Valleytronics in thermoelectric materials

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 = α2σT/(κe + κL), where α is the Seebeck coefficient, σ is the electrical conductivity, κe and κL 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. zTmax 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 zTmax at a fixed temperature. With a single parabolic band model in the nondegenerate limit, B could be expressed as:6,7

$$B = \frac{{2k_{\mathrm{B}}^2}}{{e\hbar ^3}}\left( {\frac{{k_{\mathrm{B}}m_{\mathrm{b}}^ \ast }}{{2\pi }}} \right)^{3/2}\frac{{\mu _0}}{{\kappa _{\mathrm{L}}}}T^{5/2},$$
(1)

where kB is the Boltzmann constant, ħ is the reduced Plank constant, e is the electron charge, μ0 is the carrier mobility in the nondegenerate limit, mb* 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 low-dimensional design for TE materials8,9 and the Phonon-Glass-Electron-Crystal concept10 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 κL and the other is the increase of TE power factor (PF = α2σ).12,13 The current realization of large zT enhancement (zT > 2) is mainly due to the significantly reduced κL, e.g., with all-scale hierarchical phonon scattering,14 using materials with strong anharmonicity,15 and discovering liquid-like phonon behavior.16 Some excellent reviews focusing on the thermal transport aspects of TE study17,18 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.

Fig. 1
figure1

Schematics for the applications of valleytronics on various research fields

Fig. 2
figure2

Three valley parameters in band structure that can be tuned in thermoelectric materials in the view of valleytronics. Schematics in the circles are typical illustrations for these valley parameters

Valley degeneracy

Valley degeneracy (Nv) 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 Nv should be taken into account such that the DOS effective mass (md* = Nv2/3mb* assuming each valley has the same mb* to the single valley case) is used in Eq. (1), instead of mb*. To better elucidate the role of Nv 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

$$B = T\frac{{2k_{\mathrm{B}}^2\hbar }}{{3\pi }}\frac{{C_{\mathrm{l}}N_{\mathrm{v}}}}{{m_{\mathrm{I}}^ \ast \Xi ^2\kappa _{\mathrm{L}}}},$$
(2)

where Cl is a combination of elastic constants, mI* is the inertial effective mass, and Ξ is the deformation potential. For an isotropic band structure, mI* = mb*.

Larger Nv increases DOS effective mass md*, leading to higher α at the same carrier concentration, meanwhile it has no influence on the mI* that relates to carrier mobility (μ). Therefore, large Nv generally leads to a high PF and hence good TE performance.30 However, it has to be pointed out that in the multi-valley case the scattering of carriers between different valleys (inter-valley 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 inter-valley scattering on TE performance are scarce, no practical criterion has been proposed so far to determine if a certain TE material with multi-valley band structure would suffer from inter-valley scattering. To our knowledge, optimistic view could be normally held that such influence, detrimental though it is, is not significantly influential for multi-valley TE materials, as seen from the study on Mg2(Si, Sn),32,33 SiGe alloys,34 and other good TE materials.35,36,37 Interested readers can refer to these works31,38 for more detailed discussions on inter-valley scattering.

Generally, high valley degeneracy could be found in a crystal with high symmetry when the valley extrema is located at non-Γ low-symmetry 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 Bi2Te3,39,40 PbTe,41 n-type SiGe,42 as well as the newly developed good TE materials like n-type Mg2Si,43 p-type Half-Heusler (HH),44,45 all have high crystal symmetry and thus large Nv. Another case for high Nv 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 kBT, which is also called orbital degeneracy. Many kinds of TE materials benefit from this kind of degeneracy including n-type CoSb3,46 p-type SnSe,37 n-type Mg3Sb2,47 etc. In CoSb3 skutterudites, the conduction band minimum is located at the center of the Brillouin zone (Γ point), which has an Nv 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 low-symmetry point in the Brillouin zone, which has a high Nv 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 non-parabolic feature of the primary conduction bands at large carrier concentrations,49 while the large valley degeneracy of secondary band (Nv = 12) as well as its convergence to the primary conduction bands at high temperatures (resulting in Nv = 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 CoSb3 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 Nv for well-established TE materials or to discover potentially new TE materials, using Nv as a screening criterion, either by tuning temperature or alloy composition, as discussed below.

Effect of temperature and alloying

Different temperature dependence of multi-valleys (with the valley edges located at different points in the Brillouin zone) might lead to valley convergence, thus resulting in increased Nv and enhanced PF. PbTe and similar IV–VI rock-salt 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 (Nv = 4) in the Brillouin zone, while the lower and heavier one is along the Γ-K line, labeled as Σ (Nv = 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 temperature-dependent 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.

Fig. 3
figure3

a Schematic image of the isoenergetic surface for the valence bands of PbTe; schematic diagrams of band convergence in PbTe induced by b temperature change52,55 and c alloying with certain kinds of point defects30

To date, the study of the mechanisms on temperature-induced 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 thermal-disordered lead chalcogenide structure (Pb atoms are moved further off-centered compared to the chalcogenide atoms with increasing T) was proposed and adopted to fully explain the temperature-dependent band convergence of lead chalcogenides.53 Theoretical calculation of this thermal-disordered 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 temperature-dependent angle-resolved photoemission spectroscopy study,55,56 and theoretical calculation57 (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 multi-valleys, 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 PbTe-based materials through isovalent substitutions by Mn,58,59 IIB,60 and IIA elements61 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.

N-type Mg2X-Mg2Sn (X = Si, Ge) solid solutions are typical examples illustrating how valley degeneracy can be increased through alloying.66,67 In Mg2X, above the conduction band minimum, there exists another sub-band which usually does not contribute to the transport due to large splitting energy from the band minimum (−0.4 eV for Mg2Si,68 0.16 eV for Mg2Sn69). As the two conduction band edges (one heavier and another lighter) are reversely positioned in Mg2Sn, compared with that in Mg2Si and Mg2Ge, 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 quasi-binary and quasi-ternary solid solutions of Mg2X-Mg2Sn when the alloy compositions are tuned within the convergence range.

Fig. 4
figure4

a Composition dependence of energy gaps for both the light conduction band minimum and the heavy one from the valence band maximum in Mg2Si1−xSn x solid solutions127 (inset is the schematic diagram for the band structure of Mg2Si); b Composition dependence of Eg for specific valleys in Ga x In1−xP with regard to valence band maximum72 (inset is the schematic diagram for the band structure of InP)

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 n-type Ga x In1−xP 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 Nv = 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 n-type Ga0.74In0.26P.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 change-induced phase transition from a rhombohedral structure (Bi2Se3) to an orthorhombic one (BiSbSe3) in n-type Bi2−xSb x Se3 would result in a considerable modification on the band structure: the original conduction band minimum moves from the center of the Brillouin zone (Nv = 1) to a low symmetry point (Nv = 2, also heavier mb*), leading to a large increase of md* (as seen in Fig. 5a), thus improving the PF and achieving a high zT ~1 at 800 K.74

Fig. 5
figure5

a DOS of BiSbSe3 and Bi2Se3, illustrating the structural transition-induced valley convergence in Bi2−xSb x Se3; inset is the Fermi surface for BiSbSe3 (the upper one) and Bi2Se3 (the lower one). Reproduced with permission.74 Copyright 2016, Royal Society of Chemistry. b Schematic diagram for the reduction of crystal-field splitting energy due to the pseudo-cubic structure in tetragonal chalcopyrites75

Recent study on the band structure of tetragonal chalcopyrites opens a new avenue to increase the Nv for the TE materials with low crystal symmetry (thus intrinsically low Nv), via changing the system symmetry from non-cubic to pseudo-cubic structure.75 The crystal structure of tetragonal chalcopyrites can be regarded as variants from the cubic zinc blende structure with Nv = 3 (contributed by Γ5V (Nv = 2) and Γ4V (Nv = 1)). Ordered cation substitution with two kinds of metal atoms further lowers the symmetry leading to the chalcopyrite structure type AIBIIIX2VI as well as the observed crystal-field splitting, ΔCF = Γ5V − Γ4V, which lowers Nv. 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 sub-lattice shows nearly cubic framework once η ≈ 1. And this long-range cubic-like structure regains the three-fold valley degeneracy76 and keeps the localized non-cubic lattice distortion simultaneously, leading to an enhanced TE performance. Through rational alloying, the lattice parameters could be tuned into a desired range and this unity-η design strategy (for tetragonal chalcopyrites with η ≠ 1) could be realized experimentally, like the case in Cu0.875Ag0.125InTe2.75

Size effect

First introduced in the 1990s, the idea of using low-dimensional nanostructured TE materials to realize extraordinary enhancement of zT has attracted numerous attention, due to both promising theoretical predictions8 and subsequent proof-of-principle experiments.77,78 When the size of TE materials is reduced to a critical length scale, e.g., in an ultra-thin 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~E1/2) near the band edge is altered to a step-like 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 low-dimensional nanostructures on the reduction of κ is relatively intuitive, the exact inner mechanisms for PF enhancement introduced by the low-dimensional structures are still controversial (e.g., a non-monotonous 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 low-dimensional nanostructures on the valley specifics. In the two-dimensional superlattice, this approach is also called the carrier pocket engineering as proposed in the theoretical study of n-type GaAs/AlAs 2D superlattices by Koga et al.82 The early work on 2D superlattices only consider the zT for the quantum confined 2D nanosheet77 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 Ga1−xAl x As solid solutions with composition in the valley convergence condition suffer greatly from the mobility reduction caused by inter-valley 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 inter-valley scattering between the X valleys and the (Γ, L) valleys should be reduced and high mobility is expected to be maintained.84

Fig. 6
figure6

a The schematic diagram for GaAs-AlAs superlattice; b energy offset of specific valleys in GaAs-AlAs superlattice; c layer thickness dependence of energy for different conduction valleys relative to the valence band maximum (superlattice period is fixed, dGaAs + dAlAs = 40 Å); Ll and L0 refer to the pertinent conduction valleys in longitudinal and oblique orientations relative to [111] direction. Reproduced with permission.128 Copyright 2015, Elsevier B.V

More importantly, by controlling the width of the quantum well (tuning the layer thickness of GaAs and AlAs, noted as dGaAs and dAlAs), 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 delta-function-like 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 (ED) 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 delta-like shape with certain energy width Γ around the resonant energy level in the final DOS, as seen in Fig. 7.86

Fig. 7
figure7

a Schematic diagram of the energy level of resonant dopants and the relevant change in the DOS; b calculated total DOS for Tl-doped PbTe. Reproduced with permission.90 Copyright 2011, Royal Society of Chemistry

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/mI*, where τ is the relaxation time. For resonant scattering process, the τ can be expressed as τ = τ0(1 + (2(E−ED))2), with τ 0 the minimum value of τ reached at ED.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 effect91 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 delta-like distortion of DOS.92

The typical example that demonstrates how TE performance could benefit from the resonant doping includes the Tl-doped PbTe92 and Sn-doped Bi2Te3.93 For Tl-doped 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 Na-doped PbTe without resonant state formation at given hole concentrations, and a peak zT value of 1.5 in Pb0.98Tl0.02Te was attained.92 In the case of Sn-doped Bi2Te3, the experimental proof of resonant levels was from Shubnikov-de Haas measurements combined with galvanomagnetic and thermomagnetic properties measurements. Strongly increased and nearly pinned Seebeck coefficient beyond the Pisarenko prediction was found for Sn-doped p-type single crystalline Bi2Te3 at 300 K.93 As the mobility decrease is relatively moderate, overall PF for Sn-doped Bi2Te3 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 mb*,30 which are also found from the theoretically calculated band structure diagrams.90,95 The influence of mb* 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 mb* 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 semiconductors97,98,99 has been attempted for different classes of TE materials in the aim of enhancing the PF, such as In-doped100 or Co-doped101 SnTe, Cr-doped PbTe,102 Al-doped PbSe,103 Pb-doped 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 n-type Ti-doped 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 co-doping. 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 second-order tensor, can be expressed as a scalar m* with the assumption of a single parabolic band which results in a spherical Fermi surface in k-space. Tuning the flatness of a single valley (adjusting mb*) offers us another opportunity to optimize TE properties in the framework of valleytronics. Normally, lower mb* is more favorable for TE materials, which could be seen from Eq. (2) and shown by both experiments106,107,108 and theory.11,31 The study on p-type (V,Nb)FeSb-based 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 mb* of (V,Nb)FeSb solid solutions is expected to become lower with increasing Nb content. Experimental work indeed confirms that NbFeSb has much lower mb* (1.6 me) than that of (V0.6Nb0.4)FeSb (2.5 me), as shown in Fig. 8a.111 Furthermore, in this HH material system, the decreased mb* also helps lower the optimal carrier concentration, which was too high to reach in (V0.6Nb0.4)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 Ti-doped NbFeSb.106

Fig. 8
figure8

Hole concentration dependence of a Seebeck coefficient and b PF for Ti-doped V x Nb1−xFeSb with different mb*. The inset is the schematic of band structures of VFeSb (red solid line) and NbFeSb (blue dotted line). Reproduced with permission.106 Copyright 2015, Royal Society of Chemistry

Apart from forming solid solutions to realize lower mb* in TE materials, doping would also possibly influence the mb* for a given TE material. Such a situation normally arises in cation substitution for n-type semiconductors or anion substitution for p-type semiconductors, where the resultant mb* would usually be increased in proportion to doping.112,113 For example, Pei’s work on La-doped n-type PbTe clearly demonstrates that the cation substitution strategy for n-type semiconductors increases the mb* of carriers, and thus, unfortunately, degrades the PF.107 Based on a simple Kane band model, the mb* for La-doped PbTe is found to be 20% larger than that for I-doped n-type PbTe (anion substitution), which remains almost unchanged compared to PbTe, within all the temperature range considered. The electron mobility for La-doped PbTe is so severely decreased that the optimized maximum zT is 20% lower than that of the I-doped one.107 Similar case can also be found in Li and Ga-doped p-type Mg2(Si,Sn), where the Ga-doped ones possess much higher mb* yet worse PF than the Li-doped ones.108

The enhancement for TE performance through changing the value of mb* 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 md* for a single valley mb* is the geometric average of effective masses along each principle direction: mb* = (m||*m *2)1/3 that is related to the Seebeck coefficient as well as the carrier concentration. While the inertial effective mass mI* is defined as the harmonic average mI* = 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 K-value larger than 10 is necessary to bring significant difference to the overall TE performance, as shown in Fig. 9.

Fig. 9
figure9

a K-value dependence of the effective mass with mb* equals to unit; b schematic view of valley anisotropy in band structure diagram (orange ellipsoids are the schematic diagrams for the Fermi surfaces with different K-values). Reproduced with permission.11 Copyright 2017, WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

The benefit of valley anisotropy can be understood in two aspects: on the one hand, with the increased degree of valley anisotropy (larger K-value), 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 “needle-like” or “plate-shaped”, which contributes low-dimensional transport feature and results in large α,115,116 like the case in 2D superlattice quantum wells8 or 1D nanowires.117 Consequently, the anisotropic valley offers another possibility for decoupling μ and α. The comparison between n-type PbTe and PbSe exemplifies the advantage of valley anisotropy to some extent. While the md* 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 n-type layered cobaltites,118 n-type rock-salt IV-VI 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 Fe2VAl full Heulser semiconductor, the conduction valley at X point is mainly contributed by the V eg states with dispersive band feature, above which exists another band with high anisotropy contributed by the Fe eg 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 Fe2TiSn (K ~87) and Fe2TiSi (K ~450) with good thermodynamic stability. AgBiSe2, known for intrinsically low thermal conductivity and multiple phase transitions with changing temperature,120 was found to have large valence valley anisotropy with one-dimensional “plate-shaped” carrier pockets121 based on recent DFT calculation on the room temperature phase of hexagonal AgBiSe2, 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 high-throughput 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 κL 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 inter-valley 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 two-dimensional effect with enhanced Seebeck coefficient has been realized under high vertical electrical field in oxide TE materials.124 The introduction of magnetic nano-particles 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/501310-01).

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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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Correspondence to Tiejun Zhu.

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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/s41535-018-0083-6

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