between 0 and band gap in PbS.
The correction term ΔV is adopted to arrange the reference potential between the defect-containing and pure supercells with the same size [34]. All characteristic values are recombined to the 1s core level of the atom farthest from the defect [32,35]. The formation energy depends on the chemical potential of each element, which is related to the off-stoichiometric degree (Pb-or S-rich condition). The different offstoichiometric degrees will result in different chemical potential and formation energy. The upper and lower boundary chemical potential (μ i ) is determined by the off-stoichiometric degree and the stability against precipitation of elemental Pb, S, Li, and Na: The host compounds are obtained from the sum of the chemical potentials of Pb and S: where ΔH f ðPbSÞ is the formation energy of PbS in a rocksalt structure. The second phase of Na 2 S, Li 2 S, and PbS 2 should be avoided, and the corresponding formation energy should be larger than the sum of elemental chemical potential: 3. Results and Discussion Figure 1 demonstrates the detailed information of XRD results. All specimens possess a single phase of cubic PbS. The data peak transfers to low angle range as the Li and Na content was added, which indicates that Li and Na are doped into PbS lattice. For Li-doped samples, the σ possesses a tendency to increase first and then decrease with the stoichiometry of Li increasing and reaches to its maximum 777 S/cm in Pb 0.995 Li 0.005 S, as shown in Figure 2(a). However, the Na doping presents different results. As presented in Figure 2(b), the σ has an increased trend with increasing of Na content, and the maximum σ of 1274 S/cm can be realized in Pb 0.98 Na 0.02 S. The σ is positively correlated with n H and μ n from Equation (2), which are determined by the solid solubility and the different scattering mechanisms, respectively. The continued increase in σ of Pb 1-x Na x S is mainly due to the fact that the higher solubility of Na than Li was caused by better ion radius matching (r Li + = 0:76 Å, r Na + = 1:02 Å, r Pb 2+ = 1:26 Å).
As displayed in Figures 2(c) and 2(d), different from the undoped PbS, the Seebeck coefficients (S) for all doped samples are positive, indicating that Li and Na are effective p-type dopants in PbS. For Li-doped samples, the S present the trend of first decreasing and then increasing with an increasing Li content. For Na-doped samples, the trend is reversed that the S increase first and then decrease with an increasing Na content. These diametrically opposite trends reflect the con-trary change of carrier concentration (n H ) in those materials since the S are negatively correlated with n H .
To evaluate the doping efficiency of Li and Na in PbS, the formation energy of potential defect was calculated and shown in Figures 3(a) and 3(b). The lower formation energy of Na Pb indicates the spontaneous formation of Na Pb in any conditions, which is even lower than that in V Pb . However, the Li Pb has higher formation energy under Pb-and S-rich situations. In an equilibrium theory, the defect concentration can be evaluated by the formation energy ΔH, expressed as n i = N i × e ð−ðΔH/kTÞÞ [36]. Thus, the larger formation energy of Li leads to a lower n H and a larger S (Figure 2(c)).
As presented in Figure 2(e), for Li-doped samples, a higher PF can be obtained in Pb 0.99 Li 0.01 S in a broad temperature range, and the peak value can reach 11.5 μW/cmK 2 at 450 K. The peak PF for Pb 0.99 Li 0.01 S is much higher than Pb 0.99 Na 0.01 S which is ascribed to the lower n H , namely, adjusting carrier concentration to an optimized scope. Figures 3(c) and 3(d) show the carrier mobility and carrier concentration at room temperature which are calculated by the carrier effective mass of PbS (m * = 0:38 m 0 ) [25]. According to Rowe and Bhandari's study [37], the S decreases and the σ increases as the n H increases and the PF maximizes at a suitable n H for a semiconductor. Therefore, adjusting the n H to a reasonable range is the key factor to obtain higher PF. Compared with Pb 0.99 Na 0.01 S, Li doping leads to a  The PF ave is calculated by Equation (7) in which the T h and T c are the temperatures of hot and cold ends. The PF ave represents the overall capacity and level of electrical transports over a specified wide temperature range. The PF max and PF ave of the Pb 0.99 Li 0.01 S sample are 11.5 and 9.9 μW/cmK 2 , respectively, higher than those of Pb 0.99 Na 0.01 S which are 9.5 and 7.7 μW/cmK 2 . The present results reveal that the different dopants can reach the PF max under their proper n H , which is strictly determined by the solid solubility of dopants in PbS.
Figures 4(a) and 4(b) depict the κ tot which decreases monotonically with the increase of temperature. The κ tot of Pb 1-x Li x S is lower than that in the undoped PbS, which is different from the larger content of Pb 1-x Na x S since σ is higher. The C p of Pb 0.99 Li 0.01 S and Pb 0.99 Na 0.01 S is 0.2094 J/g·K and 0.2092 J/g·K at 730 K, respectively. The C p of Li-doped samples is similar to the C p of Na-doped samples at the same content and temperature. The κ tot includes lattice thermal conductivity and electronic thermal conductivity (κ tot = κ lat + κ ele ) [22,25], where the relationship between κ ele , σ, and Lorenz number (L) described in Equation (8) indicates that the κ ele is proportional to σ [38,39]. The Lorenz number was obtained through calculating the Seebeck coefficient and integral chemical potentials [40]. Figures 4(c) and 4(d) show the Lorenz number in all samples as function of temperature. Higher L and σ lead to the larger κ ele than those in undoped PbS, as revealed through Figures 4(e) and 4(f). Thus, the reduction in κ tot is primarily caused by the decrease of κ lat .  evaluate point defect scattering caused by Li and Na doping [28,41,42]. When the temperature is higher than the Debye temperature, the point defect is an intensive scattering center to reduce the κ lat . According to the Callaway model [28,42,43], the ratio of the κ lat between the defect-containing material and host material can be written as in which κ lat and κ lat,p represent the lattice thermal conductivities in doped and parent materials, separately. The parameter u is described using in which h, Ω, v a , and θ D express the Planck constant, average atom volume, average sound velocity, and Debye temperature, separately. The imperfection scaling parameter (Γ) indicates that the phonon scattering intensity by atomic scale defects contains mass fluctuation Γ M and strain field fluctuation Γ S . The phenomenological adjustable parameter (ε) regulates the uncertainty of Γ S . The imperfection scaling parameter Γ and the phenomenological adjustable parameter ε are expressed by the following equations [42]: where ν p displays the Poisson ratio, which is calculated using the longitudinal (ν l ) and transverse (ν s ) acoustic velocities. The acoustic velocity of PbS was adopted in Poisson ratio and Grüneisen parameter (γ) calculation by the following equations: When Pb sites are replaced by Li (Na), no change happens on the position of S, Γ S = 0, which is defined by [44,45] where ΔM = M Pb − M Q , and M ðPb,QÞ = ð1 − xÞM Pb + xM Q .
where Δr = r Pb − r Q , and r ðPb,QÞ = ð1 − xÞr Pb + xr Q . Then, The calculated mass fluctuations Γ M,ðPb,QÞ and strain field fluctuations Γ S,ðPb,QÞ have been given by Table 1. The higher deviations in atomic radius and mass between Pb and Li lead to larger Γ (Pb,Li) than Γ ðPb,NaÞ , indicating more effective decreasing of κ lat by Li doping. In Figures 5(c) and 5(d), the calculated results based on the Callaway model exhibit the same trend with the experimental data. The huge deviation may result from the formation of nanostructure even though in moderate doping concentration. This phenomenon confirms that Li and Na could both play effective roles in suppressing κ lat .
The temperature-dependent ZT of PbS doped by Li and Na are presented in Figures 6(a) and 6(b). Pb 1-x Li x S samples exhibit larger ZT than Pb 1-x Na x S samples. The maximum ZT (ZT max ) in Pb 0.99 Li 0.01 S attained~0.5 when T = 730 K, which is higher than that in Pb 0.99 Na 0.01 S. The better thermoelectric performance of Li-doped samples is mainly due to the higher PF which results from the obtained proper n H range and the slightly lower κ tot from a more effective point defect scattering.
The variation trends of maximum ZT (ZT max ) and average ZT (ZT ave ), calculated by Equation (16), are consistent with PF max and PF ave , as displayed in Figures 6(c) and 6(d).
The ZT max from room temperature to 730 K and ZT ave within 423-730 K are 0.5 and 0.4 in Pb 0.99 Li 0.01 S, which is much higher than Pb 0.99 Na 0.01 S (0.4 and 0.3). The quality factor B is a parameter for estimating the optimal thermoelectric properties of a specific material according to the effective mass model, and the quality factor B is obtained by Equation (16). The weighted mobility μ w is calculated by the electrical conductivity and Seebeck coefficient according to Equation (17) [46,47].
in which m e and e are unit mass of free electron and the electron charge, respectively. F n ðηÞ represents the Fermi integral with n = 0 and is calculated by the following equations.
in which r shows the scattering factor and equals -1/2 here and η is the reduced chemical potential [46]. The calculated quality factors of Pb 0.99 Li 0.01 S and Pb 0.99 Na 0.01 S at 730 K are 0.4 and 0.2, respectively. The quality factor of Pb 0.99 Li 0.01 S is about twice higher than that of Pb 0.99 Na 0.01 S, so the ZT of Pb 0.99 Li 0.01 S is higher, which is caused by the enhanced PF by adjusting n H in a reasonable range. The thermoelectric conversion efficiencies are calculated by Equation (20) [28]:

Conclusion
This work indicates that Li doping is more effective than Na doping in thermoelectric performance optimization in PbS. The boosted thermoelectric performance of Li-doped PbS is completed by enhancing the PF through regulating n H in a reasonable range. The PF max and PF ave between 423 and 730 K of Pb 0.99 Li 0.01 S reached~11.5 and~9.9 μW/cmK 2 , which are much better compared with~9.5 and 7.7 μW/cmK 2 of Pb 0.99 Na 0.01 S. Pb 1-x Li x S samples possess slightly smaller κ lat than that of Pb 1-x Na x S because of larger mass and strain field fluctuations. At last, higher ZT max 0.5 at 730 K and ZT ave~0 .4 at 423 K-730 K can be obtained in Pb 0.99 Li 0.01 S. The calculated thermoelectric conversion efficiency~4.8% is achieved in Pb 0.99 Li 0.01 S with T h = 730 K and T c = 423 K. In the future, the ZT for Li-doped PbS can also be raised through nanostructuring, manipulating band structures, and other approaches.

Conflicts of Interest
The authors declare no competing financial interests.