ABSTRACT:

Thermoelectrics is attracting intensive research interests. The band structure information of new compounds is growing rapidly with the power of first principle calculations. However, not every thermoelectric materials candidate can be experimentally investigated due to by limited resources. Therefore, we need to develop an efficient approach of screening and selecting the most promising ones from the various band structure databases of new compounds. Here we propose the concept of pseudo-ZTs: zt_e and zt_L, where zt_e only measures the electronic influence and zt_L only scales the lattice contribution. Using zt_e and zt_L helps avoid playing the "seesaw balancing game" between the Seebeck coefficient and the conductivities. It also helps reveal deeper physics on how dimensionality, carrier concentration and band structure will affect ZT quantitatively. A range of thermoelectric materials are tested, and ~400 new compounds are calculated for predictions. The pseudo-ZTs can serve as a good guidance for thermoelectric materials search, in addition to the semi-empirical “β_SE” indicator.

Table of Content

This paper quantitatively studied how low-dimensionalization, carrier-concentration and various band-shape parameters affect thermoelectric ZT, by decomposing ZT into two pseudo-ZTs.

Broader Context: Energy crisis is an urgent global problem economically, politically, and environmentally. To address such energy crisis, we have to develop high-efficiency approaches to use clean and renewable energy, e.g. waste heat, geothermal, nuclear, and solar energy. Since all the heat-related energy resources, both fossil and green, have thermal processes and waste heat recycling problem, thermoelectric power generation is very attractive. Towards the direct high-efficiency conversion between heat flow and electricity, enhancing the thermoelectric figure-of-merit (ZT) is a complex problem and a research topic of broad interest. Modern first-principle computations have strong power in band structure prediction for new compounds, though transport calculations are still challenging and expensive. Due to limited resources, not every materials candidate can be experimentally investigated. Therefore, it will be very helpful to develop an efficient thermoelectric indicator based on band structure information to search for the most promising candidates from the various band structure databases. We here propose a framework of pseudo-ZTs to accelerate this task. 

Thermoelectric materials offer a way to interconvert heat flow and electricity. The conversion efficiency is characterized by the material specified dimensionless figure-of-merit ZT=σS²T/κ, where σ, S, T and κ are the electrical conductivity, Seebeck coefficient, temperature and thermal conductivity, respectively. Until the past two decades, it was believed that ZT could not exceed 1.¹ More recently, many novel approaches have sequentially pushed the upper limit of ZT,^2-8 including the utilization of low-dimensionalization⁹, sharp density of states ¹⁰, superlattice,^11-13 resonant states,^14,15 nanocomposites,^16,17 and pipe-shaped Fermi surfaces.¹⁸ Despite these progresses, values of ZT still remain too low for inexpensive materials to consider the prevalent use of thermoelectrics. The strong correlation between σ, S and κ results in significant difficulties on this optimization problem. For example, it is typical that materials with large σ always tend to have large κ and small S, which kills ZT. 

      The recent advancement of first-principle calculations has enabled the prediction of band structures of a largely growing number of new compound materials,¹⁹ the information of which are collected in various materials databases.^20-22 It is resource-prohibited to make efforts on each of these materials. Therefore, how to choose the most promising candidates from these databases for further improvement with experimental efforts is a crucial and urgent question.   

      To achieve this end, the semi-empirical thermoelectric indicator “β_SE” was developed,^23-25 which requires the input of estimated carrier mobility (μ) and lattice thermal conductivity (κ_L). However, the absolute values of μ and κ_L are difficult to predict, which can change substantially with crystal imperfections, such as the grain size,²⁶ defects concentration,²⁷ and grain boundary thickness.²⁸ Although accurate values of κ_L can be computed for several individual materials with expensive calculations,^29-31 the general discrepancies between theoretical calculations and experimental measurements on thermal conductivity for most materials are still obvious. Further, κ_L can be reduced significantly by novel techniques like nanocomposites synthesis.^16,17 Thus, the band structure is still the most important input for the materials search before experimental investments.

      Therefore, we here develop a framework of pseudo-ZTs to indicate the potential of good thermoelectric behavior. The pseudo-ZT prediction only needs to input the information of band structures. Further, we will see that the pseudo-ZTs can provide us with a more physical insight onto how the energy sensitivity of transport, the dimensionality, the band asymmetry, and the band gap will influence the ZT quantitatively, and suggest strategies to further improve ZT. Therefore, the pseudo-ZT system we developed here can be used along with the semi-empirical thermoelectric indicator “β_SE” ²⁵ for searching potential thermoelectric materials among the numerous candidates.

      We see that using the formula of ZT=σS²T/κ, we are not allowed to change any one single variable (σ, S or κ) with the other two fixed, e.g. it is unable to enhance ZT by increasing σ and keeping S and κ unchanged, because σ, S and κ are highly correlated. Life will be much easier if we can find a way to decompose ZT into quantities that are independent or weakly correlated to each other. To achieve this goal, we introduce the idea of splitting ZT into two pseudo-ZTs: zt_e and zt_L,

,                                               (1)

where  and . The J_n’s are dimensionless numbers defined as

 ,                                    (2)

where ε and ε_f are the reduced carrier energy (ε=E/k_BT), and reduced Fermi level (ε_f=E_f/k_BT),  f₀ is the Fermi distribution, and Ξ(ε) is the transport distribution as a function of ε.^32,33 Further, θ=Ξ(1) is a measure of the transport strength despite of its sensitivity to carrier energy. Thus, we see that zt_e only measures the influence of electronic affection to ZT, while zt_L only scales the contribution of the lattice thermal conductivity (κ_L) to ZT. Just like ZT, zt_e and zt_L are also dimensionless, which we name as pseudo-ZTs. By using the much less correlated zt_e and zt_L, we can avoid playing the “seesaw balancing game” between S and σ or σ and 1/κ, when examining the possible enhancement of ZT over different parameters, as shown in the following contents.

      First, we use the pseudo-ZTs to examine the dimensionality paradox of thermoelectrics. Since Hicks and Dresselhaus proposed that low-dimensionality will benefit thermoelectric ZT,⁹ great efforts have been made in various nanoscale materials.^34,35 It is observed that the low-dimensionalization strategy works in many but not all materials systems.³⁶ We still have many cases where the low-dimensional systems have non-enhanced ZT³⁶ compared with their bulk counterparts. Thus, it is interesting to study where this discrepancy come from, and how we can distinguish the low-dimensionality favored materials from the non-favored ones.

      Here we try to study this problem from the relation between the pseudo-ZTs and the carrier energy sensitivity of transport. The carrier energy sensitivity of transport can be characterized by

,                                                        (4)

which is generally a function of carrier energy, temperature and band valley structures. For a single carrier, it is a specific value, while for the collective behavior of multiple carriers, it is a statistical measurement of the whole system. Such statistical measurements are popularly used in diffusive transport, e.g. the carrier mobility μ is a statistical measurement on how mobile the carriers are under an external field. The overall energy sensitivity will decrease upon low-dimensionalization, because (1) the density of states will have a smaller or even negative dependence to energy, and (2) the scattering time for high energy carriers encountering possible ballistic scatterings will have negative energy dependence (τ~λ/v).

      Fig. 1 (a) shows the pseudo-ZTs as a function of carrier concentration for different values of energy sensitivity s. We can see that decreased s leads to increased zt_L but decreased zt_e. Equation (1) implies that the ultimate ZT is dominated by the smaller pseudo-ZT. Thus, for high carrier concentration systems, ZT is dominated by zt_e, while for low carrier concentration systems, ZT is dominated by zt_L, according to Fig. 1 (a). Therefore, the conclusion is that the ultimate ZT will be enhanced by low-dimensionalization only for systems with low carrier concentrations. For systems with high carrier concentrations, staying in the bulk form will be preferred. For systems with moderate carrier concentrations, the dimension of the material may not affect ZT significantly.

      To have a better understanding of the connection between the pseudo-ZTs and the β_SE indicator,²⁵ we illustrate their relation with carrier concentration in Fig. 1 (b). We can see that the β_SE indicator can well estimate ZT for the high carrier concentration region. However, when the carrier concentration is low, the dominant pseudo-ZT becomes zt_L that has an opposite trend compared to the β_SE indicator as a function of carrier concentration. Therefore, the β_SE indicator in this region is no longer a good estimation.

      Second, we use the pseudo-ZTs to give us a guidance for screening and selecting untested candidates from the databases of band structures.^20-22 We will examine the most important properties of a band structure, including band asymmetry, band gap, and band-edge alignment. Qualitative comments on how these properties may affect ZT might be found occasionally in the literature.³⁷ Here a quantitative approach on how to comprehensively evaluate a material is provided based on the pseudo-ZT concept.

      The band asymmetry characterize how different the valence and the conduction bands are from each other. While the shape of band edges have to be considered case by case for high accuracy ZT calculations, especially for the irregular shaped band structures, the ratio (γ) between the density-of-state masses of each band can be used as a primary feature parameter for the efficient materials search purpose. Fig. 2 has shown how the maximum values of zt_e and zt_L will be affected by γ.  Without loss of generality, we illustrate a case of reduced band gap ε_g=20. We see that unlike the case of s in Fig. 1, γ is a parameter that can increase both zt_e and zt_L together. In other words, the task of enhancing ZT always prefers holes and electrons to be as different from each other as possible. This explains why materials that can provide large band asymmetry with both heavy holes and light electrons, such as topological insulator,^{38, 39} quasi-Dirac-cones,^40,41 or narrow-band-gaps,⁴² are always good thermoelectric materials. This also explains why several nano-composites systems have been reported to have high ZT,^{16, 17} wherein the holes-screening grain-boundaries are effectively making holes much heavier than electrons, resulting in large γ values.   

      We have found that in contrary to conventional thought, a large band gap can enhance both pseudo-ZTs, until the band gap becomes highly correlated with lattice thermal conductivity. Our result is different from Ref. 32, where only the zt_L dominated systems are considered. To illustrate this, we use a hypothetic system with a fixed γ, so we can examine zt_e and zt_L as a function of ε_g. Without loss of generality, we illustrate a case where γ=1, as shown in Fig. 3. Fig. 3 has dissolved the traditional "fear" of large band gap, which implies small carrier concentration. It clearly shows that the maximum values of both pseudo-ZTs will increase or at least not decrease with ε_g. However, it is worth to note that the carrier concentration of large band gap insulators could become difficult to tune through doping. Furthermore, a large band gap may be positively correlated with the thermal conductivity, i.e. large band gap insulators usually have long inter-atomic distances and results in large lattice thermal conductivity, which compromises the enhanced zt_L value.

      For a multi-valley or irregular-shaped band structure that can be decomposed into a series of small parabolic or non-parabolic valleys. The shape similarities and energy degeneracy can also be quantitatively evaluated by the pseudo-ZTs. Here we illustrate this approach by comparing a one-valley band and a two-valley band. For more complex band structures, this approach can be similarly employed. For instance, assuming that we have two hypothetic band valleys denoted as V⁽¹⁾ and V⁽²⁾. We use the ratio (ζ) between the curvatures of V⁽¹⁾ and V⁽²⁾ as the primary feature parameter to characterize their shape similarity, and the difference in energy at the bottoms of V⁽¹⁾ and V⁽²⁾, i.e. ΔE=Δε∙k_BT, to characterize the degree of degeneracy between the two valleys. V⁽¹⁾ and V⁽²⁾ are fully degenerated when Δε=0. Further, we use the optimization ratio to compare the maximum pseudo-ZTs in different cases, defined as,

,                                            (5)

where the pseudo-ZTs denoted by a lower case zt stands for zt_e or zt_L, the superscript ⁽¹⁾, ⁽²⁾ and ^(1)+(2) indicates the band valley (valleys) that is (are) involved in transport, and means the greater value of and .  Fig. 4 (a) shows that multiple valleys will not help zt_e: non-zero band-edge displacement Δε can even reduce zt_e, while band dissimilarity ζ does not influence zt_e significantly. Fig. 4 (b) shows that multiple valleys can increases zt_L, only if the band-edge displacement Δε is small and the valley shape dissimilarity ζ is close to 1. The upper-limit of the optimization ratio for zt_L is the number of band valleys (2 in this case), which occurs when Δε=0 and ζ=1. This is a hint that for large carrier concentration systems, we prefer to choose materials whose band structures are with multiple similar band valleys. For small carrier concentration systems, e.g. large band gap materials, we prefer simple band structures.

      In order to see how these concepts work in real materials, we have examined a range of known room temperature thermoelectric materials, of which the various properties that we need for ZT estimations are well measured or calculated. From the above results, we see that the band asymmetry ratio γ and the reduced band gap ε_g will increase zt_e and zt_L together, so we now explore how these two parameters of band structure will influence the maximum ZT.  As we know, the ultimate ZT depends on many factors besides the band structure, e.g. scattering mechanisms, sample synthesis conditions, defects concentrations and grain size distributions, so we here calculate the upper limit of ZT based on the commonly used acoustic phonon scattering approximation, which works well enough for the searching and screening purpose. The results of ZnTe,^{43, 44} SnO₂,^{45, 46} ZnO,^{47, 48} ZnSe,^{49, 50} CoO,^{51, 52} Mg₂Ge,⁵³ Mg₂Si,⁵³ PbS,^54-56 NiO,^{51, 57, 58} and PbTe^{59, 60} are shown in Fig. 5, where we see that the upper limit of ZT is positively correlated with γ and ε_g, as expected from the pseudo-ZT analysis above. 

      Finally, we show how to use the pseudo-ZT theory for searching potential materials from the band structure databases. In Fig. 6 (a)-(d), we have calculated the values of zt_e and zt_L at the room and the high temperatures for numerous materials with known band structures²⁵ that are not yet well investigated for thermoelectrics. We propose that the ones with both high zt_e and zt_L are the most promising candidates, and deserve the priority for further studies.

      In conclusion, to accelerate the search of materials candidates for thermoelectric research from the databases of calculated band structures, we have developed a pseudo-ZT framework to consider the electronic and the lattice contributions separately. Unlike S and σ or σ and 1/κ, which always have a strong negative correlation, using zt_e and zt_L can help us avoid playing the “seesaw balancing game”. In the pseudo-ZTs method, the optimization possibility of carrier scattering mechanisms and thermal conductivity will not be suppressed as in the β_SE indicator method. Scenarios of both high and low carrier concentrations can be considered. Furthermore, we have shown that the pseudo-ZT approach can also quantitatively reveal the physics of how the dimensionality, carrier concentration, band asymmetry, band gap and band degeneracy will affect the ultimate ZT. We have also shown how the upper limit of ZT is influenced by the band asymmetry and band gap in several room temperature thermoelectric materials. Finally, we have carried out the calculations of pseudo-ZTs for various untested materials candidates from the band structure databases for future research guidance.

Fig. 1 (a) shows the pseudo-ZTs as a function of carrier concentration for different values of the energy sensitivity s. The colored solid (dashed) curves represent the values of zt_e (zt_L) for cases with different energy sensitivities (s). The change of s affect the two pseudo-ZTs, zt_e and zt_L, in opposite trends. Since the smaller pseudo-ZT dominates the overall ZT as implied from Equation (1), zt_L and zt_e dominates in low and high carrier concentration regimes, respectively. Therefore, the low-dimensionalization approach can only enhances ZT significantly in low carrier concentration materials systems. (b) The relation between the pseudo-ZTs and the β_SE indicator,²⁵ illustrated for the s=1.0 case as an example. It can be seen that the β_SE indicator can only well estimate ZT for the high carrier concentration regime. For the low carrier concentration regime, the dominant pseudo-ZT (zt_L) has an opposite trend with the β_SE indicator vs. carrier concentration. Therefore, the β_SE indicator in this regime is no longer a good estimation. Without loss of generality and validity, the single valley calculation and the arbitrary unit are used for the simplicity and convenience of illustration.

Fig. 2 Illustration of how the maximum values of zt_e and zt_L will be affected by the band asymmetry ratio (γ).  Without loss of generality, the reduced band gap of ε_g=20 is illustrated. The colored solid (dashed) curves represent the values zt_e (zt_L) for cases with different energy sensitivities (s). The legend is the same as defined in Fig. 1 (a). Unlike the parameter of s in Fig. 1 (a), γ is a parameter that can increase both zt_e and zt_L together. In other words, the task of enhancing ZT always prefers holes and electrons to be as different from each other as possible, i.e. ZT benefits from large band asymmetry ratio γ.

Fig. 3 Illustration of how the maximum values of zt_e and zt_L will be affected by the reduced band gap (ε_g=E_g/k_BT).  Without loss of generality, γ is fixed to be 1. The colored solid (dashed) curves represent the values zt_e (zt_L) for cases with different energy sensitivities as well. The legend is the same as defined in Fig. 1 (a). Contrary to conventional thought, it is shown that ε_g will enhance zt_e and zt_L together, though the increase of zt_L will not be obvious when ε_g is greater than 10. This has dissolved the traditional "fear" of large band gap. It implies that we only need to avoid selecting large band gap materials systems, of which the thermal conductivity is too high and cannot be reduced.

Fig. 4 Illustration of how the band alignment/degeneracy and band dissimilarity affect the optimization ratio pseudo-ZTs defined in Equation (5). The optimization ratio of (a) zt_e and (b) zt_L between the hypothetic cases of having both V⁽¹⁾ and V⁽²⁾ and only having V⁽¹⁾ or V⁽²⁾ contributing to transport are shown as a function of the reduced band-edge displacement Δε and the band dissimilarity ζ. Different values of s give the similar trend, and we are illustrating s=1.5 in this figure.