Abstract
The central theme of valleytronics lies in the manipulation of valley degree of freedom for certain materials to fulfill specific application needs. While thermoelectric (TE) materials rely on carriers as working medium to absorb heat and generate power, their performance is intrinsically constrained by the energy valleys to which the carriers reside. Therefore, valleytronics can be extended to the TE field to include strategies for enhancing TE performance by engineering band structures. This review focuses on the recent progress in TE materials from the perspective of valleytronics, which includes three valley parameters (valley degeneracy, valley distortion, and valley anisotropy) and their influencing factors. The underlying physical mechanisms are discussed and related strategies that enable effective tuning of valley structures for better TE performance are presented and highlighted. It is shown that valleytronics could be a powerful tool in searching for promising TE materials, understanding complex mechanisms of carrier transport, and optimizing TE performance.
Introduction
The demand for renewable energy harvesting has been growing because of the limited fossil fuels and increasing worldwide energy consumption. Thermoelectric (TE) materials, which have the capability of converting heat directly into electricity under a temperature gradient, have been regarded as an alternative option to alleviate energy shortage.^{1} Though TE devices have already been applied in deep space exploration and solid state cooling,^{2,3} the relatively low energy conversion efficiency limits their wide commercialization,^{4} which is mainly constrained by the performance of TE materials as characterized by the dimensionless figure of merit, zT = α^{2}σT/(κ_{e} + κ_{L}), where α is the Seebeck coefficient, σ is the electrical conductivity, κ_{e} and κ_{L} are, respectively, the electronic and lattice contributions to the total thermal conductivity κ, and T is the absolute temperature.^{5} Since all three physical properties (α, σ, and κ) are carrier concentration dependent, zT could reach its maximum value at an optimized carrier concentration. zT_{max} is determined by a combination of intrinsic physical parameters as well as the temperature, all of which integrated into a term called the TE quality factor, B. Higher quality factor B leads to higher zT_{max} at a fixed temperature. With a single parabolic band model in the nondegenerate limit, B could be expressed as:^{6,7}
where k_{B} is the Boltzmann constant, ħ is the reduced Plank constant, e is the electron charge, μ_{0} is the carrier mobility in the nondegenerate limit, m_{b}^{*} is the effective mass for a single valley.
Recent TE research demonstrates outstanding progress with zT values exceeding the barrier of unity easily and sometimes even above 2, due to inspirations by the lowdimensional design for TE materials^{8,9} and the PhononGlassElectronCrystal concept^{10} proposed in 1990s.^{11} The rational design strategies for zT enhancement, together with the underlying mechanisms toward high TE performance, can be classified into two categories. One is the reduction of κ_{L} and the other is the increase of TE power factor (PF = α^{2}σ).^{12,13} The current realization of large zT enhancement (zT > 2) is mainly due to the significantly reduced κ_{L}, e.g., with allscale hierarchical phonon scattering,^{14} using materials with strong anharmonicity,^{15} and discovering liquidlike phonon behavior.^{16} Some excellent reviews focusing on the thermal transport aspects of TE study^{17,18} as well as several good comprehensive TE reviews introducing the synergistic design strategies of TE materials have been published in recent years.^{1,11,19,20}
Nevertheless, there exists a threshold for zT improvement from reducing the thermal conductivity, which is limited to a theoretical minimum (amorphous limit).^{21} In contrast there is no theoretical bound to the PF, which leads to the boundless nature of zT.^{22} TE materials are mostly heavily doped semiconductors. It is only the bottom part of conduction band or the top part of valence band that determines the electrical transport properties. This specific dispersive region in the electronic band structure is named as energy valley or carrier pocket due to the morphology of fermi surface. As the α and σ are highly coupled physical parameters, adjusting the electron chemical potential with changing the doping level would only get a limited maximum PF.^{23} Therefore, it is necessary to alter the valley structure in order to improve the TE performance.
The term “Valleytronics” relates to the approaches that adjust the parameters describing the conduction valleys.^{24,25} This idea, similar to spintronics, was initially applied to construct new quantum computation devices, which tune the valley polarization to store and carry information.^{26,27} Subsequent applications of this concept include designing new optoelectronic devices and ferrovalley materials (Fig. 1).^{26,27} Here, this concept is extended to the TE field to include strategies for zT enhancement concerning the modification of valley structures, which is the main aim of this review. There are three valley parameters that allow for modification (seen in Fig. 2): the number of valleys contributing to the electronic transport (valley degeneracy), the distortion of density of states (DOS) by resonant dopants (valley distortion), and the exact shape of valleys which can be expressed as the effective mass of carriers (valley anisotropy). The approaches that enable effective tuning of valley structures for better TE performance are presented and discussed below.
Valley degeneracy
Valley degeneracy (N_{v}) is a quantitative description for the number of valleys that contribute to the carrier transport.^{28} Compared to the single valley case, TE materials with multiple valleys possess more states for carriers to occupy and contribute to transport. Thus, the valley degeneracy N_{v} should be taken into account such that the DOS effective mass (m_{d}^{*} = N_{v}^{2/3}m_{b}^{*} assuming each valley has the same m_{b}^{*} to the single valley case) is used in Eq. (1), instead of m_{b}^{*}. To better elucidate the role of N_{v} on the TE performance, it is necessary to unfold the mobility term and rewrite Eq. (1) under the deformation potential theory when acoustic phonon scattering dominants in the scattering process,^{29}
where C_{l} is a combination of elastic constants, m_{I}^{*} is the inertial effective mass, and Ξ is the deformation potential. For an isotropic band structure, m_{I}^{*} = m_{b}^{*}.
Larger N_{v} increases DOS effective mass m_{d}^{*}, leading to higher α at the same carrier concentration, meanwhile it has no influence on the m_{I}^{*} that relates to carrier mobility (μ). Therefore, large N_{v} generally leads to a high PF and hence good TE performance.^{30} However, it has to be pointed out that in the multivalley case the scattering of carriers between different valleys (intervalley phonon scattering) could bring detrimental effect on μ and leads to a deviation from the estimated zT value based on the model mentioned above.^{31} As studies on the influence of the intervalley scattering on TE performance are scarce, no practical criterion has been proposed so far to determine if a certain TE material with multivalley band structure would suffer from intervalley scattering. To our knowledge, optimistic view could be normally held that such influence, detrimental though it is, is not significantly influential for multivalley TE materials, as seen from the study on Mg_{2}(Si, Sn),^{32,33} SiGe alloys,^{34} and other good TE materials.^{35,36,37} Interested readers can refer to these works^{31,38} for more detailed discussions on intervalley scattering.
Generally, high valley degeneracy could be found in a crystal with high symmetry when the valley extrema is located at nonΓ lowsymmetry points in the Brillouin zone.^{11} Higher symmetry for a crystal in real space results in more equivalent valleys in reciprocal space, and vice versa. For instance, conventional TE materials like Bi_{2}Te_{3},^{39,40} PbTe,^{41} ntype SiGe,^{42} as well as the newly developed good TE materials like ntype Mg_{2}Si,^{43} ptype HalfHeusler (HH),^{44,45} all have high crystal symmetry and thus large N_{v}. Another case for high N_{v} is when the extrema of multiple valleys (located at different points in the Brillouin zone or from different orbitals) have comparable energy within a few k_{B}T, which is also called orbital degeneracy. Many kinds of TE materials benefit from this kind of degeneracy including ntype CoSb_{3},^{46} ptype SnSe,^{37} ntype Mg_{3}Sb_{2},^{47} etc. In CoSb_{3} skutterudites, the conduction band minimum is located at the center of the Brillouin zone (Γ point), which has an N_{v} of 3. Density functional theory (DFT) calculations combined with doping study and optical measurements reveal the existence of another set of valleys located at a lowsymmetry point in the Brillouin zone, which has a high N_{v} of 12.^{46,48} As the edge of these valleys lies about 0.11 eV above the Γ valley minimum, their contributions to the electric transport becomes more apparent at high temperatures, at which these additional valleys converge to the conduction band minimum. The electronic origin of high TE performance was previously explained by the nonparabolic feature of the primary conduction bands at large carrier concentrations,^{49} while the large valley degeneracy of secondary band (N_{v} = 12) as well as its convergence to the primary conduction bands at high temperatures (resulting in N_{v} = 15) is indeed the reason for high PF, as confirmed by theoretical calculations and experimental measurements. This could deepen the understanding of the mechanism for electronic transport in intrinsic CoSb_{3} and guide for optimization strategies.
Though it is essential to understand the contribution of high valley degeneracy to high PF and consequently enhanced zT, it is of great practical value to seek ways to realize high N_{v} for wellestablished TE materials or to discover potentially new TE materials, using N_{v} as a screening criterion, either by tuning temperature or alloy composition, as discussed below.
Effect of temperature and alloying
Different temperature dependence of multivalleys (with the valley edges located at different points in the Brillouin zone) might lead to valley convergence, thus resulting in increased N_{v} and enhanced PF. PbTe and similar IV–VI rocksalt semiconductors (like PbSe and SnTe) are among the limited class of TE materials that possess such band features. The valence band structure of PbTe consists of two distinct valleys with different temperature dependence. The higher one which determines the energy gap at room temperature is located at L point (N_{v} = 4) in the Brillouin zone, while the lower and heavier one is along the ΓK line, labeled as Σ (N_{v} = 12),^{50} as shown in Fig. 3a. The energy position of L valley lowers down with increasing temperature, while the energy position of Σ valley changes little, as seen in Fig. 3b. These two valleys become converged at high temperature, leading to enhanced PF as well as higher optimized carrier concentration. Though such temperaturedependent energy valley convergence was first proposed in early 1960s,^{51} huge success on experiments for TE investigations was reported 40 years later by Pei et al.,^{52} which inspires a new wave of research on both the optimizations of TE performance and the investigation of intrinsic mechanisms related to this valley convergence behavior.
To date, the study of the mechanisms on temperatureinduced valley convergence is still at the stage of rationalization and requires further efforts.^{12} Lattice thermal expansion is always regarded as the first reason leading to the increase of electronic band gap with increasing temperature (the lowering L valleys in energy position, seen in Fig. 3b).^{13} Moreover, a thermaldisordered lead chalcogenide structure (Pb atoms are moved further offcentered compared to the chalcogenide atoms with increasing T) was proposed and adopted to fully explain the temperaturedependent band convergence of lead chalcogenides.^{53} Theoretical calculation of this thermaldisordered structure based on ab initio molecular dynamics has been performed and finds that the convergence temperature for PbTe is about 450 K,^{54} which is consistent with the former experimental results from electrical and optical measurement with photon energies less than the fundamental gap.^{51} However, recent free carrier absorption measurements,^{55} temperaturedependent angleresolved photoemission spectroscopy study,^{55,56} and theoretical calculation^{57} (considering both lattice thermal expansion and electron–phonon interaction) reveal that the actual convergence temperature is above 700 K, much higher than the previous results. Moreover, as the Fermi energy in heavily doped TE semiconductors usually lies deep in the valence band, the secondary valence band should have considerable contribution to the carrier transport and lead to enhanced PF, even before the valley convergence actually happens.
Besides valley convergence due to different temperature dependence of multivalleys, valley convergence could also result from optimized valley positions through alloying. Many experimental works were performed with alloying to adjust the convergence temperature to desired range. The band structure modifications of PbTebased materials through isovalent substitutions by Mn,^{58,59} IIB,^{60} and IIA elements^{61} were originally studied before 1980s,^{62,63,64} and have been revisited with intensive investigation for TE applications. Compared to lead chalcogenides, the strategy of valley convergence through alloying plays a more important role in the zT enhancement for SnTe, since the room temperature energy offset between L and Σ valleys for SnTe is much higher (~0.3 eV for SnTe, ~0.1 eV for PbTe).^{65} Hence, more amounts of dopants or more effective dopants are needed to lower down the valley convergence temperature. It is reported that Mn has relatively high solubility in SnTe (>9 mol%), and that the valley convergence temperature could be adjusted to below 450 K for SnTe when doped with 9 mol% Mn, leading to 40% increase for PF compared with pristine SnTe.^{59}
Forming solid solutions between the sister compounds with inverse valley configurations in energy is an effective approach to realize valley convergence. As composition varies, the energies of converging valleys change in opposite direction such that valley convergence could be achieved at a specific composition, given that the required composition is within the solubility limit.
Ntype Mg_{2}XMg_{2}Sn (X = Si, Ge) solid solutions are typical examples illustrating how valley degeneracy can be increased through alloying.^{66,67} In Mg_{2}X, above the conduction band minimum, there exists another subband which usually does not contribute to the transport due to large splitting energy from the band minimum (−0.4 eV for Mg_{2}Si,^{68} 0.16 eV for Mg_{2}Sn^{69}). As the two conduction band edges (one heavier and another lighter) are reversely positioned in Mg_{2}Sn, compared with that in Mg_{2}Si and Mg_{2}Ge, it is possible to converge these two band edges (with splitting energy ≈0) through alloying, as revealed in Fig. 4a.^{70} A PF as high as ~50 μW cm^{−1} K^{−2} is reached in both quasibinary and quasiternary solid solutions of Mg_{2}XMg_{2}Sn when the alloy compositions are tuned within the convergence range.
Some solid solutions of III–V materials also exhibit similar band crossing features, though this class of materials is less favorable in the TE research field due to their high thermal conductivity.^{71} For example, InP and GaP have similar band features with three band edges at L, Γ, and X points, valley degeneracy for each band being 4, 1, and 3; however, the conduction band minimum of InP and GaP is located at Γ point and X point, respectively. In the ntype Ga_{ x }In_{1−x}P solid solutions, as the Ga concentration increases from 0 to 1 the conduction band minimum changes from Γ to L point at x = 0.67 and then from L to X point at x = 0.77, as seen in Fig. 4b. All three bands (L, Γ, and X) cross over at x ≈ 0.77, leading to N_{v} = 8, higher than the valley degeneracy of each parent material.^{72} zT value close to 1.1 at 873 K is predicted for heavily doped ntype Ga_{0.74}In_{0.26}P.^{73}
Symmetry change
While the electronic band structures are closely related to the symmetry of the crystal structures as mentioned above, it is very difficult to tune the system symmetry for a given TE material. Common methods involve phase transitions by tuning either composition or temperature in certain materials systems. For instance, the compositional changeinduced phase transition from a rhombohedral structure (Bi_{2}Se_{3}) to an orthorhombic one (BiSbSe_{3}) in ntype Bi_{2−x}Sb_{ x }Se_{3} would result in a considerable modification on the band structure: the original conduction band minimum moves from the center of the Brillouin zone (N_{v} = 1) to a low symmetry point (N_{v} = 2, also heavier m_{b}^{*}), leading to a large increase of m_{d}^{*} (as seen in Fig. 5a), thus improving the PF and achieving a high zT ~1 at 800 K.^{74}
Recent study on the band structure of tetragonal chalcopyrites opens a new avenue to increase the N_{v} for the TE materials with low crystal symmetry (thus intrinsically low N_{v}), via changing the system symmetry from noncubic to pseudocubic structure.^{75} The crystal structure of tetragonal chalcopyrites can be regarded as variants from the cubic zinc blende structure with N_{v} = 3 (contributed by Γ_{5V} (N_{v} = 2) and Γ_{4V} (N_{v} = 1)). Ordered cation substitution with two kinds of metal atoms further lowers the symmetry leading to the chalcopyrite structure type A^{I}B^{III}X_{2}^{VI} as well as the observed crystalfield splitting, ΔCF = Γ_{5V} − Γ_{4V}, which lowers N_{v}. Theoretical calculation reveals that ΔCF could be tuned to 0 when the crystal structure parameter η is equal to 1 (η = c/2a, where c and a are lattice parameters for tetragonal chalcopyrites).^{75} This conclusion can be easily understood if putting two unit cells together as seen in Fig. 5b: the cation sublattice shows nearly cubic framework once η ≈ 1. And this longrange cubiclike structure regains the threefold valley degeneracy^{76} and keeps the localized noncubic lattice distortion simultaneously, leading to an enhanced TE performance. Through rational alloying, the lattice parameters could be tuned into a desired range and this unityη design strategy (for tetragonal chalcopyrites with η ≠ 1) could be realized experimentally, like the case in Cu_{0.875}Ag_{0.125}InTe_{2}.^{75}
Size effect
First introduced in the 1990s, the idea of using lowdimensional nanostructured TE materials to realize extraordinary enhancement of zT has attracted numerous attention, due to both promising theoretical predictions^{8} and subsequent proofofprinciple experiments.^{77,78} When the size of TE materials is reduced to a critical length scale, e.g., in an ultrathin film of only a few nanometers thickness, the carrier transport in the direction perpendicular to the film is restricted and a quantum well in that direction is thus built. Due to the effect of quantum confinement, the original continuous energy dependence of DOS (DOS~E^{1/2}) near the band edge is altered to a steplike relationship,^{79} which helps gain a large α and improve the PF.^{9} These effects would surely influence conduction valleys for TE materials. While the influence of lowdimensional nanostructures on the reduction of κ is relatively intuitive, the exact inner mechanisms for PF enhancement introduced by the lowdimensional structures are still controversial (e.g., a nonmonotonous relationship between PF and system size was also proposed and qualitatively supported by some experiments).^{80} There have been some comprehensive reviews summarizing the current works on this aspect.^{9,81}
For better focus, we only mention here the effects of lowdimensional nanostructures on the valley specifics. In the twodimensional superlattice, this approach is also called the carrier pocket engineering as proposed in the theoretical study of ntype GaAs/AlAs 2D superlattices by Koga et al.^{82} The early work on 2D superlattices only consider the zT for the quantum confined 2D nanosheet^{77} and neglect the TE transport within the barriers (the layers to prevent the tunneling effect between neighboring 2D nanosheets and hence form potential wells). The optimization approaches (carrier pocket engineering) for GaAs/AlAs 2D superlattice step further and target on the whole system, where both GaAs and AlAs layers serve as the layers for the quantum wells and they also act as the barrier layers for each other, as shown in Fig. 6a. The energy configuration of the three valleys for bulk GaAs and AlAs can be seen in Fig. 6b. Such an reversed configuration makes it possible to reach valley convergence through forming solid solutions.^{83} However, the PF of Ga_{1−x}Al_{ x }As solid solutions with composition in the valley convergence condition suffer greatly from the mobility reduction caused by intervalley scattering.^{38} On the contrary, in the GaAs/AlAs 2D superlattice, due to the quantum confinement, the electrons in the Γ and L valleys are confined in the GaAs layers, whereas the electrons in the X valleys are confined in the AlAs layers. Since the electron wave functions for the Γ and L valleys are spatially separated from those for the X valleys, the intervalley scattering between the X valleys and the (Γ, L) valleys should be reduced and high mobility is expected to be maintained.^{84}
More importantly, by controlling the width of the quantum well (tuning the layer thickness of GaAs and AlAs, noted as d_{GaAs} and d_{AlAs}), the relative positions of three conduction valleys change in energy and effectively converge when the width is around 20 Å, as seen in Fig. 6c. As revealed by theoretical calculation, the zT of the whole system is increased by a factor of four compared with the superlattice case without the contribution of the X and L valleys.^{82}
Resonant levels
Resonant levels refer to certain impurity energy levels in the host band structure that will cause a deltafunctionlike distortion to the DOS of the host material (the background DOS).^{85,86} Different from the routine doping, the energy states of the resonant dopants (E_{D}) are located well above the conduction band minimum or right below the valence band maximum, coinciding in energy with the extended states (the electronic states responsible for electronic transports).^{87} As the resonant states has same energy with the extended states of conduction band, the resonance of these two will build up two other extended states of slightly different energies; further resonance of the developed extended states and the original ones will continuously induce more distortions, resulting in the deltalike shape with certain energy width Γ around the resonant energy level in the final DOS, as seen in Fig. 7.^{86}
TE transport properties benefit from resonant levels on two aspects, as revealed by the Mott’s equation: \(\alpha = \frac{{\pi ^2}}{3}\frac{{k_{\mathrm{B}}}}{e}k_{\mathrm{B}}T\left\{ {\frac{1}{n}\frac{{dn(E)}}{{dE}} + \frac{1}{\mu }\frac{{d\mu (E)}}{{dE}}} \right\}_{E = E_{\mathrm{F}}}\).^{86,88} On the one hand, the induced narrow peak in the DOS is favorable for achieving a large α due to stronger energy dependence of DOS. On the other hand, the resonant impurities also induce special carrier scattering process, called as resonant scattering. Under the relaxation time approximation, the mobility is expressed as μ = e˖τ/m_{I}^{*}, where τ is the relaxation time. For resonant scattering process, the τ can be expressed as τ = τ_{0}(1 + (2(E−E_{D})/Γ)^{2}), with τ_{ 0 } the minimum value of τ reached at E_{D}.^{89} The relaxation time is highly dependent on the energy difference between the carriers and the resonant levels.^{90} Therefore, the effect of resonant scattering is similar to the energy filtering effect^{91} in nanocomposites that helps increase the Seebeck coefficient. However, as the resonant scattering tends to diminish with increasing temperature, compared with the phonon scattering, such “energy filtering” is only effective at low temperature. Therefore, the enhanced PF in TE semiconductors involving resonant doping should be mainly ascribed to the induced deltalike distortion of DOS.^{92}
The typical example that demonstrates how TE performance could benefit from the resonant doping includes the Tldoped PbTe^{92} and Sndoped Bi_{2}Te_{3}.^{93} For Tldoped PbTe, optical measurements reveal that there exist the Tl impurity levels about 0.06 eV below the valence band maximum,^{94} which is a favorable energy position to form resonant levels. Theoretical band structure calculations confirm that the Tl impurity states hybridizing with Te 5p states create a distinctive “hump” in DOS, the width of which is relatively broad (~0.2 eV).^{90} Experimental work by Heremans et al. found the enhancement of Seebeck coefficient, compared with the Nadoped PbTe without resonant state formation at given hole concentrations, and a peak zT value of 1.5 in Pb_{0.98}Tl_{0.02}Te was attained.^{92} In the case of Sndoped Bi_{2}Te_{3}, the experimental proof of resonant levels was from Shubnikovde Haas measurements combined with galvanomagnetic and thermomagnetic properties measurements. Strongly increased and nearly pinned Seebeck coefficient beyond the Pisarenko prediction was found for Sndoped ptype single crystalline Bi_{2}Te_{3} at 300 K.^{93} As the mobility decrease is relatively moderate, overall PF for Sndoped Bi_{2}Te_{3} could be enhanced via this approach. In the framework of Boltzmann transport theory with the simple parabolic band model, the Seebeck coefficient enhancement at a given carrier concentration due to resonant levels can be intuitively understood by the increased m_{b}^{*},^{30} which are also found from the theoretically calculated band structure diagrams.^{90,95} The influence of m_{b}^{*} on TE performance is discussed in more details in the next section. In short, the strategy by introducing resonant levels in TE materials to obtain better valley structures would inevitably bring a tradeoff on the mobility, as larger m_{b}^{*} not only benefits for α but also lowers μ.^{96} The exact physical picture is yet more complicated and rather different from the simple consideration mentioned above. Nevertheless, further study of resonant levels as an approach of manipulating valley degree of freedom to optimize PF is worth conducting.
In addition to the two examples above, extensive experimental work concerning the resonant levels in semiconductors^{97,98,99} has been attempted for different classes of TE materials in the aim of enhancing the PF, such as Indoped^{100} or Codoped^{101} SnTe, Crdoped PbTe,^{102} Aldoped PbSe,^{103} Pbdoped BiCuSeO,^{104} etc.
Though various resonant levels can be introduced in TE materials, Seebeck coefficient may not be increased if the resonant states due to impurities are too localized and thus do not contribute to the TE transport, like in the case of ntype Tidoped PbTe.^{105} Besides, since the Fermi level determined from the resonant dopant alone may not be optimized, additional normal dopants are needed to adjust the carrier concentration (if possible) to optimize zT.^{90} Thus a broad “hump” in DOS is required to allow Fermi level adjustment by codoping. In this regard, the resonant levels induced by transition metals with their d or f orbitals coupling with the matrix states are less favorable, compared to the one with coupling from the s or p orbitals, as the former usually forms very narrow DOS peaks.
Valley effective mass and band anisotropy
The effective mass, defined as a secondorder tensor, can be expressed as a scalar m^{*} with the assumption of a single parabolic band which results in a spherical Fermi surface in kspace. Tuning the flatness of a single valley (adjusting m_{b}^{*}) offers us another opportunity to optimize TE properties in the framework of valleytronics. Normally, lower m_{b}^{*} is more favorable for TE materials, which could be seen from Eq. (2) and shown by both experiments^{106,107,108} and theory.^{11,31} The study on ptype (V,Nb)FeSbbased materials, a kind of HH alloys with excellent TE performance in the high temperature range (T > 1000 K),^{44,109} serves as a good example.^{106} DFT calculations show that VFeSb possesses a flatter valence band than that of NbFeSb.^{106,110} Therefore, the m_{b}^{*} of (V,Nb)FeSb solid solutions is expected to become lower with increasing Nb content. Experimental work indeed confirms that NbFeSb has much lower m_{b}^{*} (1.6 m_{e}) than that of (V_{0.6}Nb_{0.4})FeSb (2.5 m_{e}), as shown in Fig. 8a.^{111} Furthermore, in this HH material system, the decreased m_{b}^{*} also helps lower the optimal carrier concentration, which was too high to reach in (V_{0.6}Nb_{0.4})FeSb within the solid solubility of Ti dopant (Fig. 8b). The hole mobility gets doubled and contributes to a high PF (~45 μW cm^{−1} K^{−2} at 900 K) as well as an enhanced zT (~1.1) in Tidoped NbFeSb.^{106}
Apart from forming solid solutions to realize lower m_{b}^{*} in TE materials, doping would also possibly influence the m_{b}^{*} for a given TE material. Such a situation normally arises in cation substitution for ntype semiconductors or anion substitution for ptype semiconductors, where the resultant m_{b}^{*} would usually be increased in proportion to doping.^{112,113} For example, Pei’s work on Ladoped ntype PbTe clearly demonstrates that the cation substitution strategy for ntype semiconductors increases the m_{b}^{*} of carriers, and thus, unfortunately, degrades the PF.^{107} Based on a simple Kane band model, the m_{b}^{*} for Ladoped PbTe is found to be 20% larger than that for Idoped ntype PbTe (anion substitution), which remains almost unchanged compared to PbTe, within all the temperature range considered. The electron mobility for Ladoped PbTe is so severely decreased that the optimized maximum zT is 20% lower than that of the Idoped one.^{107} Similar case can also be found in Li and Gadoped ptype Mg_{2}(Si,Sn), where the Gadoped ones possess much higher m_{b}^{*} yet worse PF than the Lidoped ones.^{108}
The enhancement for TE performance through changing the value of m_{b}^{*} is limited due to the counteractive effects on α and μ.^{107} Nevertheless, these effects could be decoupled when the valley anisotropy is introduced. In the real circumstances, most semiconductors have anisotropic bands around the valley extrema. To the first order, the Fermi surface of these bands are often approximated as ellipsoids, characterized with effective masses along radial direction (m_{⊥}^{*}) and longitudinal direction (m_{}^{*}). The m_{d}^{*} for a single valley m_{b}^{*} is the geometric average of effective masses along each principle direction: m_{b}^{*} = (m_{}^{*}m_{ ⊥ }^{*2})^{1/3} that is related to the Seebeck coefficient as well as the carrier concentration. While the inertial effective mass m_{I}^{*} is defined as the harmonic average m_{I}^{*} = 3(m_{}^{*−1} + 2 m_{⊥}^{*−1})^{−1} and related to the electronic transport behavior through mobility.^{114} Generally, the valley anisotropy is quantitatively defined as K = m_{}^{*}/ m_{⊥}^{*}. And a Kvalue larger than 10 is necessary to bring significant difference to the overall TE performance, as shown in Fig. 9.
The benefit of valley anisotropy can be understood in two aspects: on the one hand, with the increased degree of valley anisotropy (larger Kvalue), the Fermi surface in the Brillouin zone tends to elongate in certain directions to get both light and heavy channels, and the charge carriers would prefer to drift in the direction with lighter effective mass, gaining carrier mobility compared to that in an isotropic band. On the other hand, the original ellipsoid morphology turns to either “needlelike” or “plateshaped”, which contributes lowdimensional transport feature and results in large α,^{115,116} like the case in 2D superlattice quantum wells^{8} or 1D nanowires.^{117} Consequently, the anisotropic valley offers another possibility for decoupling μ and α. The comparison between ntype PbTe and PbSe exemplifies the advantage of valley anisotropy to some extent. While the m_{d}^{*} for both compounds is similar, the conduction valley of PbTe (K ≈ 8) is more anisotropic than that of PbSe (K ≈ 2), resulting in a higher μ and PF.^{31} Other TE materials that obviously benefits from this anisotropic valley morphology include ntype layered cobaltites,^{118} ntype rocksalt IVVI compounds,^{119} etc.
To date, the efforts of manipulating the valley anisotropy to realize an enhanced TE performance are still limited to theoretical calculations. For example, in Fe_{2}VAl full Heulser semiconductor, the conduction valley at X point is mainly contributed by the V e_{g} states with dispersive band feature, above which exists another band with high anisotropy contributed by the Fe e_{g} states. First principles calculations find it possible to move the dispersive band at X point upper through substitutions of V and Al atoms, thus enabling the highly anisotropic valley dominate the electron transport.^{115} The PF values of 4–5 times larger than that in classical TE materials at room temperature are predicted for Fe_{2}TiSn (K ~87) and Fe_{2}TiSi (K ~450) with good thermodynamic stability. AgBiSe_{2}, known for intrinsically low thermal conductivity and multiple phase transitions with changing temperature,^{120} was found to have large valence valley anisotropy with onedimensional “plateshaped” carrier pockets^{121} based on recent DFT calculation on the room temperature phase of hexagonal AgBiSe_{2}, for which a high zT value of 0.7 at 300 K is also predicted. Further, high valley anisotropy can also be applied as a guideline in highthroughput calculation screening for new potential TE materials.^{114}
Summary and outlook
We have reviewed the recent progress in the research of TE materials from the perspective of valleytronics, which focuses on strategies to improve the TE power factor in a more intrinsic manner. While a lot of research on TE materials is focused on decreasing κ_{L} to the theoretical minimum,^{14,122,123} this review highlights the importance of engineering the valley degree of freedom to further improve TE performance. Valley convergence, resonant levels, and valley anisotropy are concluded as effective ways to engineer band structure in TE materials. Compared with the other two, valley convergence seems to be more practical and controllable for various materials systems, if intervalley scattering is not severe. For the resonant levels, special attention should be paid to the mobility reduction besides the enhancement of Seebeck coefficient, in order to have a complete view about whether certain resonant dopants benefit to the overall TE performance. The effect of valley anisotropy, as well as how to tune such a band structure feature, has been less studied, and most of the relevant work are theoretical calculations, which either help further understanding of some existing good TE materials or predict new materials with promising TE properties.
Apart from the strategies mentioned above, external field tuning, which is usually applied to manipulate the valley degeneracy in 2D transition metal dichalcogenides, is yet rarely used to modify TE materials.^{25,26,27} Recent study shows that twodimensional effect with enhanced Seebeck coefficient has been realized under high vertical electrical field in oxide TE materials.^{124} The introduction of magnetic nanoparticles in TE materials also favors the zT, indicating that the inner built magnetic field would help manipulate the electron transport to obtain higher TE performance.^{125} On the other hand, as the manipulation of all the three valley parameters are intrinsically constrained by the crystal structures (symmetry), extraordinary strategies that enable altering of the crystal symmetry, such as “directional distortion”,^{126} would be especially helpful, though also quite difficult. Deep understanding on the correlations among chemical boding, crystal symmetry, and energy valley structure, as well as the response of the valley parameters under various external fields, is required in order to realize meaningful modification. In all, more revolutionary discovery could be anticipated in the application of valleytronics to TE materials.
Valleytronics, which study the influence and manipulation of the band (or valley) morphology on the TE performance, could be a powerful tool in terms of searching new promising TE materials, understanding complex mechanism of carrier transport, and optimizing TE performance.
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Acknowledgements
This work was supported by the National Science Fund for Distinguished Young Scholars (No. 51725102) and Natural Science Foundation of China (Nos. 11574267 and 51571177). Y.T. acknowledges the funding support from Swiss Federal Office of Energy (contract number: SI/50131001).
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J.Z.X. and Y.L.T. drafted this review. Y.T.L. offered important idea of the figures. X.B.Z. and H.G.P. gave essential criticisms and suggestions. T.J.Z. created the central idea of this review, constructed the framework, and revised it critically. All authors contributed to the writing of the manuscript.
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Xin, J., Tang, Y., Liu, Y. et al. Valleytronics in thermoelectric materials. npj Quant Mater 3, 9 (2018). https://doi.org/10.1038/s4153501800836
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