Charge and phonon transport in PbTe-based thermoelectric materials

PbTe is a typical intermediate-temperature thermoelectric material, which has undergone extensive developments and achieved excellent high thermoelectric performance. In this perspective we summarized several strategies that were successfully applied in PbTe-based thermoelectric materials through manipulating charge and phonon transports, such as optimizing carrier density to tune Fermi level, tailoring band structure to enhance effective mass, and designing all-scale hierarchical architectures to suppress phonon propagation. Meanwhile, due to the different features of conduction and valence bands, we separately introduced the approaches to enhance performance of p-type and n-type PbTe. In p-type PbTe, the strategies of band convergence, band alignment and density of state (DOS) distortion are more effective to achieve high electrical transport properties. By contrast, flattening conduction bands and introducing deep impurity level are more suitable for n-type PbTe. Lastly, several potential strategies were proposed to further improve the thermoelectric performance of PbTe-based materials, which might be extended to other thermoelectric systems.


INTRODUCTION
With more than 60% of input energy being lost, most of which as waste heat, there is a compelling need for high performance thermoelectric materials that can directly and reversibly convert heat to electric energy. With continuing vigorous developments over the past decade, encompassing diverse strategies to enhance the power factor and reduce the thermal conductivity, the field of thermoelectrics has truly reached its Renaissance era. However, practical realization of thermoelectric materials is limited by hitherto low dimensionless figure of merit, ZT, which is governed by Carnot efficiency of waste heat conversion. 1 Namely, ZT = S 2 σT/ κ, where S, σ, T, and κ denote the Seebeck coefficient, electrical conductivity, working temperature in Kelvin, and thermal conductivity, respectively. 2 Apparently, a good thermoelectric material simultaneously demands large Seebeck coefficient, high electrical conductivity and low thermal conductivity. However, the complex relationship among these thermoelectric parameters makes it difficult to enhance the final ZT value. Therefore, countless efforts are aimed to improve the ZT value through balancing these interdependent thermoelectric parameters.
Among well-known thermoelectric materials, PbTe is recognized as a very promising compound for power generation in intermediate-temperature range. PbTe-based thermoelectric materials have successfully served in several NASA space missions. 17 In 1960s, due to the technological limits, the thermal conductivity was overestimated, thus the maximum ZT (ZT max ) values~0.7-0.8 in PbTe were underestimated. After laser flash method was provided to determine thermal diffusivity, 51 the ZT max values in both n-and p-type PbTe were recognized to be as high as~1.4. 52 Generally, the thermoelectric performance in PbTebased materials experienced a large progress especially in recent decades, as shown in Fig. 1. The ZT max values of p-type PbTe are boosted from ∼1.0 to ∼2.5, 53 and the ZT max values in n-type PbTe reach~1.8-2.2. 16,54,55 Notably, it is necessary to maximize the efficiency for a thermoelectric device through developing the well-matched performance for both p-and n-type PbTe.
Although PbTe is a simple binary compound, its performance stands out among intermediate-temperature thermoelectrics. PbTe possesses several outstanding features that are typical requirements for thermoelectrics, such as high Seebeck coefficient due to its complicated electronic band structures, superior electrical conductivity from high-symmetry cubic crystal structure, and low thermal conductivity caused by strong anharmonicity due to the Pb local off-center. [56][57][58] To date, PbTe is still a research hot spot in thermoelectric community. After a good perspective summary in 2011, 17 a large amount of achievements have been reported in PbTe-based thermoelectrics, which are evidenced by ZT max~2 .2 in p-type PbTe-SrTe in 2012, 59 ZT max~2 .0 in p-type PbTe-MgTe in 2013, 60 ZT max > 2.0 in p-type PbTe-PbS in 2014, 61 ZT max~2 .5 in p-type PbTe-SrTe in 2016, 53 ZT max~1 .8 in n-type PbTe-InSb, 54 and ZT max~1 .8 in n-type PbTe-Sb-I in 2017, 55 et al. Therefore, it is essential to update these research progresses in PbTe-based thermoelectrics.

OPTIMIZATIONS OF CARRIER DENSITY
PbTe possesses the NaCl-type crystal structure with space group of Fm-3m, which exhibits high symmetry. The unit cell is facecentered cube and all sites in the lattice have six coordinated atoms, as shown in Fig. 2a. The chemical bonds in PbTe are mixed ionic-covalent bonds, thus it is usually classified as polar semiconductor. The electronic band structure of PbTe in Fig. 2b shows a typical semiconductor with a direct bandgap~0.3 eV. 65 Distinct differences between valence band and conduction band can be easily observed, and the offset of light valence band (L) and heavy valence band (Σ) is~0.15-0.20 eV, 69 while a larger band offset~0.45 eV exists in conduction band. 70 To obtain p-type PbTe, theoretically, alkali metal in group-IA (Li, Na, K) can work as an acceptor dopant in Pb site and produce hole carrier. Experimentally, most of the advanced p-type PbTe materials in Fig. 2c are doped with Na and K, and the optimal hole carrier density is in the range of 3-40 × 10 19 cm −3 . In n-type PbTe, elements of I and Br (doping on Te site), Sb, Bi, Al, Ga, and In (doping on Pb sites) are usually chosen as donor dopants, and the electron carrier density in superior n-type PbTe is in the range of 4-40 × 10 18 cm −3 , shown in Fig. 2d. Notably, the optimal carrier density in p-type PbTe is an order of magnitude higher than that in n-type PbTe.

MANIPULATIONS ON VALENCE BAND STRUCTURE IN P-TYPE PBTE
In physics, quantum is the minimum unit to characterize physical properties that involve interaction. In thermoelectric materials, charge is a quantum unit of carriers that is used to estimate the electrical transport properties. Since the quantum charge transports in crystalline materials are closely relevant to electronic band structure, manipulation on electronic band structure is an effective strategy to optimize electrical transport properties. Manipulating approaches on electronic band structure include band  62 PbTe-Mg:Na (band convergence), 60 and PbTe:Tl (resonant level); 64 d graphic presentation of band alignment; e the ratio of carrier mobility to lattice thermal conductivity as a function of SrTe content in PbTe-Sr:Na; 59 f temperature-dependent ZT values of PbTe-Se:Na, PbTe-Mg:Na, PbTe:Tl, and PbTe-Sr:Na convergence, DOS distortion and resonant state, band alignment between matrix and second phase, band flattening, and introducing impurity level, etc.
Band convergence In p-type PbTe, the small offset between light valence band (L) and heavy valence band (Σ) facilitates charge redistribution from single band to multiple bands. Elements alloying could enclose the light and heavy valence bands, called band convergence. Converging valence bands could enhance effective mass through increasing valley degeneracy numbers. 3,71 Recently, the approach of band convergence was revisited along with nanostructuring technology, which is one powerful synergistic strategy to enhance thermoelectric performance in p-type PbTe. 62,72 Band convergence can achieve a high valley degeneracy number (N v ) to enlarge the effective mass (m * ), defined as denotes the local band effective mass. Because the carrier mobility μ is proportional to 1=m Ã b , the mobility is nominally unaffected by N v . Even though the carrier mobility may be deteriorated by the inter-valley scattering, the enhancement in effective mass overcomes the loss in carrier mobility, resulting in an enhanced net ZT.
It should be noted that the multiple electronic bands can be activated only when the band energy offsets are small enough and comparable to several k B T, where k B is the Boltzmann constant. In p-type PbTe, the light (L) and heavy (Σ) bands in valence band are~0.15-0.20 eV, as shown in Fig. 3a, and the small energy offset makes the multiple valence bands easily converge. Typically, several superior p-type PbTe-based systems are achieved through manipulating electronic band structures, and typical examples are PbTe-PbSe, 62 PbTe-PbS, 61,63 PbTe-MgTe, 60 PbTe-SrTe 53,59 etc. In the valence bands of PbTe, valley degeneracy at L point is 4, while the valley degeneracy at Σ point is 12. 62 Therefore, when two valence bands jointly carry carriers at a close energy range, the effective mass m* can get a dramatic enhancement, as shown in Fig. 3c. It is readily seen that the effective masses in PbTe-PbSe and PbTe-MgTe systems are larger than the Pisarenko line that is derived from the single parabolic band (SPB) model with an effective mass m* = 0.3m e , evidencing a larger effective mass contributed by multiple valence bands.
DOS distortion and resonant state Another alternative mechanism to increase effective mass and Seebeck coefficient is to produce distortion energy level into the conduction or valence bands. The effect of locally increased DOS to improve Seebeck coefficient can be explained by the Mott expression as follows: 64 where q is unit charge, k B is Boltzmann constant, σ(E) denotes energy-dependent electrical conductivity, n denotes carrier density, μ is carrier mobility and T is working temperature. From Eq. (1), we can understand that the enhancement in Seebeck coefficient induced by resonant level arises from two mechanisms: (1) an excess density of states n(E) near Fermi level, and (2) increased energy-dependent μ(E), known as "resonant scattering". 73 The first mechanism is an intrinsic feature that is slightly temperature-dependent, and it will predominate with rising temperature. The second mechanism only works well at cryogenic temperature when the phonon-electron scattering is weak. When phonon-electron scattering dominates in elevated temperature range, "resonant scattering" plays an ignorable role in enhancing Seebeck coefficient. Therefore, resonant state is very sensitive to temperature and it can only play a significant role in enhancing Seebeck coefficient near and below room temperature range. 73 Fig. 3b schematically presents the effects of resonant level on the electronic energy distribution in PbTe matrix, which could cause a sharp increase of DOS near Fermi level. This phenomenon has been observed in p-type Tl-doped PbTe system, 51 and the Tl resonant level in valence band of PbTe indeed contributes to a distinct increase of Seebeck coefficient shown in Fig. 3c. When coming to carrier mobility issue, it should be noted that DOS distortion will cause a larger carrier mobility deterioration than that caused by band convergence. The DOS distortion enhances the total effective mass through producing larger local band effective mass, however, the band convergence enhances the total effective mass through increasing valley degeneracy number.
Band alignment between PbTe matrix and nanostructured second phases It is well-known that introducing defects in a host material not only intensifies phonon scattering, but also deteriorates carrier mobility. When nanostructured second phase exists in matrix, it is crucial to maintain high carrier mobility. One of the most effective approaches is to introduce the exotic phase with small band offset compared with matrix, namely band alignment. Actually, band alignment works very well in solar cell community to maintain carrier mobility, 74,75 which was firstly well elucidated in PbTe system. 65 Fig. 3d schematically presents the function of band alignment in p-type PbTe-SrTe. 65 Because the energy of valence band in SrTe is comparable to that of p-type PbTe, when second phase SrTe precipitates out in p-type PbTe matrix, the carrier mobility will be slightly deteriorated but phonon scattering is intensified by precipitates, which is considered as a typical example of well-known "phonon glass-electron crystal" proposed by Slack. 76 To quantitatively evaluate the contribution of band alignment to thermoelectric performance, one can calculate the ratio of carrier mobility to lattice thermal conductivity (μ/κ lat ), as shown in Fig. 3e. The large increase in μ/κ lat indicates that SrTe phases in p-type PbTe matrix scatter phonon stronger than carrier. The final enhancements in ZT values for p-type PbTe by synergistic approaches are well elucidated in Fig. 3f, which reveals that the methods of band convergence, resonant level and band alignment are pretty effective to enhance the electrical transport properties in p-type PbTe. More importantly, these approaches are also promising to be extensively applied in thermoelectric community as a general route.

MANIPULATIONS ON CONDUCTION BAND STRUCTURE IN N-TYPE PBTE
Flattening band structure In n-type PbTe, conduction band flattening can distinctly enlarge the effective mass, thereby optimizing Seebeck coefficient. 77 For an SPB, the local band effective mass m Ã b near the band edge is sensitive to the band shape. From the energy dispersion relationship, the m Ã b is defined by following equation: where ћ is reduced Planck constant, E(k) and k denotes energy dispersion function and wave vector, respectively, in the reciprocal space. The Eq. (2) demonstrates flat band shape will lead to high effective mass, as shown in Fig. 4a. Experimentally, elements S or Mn alloying in n-type PbTe could produce an obvious band flattening and thereby an enhancement in carrier effective mass in Fig. 4b. Typically, the ZT max values of n-type PbTe are largely boosted from~1.1 to~1.4 at 923 K and~1.6 at 723 K through flattening conduction bands. 66,77 In fact, as suggested by Kane band model, the energy dispersion is closely related to the bandgap E g , and broadening bandgap will flatten conduction band in PbTe following the relationship below: 69,78 It is noteworthy to point out that carrier mobility μ will also be inevitably deteriorated due to the relationship of μ ∝ 1/m b * after conduction band flattening. Therefore, it should be careful to balance carrier effective mass and carrier mobility. Typically, combination of lower effective mass and higher carrier mobility in Fig. 4c leads to a high ZT max value in I-doped PbTe (~1.4 at 723 K), which is 40% higher than that in La-doped PbTe (~1.0 at 723 K), 79 as shown in Fig. 4d.
Deep impurity level Impurity levels caused by adding group-IIIA elements (Ga, In, Tl) in PbTe are systematically investigated. 80,81 At room temperature, Ga and In elements can from a deep impurity level in the forbidden bandgap. On the contrary, Tl impurity level will enter into the valence band to form a resonant level in p-type PbTe, as we discussed above. 64 Fig. 5a schematically describes the deep impurity level in n-type PbTe produced by introducing Ga and In. Interestingly, Ga and In (+3) substitutions on Pb (+2) sites do not boost the carrier density of PbTe. At low temperature, the impurity levels produced by Ga and In could trap free electrons and work as a charge reservoir. 67,82 With rising temperature, the trapped electrons will be released from the deep impurity levels into conduction band, finally increasing the carrier density as shown in Fig. 5b. In the n-type PbTe with In doping, the carrier density undergoes an order of magnitude enhancement from~5 × 10 18 cm −3 at 300 K to~2.8 × 10 19 cm −3 at 773 K in Fig. 5c, exhibiting a kind of dynamic doping behavior. As for the PbTe with a single conduction band, this anomalous variation in carrier density is related to the changing charge states of Ga and In. With increasing temperature, the mixed charge states (+1 and +3) of Ga and In will change to fully +3 states. 80 Consequently, the dynamic doping extends ZT max in a larger temperature range, resulting in a large average ZT (ZT ave ). 67,82 The ZT ave over the entire working temperature range is important as it determines the thermoelectric conversion efficiency. 22 Similarly, Cr, 83 Fe, 84 Ti, 85 Sc 86 etc. can introduce impurity states in PbTe, however, these resonant impurity levels in conduction band are localized and contribute less to the carrier transports. 85

CHARGE TRANSPORTS IN PBTE-BASED SUPERLATTICE
Theoretical prediction shows that low-dimensional materials such as superlattice thin films and nanowires can significantly enhance thermoelectric performance. 7,9 Quantum confinement offers one special way to manipulate carrier transports because of the close relationship between electronic band structure and dimensionality. For the thermoelectric materials with quantum dots, the band offset and/or strain field between quantum dots and matrix could act as a confining potential. Therefore, the electronic density of states (dN/dE) has the shape of a Dirac delta function 87 where ε i is discrete energy levels and δ is the Dirac function. Figure  6a schematically shows the PbSeTe/PbTe superlattice structures prepared by molecular beam epitaxy (MBE). The field emission scanning electron microscope (FE-SEM) in Fig. 6b shows individual quantum dots in white massively distribute in planar arrays. Compared with the state-of-the-art PbTe bulks, the artificial superlattice structures could produce much larger Seebeck coefficient in Fig. 6c. The increased Seebeck coefficient arises from larger effective mass caused by the band structure modification due to quantum confinement. Figure 6d elucidates that the ZT max value can be enhanced through designing PbSeTe/ PbTe superlattice, indicating that reducing dimensionality of a given material is a potential possibility to enhance thermoelectric performance.

MANIPULATIONS ON PHONON TO REDUCE LATTICE THERMAL CONDUCTIVITY
Apart from electrical transport properties, low thermal conductivity is required to preserve large temperature gradient, which is a vital factor to realize high thermoelectric conversion efficiency. Heat transport is regarded as elementary vibrational motion, and the atom oscillation can be quantitatively evaluated using a quantum description "phonon". Phonon propagation and dispersion in crystalline materials are closely related to the mean free path (MFP), which ranges from atomic-scale, nano-scale to mesoscale. Since the MFP of PbTe is predominated in nano-scale, therefore, nanostructuring is widely applied to reduce lattice thermal conductivity in PbTe. 16,59 When the defect sizes are comparable to the MFP range, the phonon scattering can be dramatically intensified and therefore lead to a very low lattice thermal conductivity. 54,55 Based on this idea, artificially designing defects with all-scale hierarchical architectures are put forward to considerably reduce lattice thermal conductivity. Figure 7a is a schematic description of all-scale hierarchical structures including atomic-scale point defects, nanoscale precipitates and mesoscale grain boundaries. The contributions of reduced lattice thermal conductivity from these all-scale defects can be quantitatively estimated by: 88,89 where v is average phonon-group velocity, T is working temperature, θ D is Debye temperature, x = hω/κ B T, and τ is relaxation time. Here, the relaxation time is treated as an integration of various process including Umklapp process (τ U ), normal process (τ N ), grain boundary process (τ G ), nano-precipitate process (τ NP ) and point defect process (τ PD ): 89,90 When all above mechanisms contribute to the scattering of phonon propagation, the lattice thermal conductivity is accumulatively suppressed as shown in Fig. 7b. The lattice thermal conductivity can be described as κ l = 1/3Cvl, where C denotes heat capacity per unit volume, v is average speed of lattice vibration, l is mean free path. When l reaches interatomic distance, the lattice thermal conductivity can approach the theoretical  minimum κ min . It should be pointed out that the MFP range is related to the lattice-vibration frequency, which is temperaturedependent, implying that more scattering mechanisms may be involved in a given temperature.
Usually, nano-precipitates experience continuous steps of nucleation and growth, thus nano-precipitates range from several nanometers to hundreds of nanometers. Both eutectic phase diagram and spinodal decomposition phase diagram in Fig. 8 show two representative ways to obtain nanostructures B in A matrix. In Fig. 8a, the A and B will be homogeneously melting at high temperature, when B exceeds its solubility limit in A, it will precipitate out as nanostructures, which are well elucidated in PbTe-MTe (M = Mg, Ba, Sr, Ca and Cd) systems. 59,60,[91][92][93] Fig. 8b exhibits a miscibility gap where thermodynamic phase separation will occur through metastable nucleation and growth, which is called spinodal decomposition process. Spinodal decomposition can also produce different nano-precipitates, which is similar to that of eutectic process, as shown in the right column of Fig. 8.
Regardless eutectic phase diagram and spinodal decomposition, when A and B possess similar crystal structure and comparable lattice parameter, a coherent or semi-coherent interface between matrix and nano-precipitates will be formed, which will reduce the carrier scattering and facilitate carrier transport between phase interfaces. 94 The nano-precipitates with coherent or semicoherent interfaces have been widely observed in PbTe-PbQ (Q = Se, S) solid solutions. 61,63,95,96 Interestingly, the shapes and sizes of nano-precipitates in a given thermoelectric system are also sensitive to the processing parameters during synthesizing, such as cooling rate, annealing temperature and annealing time, etc. 53,97 Internal structure designs in PbTe-based thermoelectric materials Most of superior PbTe-based thermoelectric materials are obtained by reducing lattice thermal conductivity through scattering phonons via designing special microstructures. To match the high-performance p-type PbTe, the approach of Charge and phonon transport in PbTe-based thermoelectric materials Y Xiao and L-D Zhao designing microstructures in n-type PbTe is highly sought since conduction bands are less complex than its valence bands. Recently, a record high ZT max~1 .8 at 773 K was obtained in n-type PbTe-InSb with special nanostructures, 54 as shown in Fig. 9a. Both electrical and thermal transport properties are improved in nanostructured n-type PbTe. Seebeck coefficients are enhanced due to carriers filtering by the energy barrier caused by nanostructures, which also contribute to extremely low lattice thermal conductivity due to strong phonon scattering. 54 Sb doping in n-type PbTe (Fig. 9b) also exhibits very interesting behaviors. Sb-contained endotaxial nano-precipitates can be clearly observed at room temperature, but Sb will dissolve into PbTe matrix with rising temperature as shown in Fig. 9c. 55 The solubility limit of Sb in PbTe is temperature-dependent, which could produce the dynamic doping and phonon scattering, thus both electrical and thermal transport properties are improved. 55 The idea of all-scale hierarchical architectures is confirmed as an effective approach to enhance thermoelectric performance in ptype PbTe-SrTe through reducing lattice thermal conductivity. 59 The hierarchical architectures could also be realized in n-type PbTe just through introducing Cu, 98 as shown in Fig. 9d-f. The lattice thermal conductivity could be reduced as low as 0.38 Wm −1 K −1 in PbTe-5.5% Cu 2 Te. Interestingly, as shown in Fig. 9f, massive Cu interstitials would induce local lattice distortion. Moreover, apart from the hierarchical architecturing, these Cu interstitials are inclined to vibrate in a large displacement, which could further suppress the lattice thermal conductivity. Figure 10 shows the temperature-dependent lattice thermal conductivity and ZT values in several typical PbTe systems. From this comparison, it is obviously seen that the sample with all-scale defects possesses the lowest lattice thermal conductivity, which is comparable to the theoretical minimum value (κ min0 .36 Wm −1 K −1 ) in Fig. 10a. The ZT max value achieves~2.2 at 915 K in p-type PbTe-Na-4%SrTe and~1.8 at 773 K in n-type PbTe 0.998 I 0.002 -3%Sb, as shown in Fig. 10b. 55,59

PARADIGMS OF SYNERGIC STRATERGIES TO HIGH THERMOELECTRIC PERFORMANCE
Here we introduce an interesting high performance p-type PbTe-SrTe, in which the charge and phonon transports have been well manipulated using synergic strategies. 53 The DFT calculation in Fig. 11a reveals that SrTe can effectively enlarge the bandgap and concurrently converge two valence bands. As shown in Fig. 11b, the light conduction band keeps at a fixed position, while the offset between two valence bands is reduced with increasing SrTe fraction. The valence band convergence contributes to a large Seebeck coefficient, and the enlarged bandgap can suppress the bipolar diffusion. In addition, carrier mobility is slightly deteriorated by SrTe precipitates due to valence band alignment between SrTe and PbTe, as shown in Fig. 11c. Furthermore, a special non-equilibrium synthesis process is used in PbTe-SrTe with ice water quenching, which produces small grains with size ranging from 1 to 3 μm in Fig. 11d. Meanwhile, nano-precipitates are ubiquitously observed in Fig. 11e. Through using synergistically optimizing charge and phonon transports, a record high ZT max value~2.5 can be realized in PbTe-8%SrTe, as shown in Fig.  11f. The synergistic strategy is also successfully used in PbTe-MgTe, 60 PbS-CdS, 18 SnTe-CdTe, 99 (Ti, Zr)NiSn, 100 etc. These cases indicate that synergistically manipulating charge and phonon transports is promising to be extended to other thermoelectric materials.

SUMMARY AND OUTLOOK
In the past decades, the emergence of new strategies enriched the story of PbTe-based thermoelectric materials. A high performance can be realized in p-type PbTe-SrTe system through synergistically manipulating charge and phonon transports. One can see that most reports focused on p-type PbTe, but few reports on n-type PbTe, which is urgent to be developed to match its high performance p-type counterpart. The high performance n-type PbTe is expected to be realized through using following approaches: obtaining higher carrier mobility through reducing intrinsic vacancies; scattering all-length phonons through designing all-scale hierarchical architectures; facilitating charge transports through aligning conduction bands; suppressing the bipolar diffusion through enlarging bandgap; enhancing effective mass through tailoring conduction band structures.