How efficient are thermoelectric materials? – An assessment of state-of-the-art individual and segmented thermoelectric materials

To ef ﬁ ciently convert heat into electricity using thermoelectric energy harvesting, it is essential to employ newmaterialstrategies.Oneeffectiveapproachissegmentation,wherecompatiblematerialsoptimizedfor different operating temperatures are combined to improve thermoelectric ef ﬁ ciency. Despite reports of severe ef ﬁ ciency reductions when segmenting incompatible materials, an updated assessment of the compatibility across state-of-the-art thermoelectric materials is missing. Here, we employ a numerical modeltoassess howef ﬁ cientlynon-segmentedandsegmentedhigh-performingthermoelectric materials canconvertheatintoelectricity.Forthenon-segmentedmaterials,ef ﬁ ciencyreachesup17.9%at D T ¼ 615K without heat losses and contact resistances. Losses due to self-incompatibility were generally found to be small for most materials ( < 0.6% points) with few exceptions. In contrast, segmented thermoelectric legs wereverysensitivetocompatibilityeffects.Segmentation isfoundtoonlyboosttheef ﬁ ciencysigni ﬁ cantly for D T > 300 K. Here, we ﬁ nd ef ﬁ ciencies of up to 24% for p-type AgSb 0.94 Cd 0.06 Te 2 /Pb 0.98 Te 0.02 -8%SrTe and 19% for n-type Bi 1.8 Sb 0.2 Te 2.7 Se 0.3 /Pb 0.93 Sb 0.05 S 0.5 Se 0.5 . We further map out the maximum tolerable contact resistances before segmentation becomes detrimental. By providing an overview of the achievable energy conversion ef ﬁ ciencies, this study highlights the present state of thermoelectric energy conversion and critically assesses the prospects of segmentation. © 2024 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).


Introduction
Thermal energy harvesting technologies have become increasingly important due to the vast amounts of waste heat stemming from industrial and domestic processes.One type of thermal energy harvester relies on thermoelectric (TE) materials to convert waste heat directly into electricity.Typically, the heat conversion efficiency of TE materials via the thermoelectric effect is evaluated using the dimensionless figure of merit zT ¼ sS 2 k T, where s and k are the electrical and thermal conductivities, respectively, and S is the Seebeck coefficient.Thermoelectric research has long been devoted to improve the performance of TE materials by enhancing the figure of merit via (a) increasing the power factor (PF ¼ sS 2 ) and (b) reducing the thermal conductivity.In the last decade, great advances in zT have been made with values far exceeding two in several material classes by employing strategies such as nanostructuring, alloying/doping, and strain engineering [1e4].Additionally, several significant challenges associated with TE materials have largely been improved, including scalability, brittleness, and resource shortage [5,6].
Metallic tellurides (chalcogenides) are the best-performing TE materials at room and medium temperatures.These compounds comprise numerous compositions and structural variations, such as Bi 2 Te 3 and AgSbTe 2 -derivatives, which can display peak zT values between one and two at a temperature slightly below 400 K [7,8].Due to the thermal instability of tellurides at high temperatures and their scarcity on Earth, there is an active effort toward finding alternative TE materials with a broader thermal window.For instance, zintl compounds particularly based on Mg 3 Sb 2 have attracted recent attention as a possible candidate to compete with tellurides at room and medium temperatures (peak zT ~1e2) [9e14].A collection of high-performing TE materials exhibiting exceptionally high zT approaching three, have been identified in the medium to high-temperature range (400e900 K), including compounds like SnSe [15,16], skutterudites [17], clathrates [17,18], or half-heuslers [19e21].Other TE materials like FeGa 3 -types [22,23] and oxides [24,25] also offer a competitive zT at high temperatures combined with elemental abundances.
In parallel to developing high-zT materials, device fabrication has also been extensively investigated, emphasizing the need to lower contact resistances [26e32] and exploring different device designs [33,34].In a conventional thermoelectric generator (TEG), a pair of p-type and n-type semiconductor legs is joined to form a unicouple, where each leg generates a voltage difference for a given thermal gradient via the Seebeck effect.Typically, the TE legs are connected electrically in series but thermally in parallel, such that numerous unicouples can be brought together to increase the voltage output.The optimal operating temperature of the constituent materials that make up either the n-or the p-type leg is determined by zT and the thermal stability of the leg, where the effects of a prolonged thermal gradient can irreversibly change the chemical structure of the TE material over time.Hence, every TE leg displays an optimum zT value within a specific temperature range, making the combination of materials in a single leg (segmentation) an attractive strategy to boost efficiency.This approach enables each leg of the unicouple in the TEG to function at its best when a larger temperature difference is applied, resulting in increased output power and efficiency compared to non-segmented legs exposed to the same temperature difference.For example, it has been shown experimentally that an optimized segmentation between Bi 2 Te 3 and germanium telluride (GeTe) alloys can achieve an efficiency of 13.6% at DT ¼ 493 K, which is among the highest measured values amongst segmented materials operating within the same temperature range [35].Even good prospective modules with segmented skutterudite-based thermoelectric legs can be found in the literature, as well as generators made up of newly discovered and high-zT materials like GeTe and Mg 2 Sb 3 [36].
However, finding the compatibility between TE materials, i.e., how well a set of thermoelectric materials can maintain optimal performance over a range of temperatures, is not trivial, as shown by Snyder and Ursell [37], and improper segmentation may even harm the efficiency.The compatibility factor was thus introduced in the early 2000s [37,38] to describe the optimal electric current necessary to achieve the highest efficiency.If the reduced current density matches the compatibility factor at any temperature along a TE leg, the maximum efficiency is obtained, which can be conveniently calculated from zT alone.However, in general, the reduced current density deviates from the compatibility factor, which reduces efficiency for both non-segmented and segmented legs.In particular, for segmented TE legs where each segment possesses a widely different compatibility factor, a significant mismatch between the reduced current density and compatibility factor is typically observed leading to a considerable reduction of efficiency [37].For this reason, to efficiently match segments, the compatibility factors of each segment should not exceed a factor of two as a rule of thumb [37,38].A few examples of improper segmentation were reported in Ref. [38], such as a TE leg consisting of p-TAGS materials ((AgSbTe 2 ) 0.15 (GeTe) 0.85 ) and p-PbTe, yielding an efficiency even lower than that of p-TAGS alone at DT ¼ 500 K.
Additionally, Ngan et al. [39] performed an extensive investigation using a 1D model to numerically evaluate the performance of segmented TE legs and unicouples.For instance, at DT ¼ 600 K, segmenting a leg with the three materials Bi 2 Te 3 , clathrate and MnSi 2, results in a reduction of almost 5% points in efficiency compared to using only two segments since MnSi 2 acts as a nonmatching compound [39].Ngan et al. also assessed the most effective segmented thermoelectric legs at three distinct temperature gradients using state-of-the-art materials available in 2013.
For instance, at a temperature difference of DT ¼ 400 K, the p-type leg composed of materials in the family Bi 2 Te 3 /PbSrTe achieved an impressive efficiency of ~15%; at the same time, the n-type combination of Bi 2 Te 3 and skutterudite delivered a notable ~13% efficiency.
Many effects may; however, negatively impact the final performance of the thermoelectric legs.While convective and radiative heat losses can be significantly reduced by proper insulation, contact resistances have hindered the further development of segmented thermoelectric legs.Here, the thermoelectric performance is lowered by (1) the contact resistances at the interfaces to the electrodes and (2) the interface between the two thermoelectric segments; usually stemming from interfacial surface roughness or intermixing of elements.The interfaces between the thermoelectric segments have been addressed in a variety of TE legs [28,31,32,40e42] and have been reported to yield a similar order of magnitudes as the interfaces to the electrodes [27,39,43e48]: In the literature, reasonably low values for the specific electrical and thermal contact resistance have been reported (<10 À7 -10 À8 Ohm m 2 and 10 À6 -10 À8 m 2 W/K).Bjørk et al. (2015) [27] further numerically examined how the efficiency was affected by the electrical and thermal contact resistances between two thermoelectric segments for many segmented thermoelectric legs.Here, the authors identified a universal trend allowing an estimation of the maximum electrical and thermal contact resistance between material segments that can be tolerated before segmentation becomes worse than simply using the hot-side material alone.
Ouyang et al. [49] further evaluated segmented p-and n-type legs using 3D simulations by selectively choosing the best TE materials including the impact of contact resistances between the thermoelectric segments and the electrodes.Their results using geometrically asymmetric-sized unicouples, i.e. legs with a larger p-type area than that of n-type, showed that these modules could yield high efficiencies up to 20.9% when using p-(Bi,Sb) 2 Te 3 / MgAgSb/PbTeS/SnSe and n-CuBi2Te3Se/AgPbSbTe/SiGe segmented legs.As thermal and electrical contact resistances reached specific thresholds, the efficiency and output power in the TEG module were shown to decline.However, TEG performance also exhibited a plateau below these threshold values, suggesting that if interfaces can be well controlled in each segmented leg, contact effects will not have a significant impact.Recently, Ryu et al. [1] further examined a more comprehensive library of materials using a highthroughput method, where they found segmented legs reaching a high efficiency of up to 24% at DT ¼ 800 K.While the authors investigated TE efficiencies incorporating the temperature dependence of the thermoelectric properties, they did not consider compatibility matching between materials.Hence, despite numerous reports of segmentation, an updated compatibility assessment across state-of-the-art thermoelectric materials is needed.
To update the thermoelectric library of segmented materials and investigate new combinations with optimum conditions, we employ a 1D numerical model with a two-fold goal: (1) Evaluating the efficiency and degree of self-compatibility of non-segmented pand n-type TE legs using a library of high-zT materials.(2) Evaluating the efficiency and compatibility of segmented TE legs composed of two material segments with state-of-the-art zT values.This approach allows us to enhance thermoelectric materials and systems with updated high-performing materials.

Methods
We collected data on the thermoelectric properties of some of the highest zT materials for various temperatures in the bestperforming TE material classes reported so far.The materials are grouped under well-recognized family categories with their chemical composition, material class, and peak zT value displayed in Table 1.Although the area of thin-film thermoelectrics and lowdimensional systems has seen significant growth in recent decades [1,2,4,50,51], only bulk materials were reported in this paper.
We use a 1D model [37] to assess the efficiency and compatibility of non-segmented and segmented legs.This analysis is carried out as a function of an applied temperature gradient.The 1D model accounts for the most significant thermoelectric phenomena, namely the Peltier, Seebeck, and Thomson effects.It also considers the compatibility of the TE legs, heat conduction, Joule heating, and the temperature dependence of the intrinsic material properties, including resistivity (r ¼ 1=s), Seebeck coefficient (SÞ, and thermal conductivity (kÞ.These properties and their temperature dependences can be found in Supplementary Section S1 for the selected material library.The temperature difference (DT) is defined as an input parameter by fixing a constant cold-side temperature (T c ¼ 300 K) and varying the hot side (T h ) temperature of the leg, thereby creating a heat flow across the TE leg.
It should be noticed that thermal and electrical contact resistances as well as heat losses are excluded from the model.Their detrimental effects on the thermoelectric performance have been assessed elsewhere [1,27,49].In real thermoelectric devices, thermoelectric legs are also 3D, meaning heat flows in all directions instead of along the TE leg, as assumed in the 1D model.For a more complete understanding between heat transfer and 3D geometry, the reader is referred to Refs.[49,52].
The 1D model is expressed only in terms of temperature and the intrinsic variables of the material, as already discussed in Refs.[37,53].Instead of being a function of current and position, the 1D model introduces the so-called reduced current density u ¼ J/(k DT), which is the ratio between the electrical current density (J) and the heat flux from heat conduction.Once the starting conditions of u are defined at the hot side, u h ¼u(T¼T h ), eq. ( 1) will determine the temperature profile of u throughout the TE leg [53]: The solution to the reduced current density can be used to extract several properties, such as the thermoelectric efficiency, the heat flow along the leg, and the temperature profile, as described in more detail in Refs.[37,53].The efficiency of the TE leg can be calculated as eq.( 2) [53], where the subscripts refer to the hot (h) and cold (c) side of the TE leg.However, eq. ( 1) cannot be solved analytically due to the dependency of the material parameters on temperature, and we solve it recursively following eq.( 3) [53], where the n-subscript denotes the step in temperature (T n and T nÀ1 Þ and rk and S denote the average of rk and S between the steps, respectively.Here, for a fixed hot side temperature, T h , we select a starting condition u h ¼ u nÀ1 , that is employed to calculate the subsequent values of u n , thereby recursively generating a profile of uðTÞ that allows finding a value for u c .By varying the starting u h condition, many uðTÞ profiles over temperature can be generated according to eq. ( 3).We then evaluate the efficiency for each specific u h using eq.( 2) and select the uðTÞ profile that yields maximum efficiency.This approach is equivalent to optimizing the power transfer between electronic circuits when performing impedance matching.We further solve eq. ( 1) and ( 2) as a function of the hot side temperature (T h ) while keeping the cold-side temperature constant at 300 K. Here, T h was varied sequentially with a step size of 5 K over the full temperature range of the materials.Note that the profile of the optimum u (T) is determined by eq. ( 1) at every T h .Thus, uðTÞ may differ significantly from the ideal reduced current density (s) defined by the compatibility factor, which is determined solely by intrinsic properties of the material [37].Consequently, a drop in efficiency in the thermoelectric leg can be caused by the u-profile differing significantly from the compatibility factor either in a single material or when segmenting two dissimilar materials.Eq. ( 1) is therefore used as well to model segmented legs, where the material properties along the legs are changed abruptly from one material to the other for a given interface temperature (T i ).In this report, the discussion is limited to segmented pairs, where we denote the hot (cold) segment as the material in the leg that operates at higher (lower) temperatures.
A rule of thumb to achieve beneficial leg segmentation between the cold and hot side materials lies in choosing compounds with compatibility factors differing by less than a factor of 2 [37,53].In the ideal case where the profile of u matches to that of s, one obtains the theoretical maximum efficiency of the TE leg (h (u ¼ s) ): This expression is traditionally used to evaluate the efficiency of thermoelectric materials based only on temperature and zT.However, eq. ( 5) does not consider the mismatch between u (T) and s(T) that we show here which can result in significant overestimations of the efficiencies in both non-segmented and segmented legs composed of present state-of-the-art materials.
As the contact resistances and heat losses are excluded in the 1D model, the efficiency is independent of the length and crosssectional area of the TE leg.However, following Bjørk et al. (2015), the maximum tolerable values for the specific electrical (r max c;el ) or thermal contact resistance (r max c;th ) can be extracted for segmented thermoelectric legs with two segments according to eq. ( 6) [27]: Here, r leg denotes the average resistivity over the full segmented leg of length L, g e=th are universal contact resistance functions.Thus, one can find the maximum tolerable contact resistances through eq. ( 6), using the linear universal function expressions, g e ¼ 0:72, h gain þ 7:09 and g th ¼ 0:56,h gain þ 2:32, where h gain is defined as the relative efficiency gain calculated in the absence of contact resistances, i.e. h gain ¼ ðh segmented;leg À h hot;leg Þ=h hot;leg .We note that the maximum tolerable contact resistances are independent of the cross-sectional area and linearly proportional to the length of the leg.The linear functions g e=th were found to describe a large number of segmented thermoelectric devices in a universal manner up to h gain $ 50%, whereas segmented legs with larger gains were unaccounted for.

Results
To illustrate the concept of the simulations, Fig. 1aec show a non-segmented n-type leg within the parent family of n-PbTe, a material reported to have a large zT > 1.5 between 600 K and 800 K [54] (see Table 1 for more details).Fig. 1c displays the efficiency calculated using the 1D model as a function of the hot side temperature with a fixed cold-side temperature of 300 K.The efficiency reaches a maximum of h ~7% (solid line) compared to h (u ¼ s) ~8% (dashed line) corresponding to the reduced current density being equal to the compatibility factor at all temperatures, which is identical to calculating the efficiency directly from zT.Both efficiencies increase with the hot side temperature, but a significant discrepancy between h and h (u ¼ s) also develops.Fig. 1a and b display the optimal reduced current density at hot side temperatures of 530 K and 780 K, together with the compatibility factor of the material.When the single leg is subjected to a moderate temperature gradient (T h ¼ 530 K, DT ¼ 230 K), the material operates close to its maximum thermoelectric efficiency since the optimal current density u is close to the compatibility factor s, as observed in Fig. 1a.For larger gradients (T h ¼ 780 K, DT ¼ 480 K), the difference between h and h (u ¼ s) is caused by the large distance between u and s as shown in Fig. 1b.The nearly constant profile of u cannot accommodate the strongly temperature-dependent compatibility factor of n-PbTe, resulting in a 1.2% points loss in efficiency (more information on the losses can be found in Supplementary Section S2).This loss thus originates from what we refer to as self-incompatibility effects.
For the case of a segmented leg, we illustrate the detrimental effect of segmentation on the efficiencies in Fig. 1g using two materials of the parent family p-Bi 2 Te 3 and p-Cu 2 S as the cold and hot sides of the leg, respectively.Here, p-Bi 2 Te 3 has shown a zT of nearly 2 close to room temperature at 320 K [7], whereas p-Cu 2 S reaches only a large zT > 1 at much higher temperatures close to 650 K [55] (see Table 1).Owing to the larger zT of p-Bi 2 Te 3 at moderate temperatures compared to p-Cu 2 S, we fix the interface temperature T i between p-Bi 2 Te 3 and p-Cu 2 S to 480 K, which corresponds to the highest temperature reported for the chosen compound in the category of p-Bi 2 Te 3 [7].The choice of the interface temperature between two segmented materials can significantly impact the resulting efficiency, as described in Supplementary Section S3.In this example of a segmented TE pair, a beneficial or negative effect from segmentation can be found depending on the applied temperature gradient the leg is subjected to.Starting with a nonsegmented p-Bi 2 Te 3 leg, Fig. 1d displays the behavior of u and s for a low-temperature gradient (T h ¼ 400K, DT ¼ 100 K) where only a small efficiency drop due to self-incompatibility can be observed (Fig. 1g).As the leg is segmented with p-Cu 2 S and a relatively higher temperature gradient is applied (T h ¼ 580 K, DT ¼ 280 K), see Fig. 1e, the nearly constant profile of u deviates from the dissimilar compatibility factors of p-Bi 2 Te 3 and p-Cu 2 S. Here, s differs by up to an order of magnitude between the two materials, which impacts the overall efficiency negatively, as seen in Fig. 1g.Even a nonsegmented p-Bi 2 Te 3 leg exposed to a more moderate temperature gradient of DT ¼ 180 K provides a higher efficiency compared to a segmented p-Bi 2 Te 3 /Cu 2 S leg as the temperature of hot side temperature reaches 740 K (DT ¼ 440 K).However, when the temperature gradient is magnified (T h ¼ 800 K, DT ¼ 500 K), the overall match between u and s improves, as displayed in Fig. 1f, finally providing a beneficial segmentation.Even in this case, the final efficiency achieved (11%) at T h ¼ 800 K does not largely exceed that of the single material p-Bi 2 Te 3 (9%) at 480 K.This significantly hampers the potential for segmenting these materials, especially when considering the added complexity in manufacturing, the extra electrical and thermal contact resistances that are introduced, and the accelerated degradation at elevated hot side temperatures.
To assess the reduction in efficiency due to self-compatibility effects and provide guidance to beneficial segmentation, we have collected the figures of merit (Fig. 2a and b) and calculated the corresponding compatibility factors (Fig. 2c and d) for a range of the state-of-the-art thermoelectric p-type and n-type materials reported in the literature.Their specific compositions are described in Table 1, and their electrical conductivity, Seebeck coefficient, and thermal conductivity can be found in Fig. S1 in the Supplementary Material S1.
A key obstacle in constructing highly efficient and scalable TE modules lies in the scarcity of raw materials, such as tellurium; and their toxicity, like lead [2,5].Alternative TE materials included in this study introduce elements like sulfur, silicon, oxygen, selenium, or magnesium, which benefit from higher abundance, lower prices and better safety profiles [2,5] despite having lower figures of merit in most cases (Fig. 2a and b).Another key concern is the thermal stability of TE legs subjected to large temperature gradients during operation.Common mechanisms of TE-degradation are sublimation of volatile dopants, diffusion within the TE materials, and diminished contact performance.For instance, the volatility of certain elements and detrimental cation or vacancy diffusion at high temperatures is known for many materials, where efforts to reduce ion migration and modification of TE leg geometry have been investigated [2,66,67].One measure to protect against diffusion and sublimation of elements is, for example, to fill the space of TE modules with inert gas but also cover their sides with materials that weaken sublimation and diffusion at high temperatures [67].The stability, price, and availability of these materials lies, however, beyond the scope of the present study, and the reader is referred to other publications for more in-depth studies hereof [2,5,6,66e68].
The materials in this study are not selected to broadly represent the respective material classes.Instead, they are examples selected due to their high figure of merit and well-recognized published performance.In the low-temperature range (300e400K), p-type Bi 2 Te 3 remains a champion material with a zT reaching 1.86 near room temperature [8,15].Similar alloys, namely AgSbTe 2 materials, show a competitive figure of merit, around 2.6, at the high end of this low-temperature region (T h ¼ 573 K) [56].Competing with tellurides at room and medium temperatures, zintls materials like n-Mg 2 Sb 3 with several doped elements have achieved high zT peak values of 1.1 at 357 K [13] and 1.8 at 650 K [14].In this medium/high temperature span (600e800 K), SnSe was recently shown to have a zT above 3 at 783 K [16] as a p-type material and near 3 at 773 K in the n-type case [15].On the other hand, a zT of around 1.8 and 1.25 at T ¼ 773 K and 720 K was found in n-type PbTe [54] and zintl materials [69], respectively.For even higher temperatures (900e1200 K), p-PbTe-SrTe materials show a remarkable zT of 2.5 at T ¼ 923 K [57], while recently reported n-and p-type half-Heusler's [19,20] dominate in the extremely high-temperature regime with a zT around 1 between 900 up to 1200 K.
Analogous to zT, the compatibility factor (s) is displayed in Fig. 2c and d, showing a strong temperature dependence, particularly for the p-type materials Bi 2 Te 3 , AgSbTe 2 and SnSe and n-type materials BiSbTeSe, Zintl-1, Zintl-2, PbTe, PbSe, and Cu 2 S. The overall compatibility factors can vary by more than an order of magnitude between the materials, further motivating the present study on assessing efficient segmentation of high-performing TE materials.In Supplementary Section S3 (Fig. S3), we further provide the compatibility factor averaged across the full temperature range for all p-and n-type materials as well as a simplified guide for selecting compatible materials.
The efficiency of a single thermoelectric leg calculated using the 1D model for the collected library of materials is shown as a function of the hot side temperature for p-and n-type in Fig. 3a and  b.Among the p-type materials that exhibit the highest efficiencies, materials such as AgSbTe 2 (14.7%),SnSe (16.4%),PbSrTe (17.9%), and half-Heuslers-2 (15.3%) reflect their high value of zT.Cu 2 S is an interesting case with high zT ¼ 1.74 at 992 K, which far exceeds the zT of the half-Heusler materials.However, the resulting efficiency of Cu 2 S at 1000 K is significantly lower than for the half-Heuslers, owing to the drop in zT between 300 and 600 K.Among the n-type materials, BiSbTe (11.4%),Zintl-2 (15.0%), skutterudite (14.7%), and half-Heusler (9.32%) display the best efficiencies at low, medium, and high temperatures.In addition, it is worth noting, the high performance at moderately low temperature gradients of the ntype material Zintl-1 which is highly competitive with n-type BiSbTe.
The dashed line in Fig. 3a and b display the maximum efficiency for the ideal situation where u ¼ s.The difference between these curves highlights the self-incompatibility of each individual material.Noticeably, there is a monotonic growing efficiency drop from the lack of self-compatibility with increasing temperature gradient.For most materials, the efficiency drop due to selfincompatibility is small (<0.6% points).However, a few materials, such as the high zT p-type half-Heusler-1 at T h ¼ 1200 K and n-type PbTe at T h ¼ 762 K, exhibit more notable efficiency losses of 0.95 and 1.17% points, respectively.These results reflect that a strong temperature-dependent compatibility factor will often yield larger self-incompatibility losses.A more detailed view of the efficiency loss for each material is provided in Supplementary Section S2.
We now focus on segmented legs comprising two segments where the boundary between the segments is characterized by an interface temperature (T i ).In the 1D model, the interface temperature can be directly translated into the spatial domain [53], which then describes to which extent the cold-side and hot-side segments take up the length of the leg.The interface temperature strongly impacts the resulting performance.Hence, we numerically optimize T i and current density profile u, to achieve the largest efficiency for all considered material pairs and hot side temperatures.In Supplementary Section S4, we compare this with two simpler approaches to finding T i , namely, (1) T i is selected to be the temperature that maximizes zT everywhere in the leg [39] and (2) T i is selected as the temperature where the s-profiles of the two segmented materials intersect [37,53].The first approach yields interface temperatures and efficiencies close to the optimum values for nearly all materials.In contrast, the second approach yields significant deviations in interface temperature and resulting efficiency.
For the cold segment, we chose either Bi 2 Te 3 or AgSbTe 2 in ptype thermoelectric legs and BiSbTeSe or Zintl-1 in the n-type legs.These materials were selected due to their high efficiency near room temperature.For the non-segmented p-type legs, Bi 2 Te 3 and AgSbTe 2 reach 9.1% and 14.7% in efficiency at their highest reported T h ~476 K and ~570 K, respectively (see Fig. 3a).Conversely, n-type non-segmented legs made up of BiSbTeSe and Zintl-1 reach efficiencies of 11.4% and 8.75% at T h ~574 and 516 K, respectively (see Fig. 3b).However, these high-performing materials at low temperatures possess strong temperature-dependent compatibility factors (see Fig. 2c and d), which may hamper beneficial segmentation.The compatibility factors of p-Bi 2 Te 3 , p-AgSbTe 2 , as well as n-BiSbTeSe and n-Zintl-1 change by a factor of 2.9, 1.3, 3, and 1.8 across their reported temperature range, respectively.Nonetheless, when evaluating the average compatibility factor of these cold segments across their entire temperature range, they still show the potential to be efficiently segmented with most materials (see Supplementary Section S3).Fig. 4 displays the resulting efficiency when combining the selected cold-side segments with the material library.The results are shown for three hot-side temperatures T h ¼ 700 K, 800 K, and 900 K (bottom, middle, and top panel) while fixing the cold-side at T c ¼ 300 K.For all hot-side temperatures and material combinations, we numerically optimize the interface temperature and the reduced current density profile.We translate the interface temperature to the spatial domain and display the relative lengths of the hot-and cold-side materials with the colored bars in Fig. 4. (see Supplementary Section S5 for further information on the relative lengths of the segments) Overall, the relative length of the hot segment material increases with the hot side temperature.For instance, when p-type SnSe is segmented with p-type Bi2Te3 at T h ¼ 700 K shown in the bottom panel of Fig. 4a, both the gray and blue bars have equal heights, implying that the optimal leg has roughly 50% p-SnSe and 50% of p-Bi2Te3.As T h is raised to 900 K, the relative length of p-SnSe in the leg becomes 61% of the total leg.In contrast, for n-type legs with segmented n-type BiSbTeSe/SnSe at T h ¼ 700 K (bottom panel of Fig. 4b), the optimal solution is composed mostly (65%) of   the n-type BiSbTeSe segment.Note that if the solution only contained the hot-side material without segmentation, the optimum interface temperature would be reduced to 300 K, and only a single material color would be present.This is not the case for any material combinations except for p-Bi2Te3/clathrate as shown in Supplementary Section S4, signifying the gain in efficiency with respect to the hot side material alone by adding a high-performing cold-side material to the hot segment.
The largest segmentation gains were found when comparing the efficiencies of the segmented legs with those of the individual hot side materials (see Supplementary Section S6 for an overview hereof).Here, non-segmented p-type Cu 2 S and n-type FeGa 3 displaying efficiencies of 4.1% and 2.1% at T h ¼ 700 K (Fig. 3a and b), can be enhanced by 10.7 and 7.6% points when segmented as p-AgSbTe2/Cu 2 S leg and a n-Zintl-1/FeGa3 leg, respectively.The segmentation gain with respect to the efficiency of the individual hot segment can, however, be marginal in other cases, such as for n-BiSbTeSe/SnSe (0.90% points) and n-Zintl-1/Zintl-2 (1.33% points) at T h ¼ 700 K.
In contrast, efficiency benefits from segmentation can be challenged if we now compare the segmentation gain at T h ¼ 700e900 K with the efficiencies of non-segmented cold-side materials between 476 and 678 K (i.e.maximum reported temperature for the cold-side legs).For instance, non-segmented legs of n-type BiSbTeSe can achieve an efficiency of 11.4% at its highest T h ~574 K (dashed line in Fig. 4b); however, a segmented n-type leg of n-BiSbTeSe/HH yields 9.76% at T h ¼ 700 K (Fig. 4b), making the non-segmented leg with a lower temperature gradient a better choice.Note that segmented n-type BiSbTeSe/HH still provides gains in efficiency compared to its hot side counterpart operating independently (n-HH yields 4% efficiency at T h ¼ 700 K i.e. a gain of 5 pp when segmented).Thus, using a single material leg with BiSbTeSe and a reduced thermal gradient would be the preferred choice in this case.Otherwise, adding, for example, a passive material to the non-segmented leg can also be an alternative solution to keep the cold segment operating in its thermal stability window.
Interestingly, this trend is frequently observed at Th ¼ 700 K, where many segmented legs do not exhibit a strongly increased efficiency compared to that achieved by the non-segmented cold-side material (see Fig. S5).
In general, efficiencies for the p-types are larger than for n-type materials due to their higher average zT in this p-type material selection.When highlighting the case of T h ¼ 900 K (DT ¼ 600 K), the p-type segmented legs that reach the highest efficiencies are the following combinations: p-AgSbTe 2 /PbSrTe (23%), p-Bi2Te3/ PbSrTe (21.5%), and p-AgSbTe2/HH-1 (19.6%).This value exceeds the maximum efficiency of less than 20% at T h ¼ 900 K, calculated using the same methodology in Ref. [39] and of the best segmented legs found in Refs.[1,49] with state-of-the-art materials.The bestperforming n-type segmented legs were n-Bi2Te3/PbSe (18.9%) and n-Zintl-1/PbSe (18.4%) at T h ¼ 900 K, also with values exceeding those found in Refs.[1,39,49].
Segmentation introduces; however, an additional interface in the TE leg with associated electrical and thermal contact resistances, which is highly material-specific.To provide guidance on the maximum electrical (r max c;el ) and thermal (r max c;th ) contact resistances that segmented thermoelectric legs can tolerate before segmentation becomes detrimental e compared the nonsegmented hot-side material e we have plotted Fig. 5a and b for p-and n-type materials at T h ¼ 700 K (DT ¼ 400 K) following the work by Bjørk et al. [27] (see eq. ( 6)).We note that the maximum tolerable values increase linearly with the length of the leg and we hence fix the length of the leg to L ¼ 1 cm.In addition, as Bjørk's model was only validated for a limited range of segmented efficiency gains.Hence, the maximum tolerable contact resistances are only shown for material pairs within this validated range.For more information on the relative efficiency gains, see Supplementary Section S6.
Above these maximum tolerable contact resistances, the selected segmented pairs will not benefit from any gain in efficiency at T h ¼ 700 K compared to just using the hot-side materials alone at the same temperature.For example, in Fig. 5a, p-PbSrTe acting as a single leg shows an efficiency of 11.8 % at T h ¼ 700 K; however, when segmented as a pair p-Bi 2 Te 3 /PbSrTe, the gain in efficiency with respect to the hot side is 3.9% points.Hence, this segmented leg shows an increase of Dh gain ~33 % with respect to the hot side, which translates into maximum tolerable contact resistances of r max c,el ¼ 1.3 Â 10 À7 U m 2 and r max c,th ¼ 7.6 Â 10 À8 m 2 W/K.When using an alternative cold-side, the segmented leg p-AgSbTe2/PbSrTe achieves a higher efficiency of 18.4 % at T h ¼ 700 K, i.e. a gain of 6.59% points with respect to the hot side alone (Dh gain ~56 %).Therefore, it will possess a larger tolerable contact resistance, namely, r max c,el ¼ 4.4 $10 À7 U m 2 and r max c,th ¼ 2.5 $ 10 À7 m 2 W/K.In practice, the maximum tolerable electrical contact resistances found in Fig. 5a and b are on the same order of magnitude as the typical low experimental values found in the literature with r c,el ~10 À8 to 10 À6 U m 2 [27,39,43e48].However, the estimation of the tolerable thermal contact resistance is conservative in most cases, which is typically reported to be in the range of r c,th ~10 À6 -10 À3 m 2 W/K [41,43e45].Fig. 6a and b display a map of efficiencies as a function of hot side temperature for the best-performing non-segmented and segmented legs.The segmented and individual material legs with lower efficiencies than those in Fig. 6a and b are not shown here for clarity in favor of the best-performing legs.Attached to each curve are bars representing the optimized relative lengths of the cold and hot side materials.Once again, p-type materials in Fig. 5a show overall higher efficiency compared to the n-type materials in Fig. 5b.For reference, the highest efficiency curves are lower than the Carnot efficiencies, h Carnot ¼ 1-T c /T h , by a factor of roughly 0.35 for the p-types and 0.29 for the n-type materials.
Among the p-type legs at low hot side temperatures, AgSbTe 2 is a high-performing non-segmented leg, reaching a maximum efficiency of 14.7% at T h ¼ 570 K. Up until that temperature, the maximum efficiency of this non-segmented leg is larger than what could be achieved with any segmented leg.As a non-segmented leg also benefits from fewer interfaces, segmentation would only be detrimental in this low-temperature region (DT < 300 K) due to the addition of adverse thermal and electrical contact resistances.At higher temperatures (DT > 300 K), several segmented legs provide outstanding optimized efficiencies, such as p-AgSbTe2/PbSrTe (24% at T h ¼ 915 K) and p-AgSbTe2/HH-2 (20.7% at T h ¼ 972 K), keeping maximum specific contact resistances according to Fig. 5a below r c,el ¼ 4.4 $10 À7 U m 2 and r th,el ¼ 2.5 $ 10 À7 m 2 W/K and r c,el ¼ 6.7 $10 À7 U m 2 and r c,th ¼ 3.0 $ 10 À7 m 2 W/K, respectively.These high efficiencies are partly due to (1) the large thermal gradient that the leg is subjected to, (2) their individually high zT, and (3) a decent overall compatibility of the combined materials.A striking feature is that increasing the hot side temperature beyond 915 K does not show any significant increase in efficiency.This is in close agreement with the findings from Ryu et al. [1], owing to the lack of compatible high-zT materials at these temperatures and low average zT values.
Among the n-type legs, non-segmented BiSbTeSe achieves the highest efficiency, up to 11.4% at T h ¼ 574 K.We also highlight the high performance up to T h ¼ 715 K of Zintl-2 as a single leg with an efficiency of 15.1% exceeding that of many segmented pairs even at higher temperature gradients.The segmented pair n-Zintl-1/Zintl-2 simultaneously achieves 16.4%; however, with maximum tolerable contact resistances of r c,el ¼ 6.0 $10 À8 U m 2 and r c,th ¼ 3.1 $ 10 À8 m 2 W/K according to Fig. 5b.The monotonic efficiency increase with temperature continues when segmenting BiSbTeSe with PbSe, reaching a 18.9% at 900 K.Note that this segmented pair at a lower T h ¼ 700 K displayed a maximum tolerable contact resistance of r c,el ¼ 1.7 $10 À8 U m 2 and r c,th ¼ 9.8 $ 10 À8 m 2 W/K.In the medium temperature region (300 K < DT < 550 K), the non-segmented nskutterudite material presented a good alternative to segmented legs, reaching an efficiency 14.5% at T h ¼ 820 K.For a higher gradient, the pair n-BiSbTe/Oxide displayed 12.4% at T h ¼ 970 K and n-Zintl-1/Half-Heusler with 13.1% at T h ¼ 1104 K.As observed, only segmentation of n-BiSbTeSe/PbSe gave a noticeable boost in efficiency out of all the n-type material combinations considered.
When considering the chosen material library, we can deduce that if T h < 600 K (DT < 300 K), segmentation of thermoelectric materials does not provide significant efficiency gains over nonsegmented legs.Hence, single material legs with high figures of merit, such as p-type Bi 2 Te 3 and AgSbTe 2 or n-type BiSbTeSe and n-Zintl-1, can be used instead.On the other hand, more noticeable efficiency gains are found for larger thermal gradients at DT > 300 K, with segmented combinations that display good compatibility, such as p-AgSbTe 2 /PbSrTe and n-BiSbTeSe/PbSe.While the results of these simulations can only be considered as an upper boundary for efficiency, maintaining these high-temperature gradients on the TE legs will bring about other issues, such as diffusion or intermixing at the interface, together with chemical instability of the materials; effects are not considered in this study.In addition, we stress again on other important factors to making scalable TE legs, such as (1) available and abundant materials, (2) large thermal and cycling stability, and (3) reduced thermal and electrical contact resistances of these materials.

Conclusion
The efficiency of state-of-the-art thermoelectric materials was calculated using the 1D model, considering their compatibility factors.Non-segmented materials displaying large changes in the compatibility factor will exhibit the largest loss in efficiency due to self-compatibility effects.At medium-temperature gradients, noticeable efficiency losses can be seen in p-Bi2Te3 (0.53% points) and n-BiSbTeSe (0.60% points), but in general it is overall low.In comparison, at high-temperature gradients, we find materials with a more noticeable efficiency loss, such as p-PbSrTe (0.52% points), p-half-Heusler-1 (0.95% points), or n-PbTe (1.17% points).Among the single legs that exhibit the highest efficiencies at DT > 300 K, p-SnSe (16.4%) and p-PbSrTe (17.9%) are very competitive at high temperatures, together with n-Zintl-2 (15.0%) and n-skutterudite (14.7%).
When pairing materials into a TE leg, a careful choice must be made by looking at their compatibility factors.The efficiency gain of segmentation is marginal at DT < 300 K when compared to the efficiency of high-performing non-segmented legs.Above DT > 300 K, e.g. at T h ¼ 900 K, segmented pairs like p-AgSbTe 2 /PbSrTe (23%), p-Bi 2 Te 3 /PbSrTe (21.5%), p-AgSbTe 2 /PbSrTe (19.6%), or n-Bi 2 Te 3 /PbSe (18.9%) are examples of beneficial segmentation boosting the efficiency compared to the individual materials.The maximum efficiency achieved by segmentation was p-AgSbTe 2 / PbSrTe (24.0%) at T h ¼ 915 K and n-Bi 2 Te 3 /PbSe (18.9%) at T h ¼ 900 K.While the results of these simulations can only be considered as an upper boundary for efficiency, they also highlight a path for significant improvements in the current TE performance and the need for a careful selection of materials when segmentation is done in a TE pair.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Fig. 1 .
Fig.1.A thermoelectric leg composed of one single material will experience different reduced current densities when T h is varied from (a) a low T h (b) to high T h ; (c) The performance of the material in terms of efficiency (solid line) and theoretical maximum efficiency if u ¼ s (dashed line) illustrates the idea of self-compatibility.For a segmented leg with two materials with non-matching compatibility factors, sweeping T h determines the optimized profile of u in (d), (e), and (f), which can result in a negative or positive contribution to the efficiency of the overall leg, as shown in (g).

Fig. 2 .
Fig. 2. Figure of merit (zT) for all individual thermoelectric (a) p-type and (b) n-type materials in Table 1.The compatibility factor of these materials is shown in (c) and (d) for p-and n-type materials, respectively.

Fig. 3 .
Fig. 3. Self-compatibility of individual (a) p-type and (b) n-type materials grouped by color.The solid line represents efficiency as a function of the hot side temperature when taking self-compatibility into account.In contrast, the dashed line shows the theoretical maximum efficiency obtained when the reduced current density equals the compatibility factor at all temperatures.(For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article).

Fig. 4 .
Fig. 4. Efficiency for segmented legs at three different hot side temperatures (T h ¼ 700,800, and 900K) for (a) p-type and (b) n-type materials.For the p-type legs, the cold-side material is fixed to either p-Bi 2 Te 3 (dark gray) or AgSbTe 2 (light gray) and many hot side materials are evaluated.For the n-type legs, we fixed the cold-side segment to BiSbTeSe (dark gray) or silicide (purple).The interface temperature (Ti) is picked such as to optimize the maximum efficiency.Each segmented bar reflects the relative height of each material segment providing the highest efficiency.The efficiency of the cold-side legs is also shown as horizontal dashed lines for reference.(For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article).

Fig. 5 .
Fig. 5. Maximum tolerable specific electrical (r max c;el ) and thermal (r max c;th )contact resistances at T h ¼ 700 K in electrical (blue) and thermal (orange) resistivity for segmented (a) p-type legs using either p-Bi2Te3 (diamonds) or p-AgSbTe2 (stars) as the cold-side and (b) n-type legs using n-BiSbTeSe (diamonds) or n-Zintl-1 (stars) as the cold-side.The total length of the segmented leg in these calculations was L ¼ 1 cm.(For interpretation of the references to color in this figure legend, the reader is referred to the Web version of this article).

Fig. 6 .
Fig. 6.Efficiency of single and segmented legs at different hot side temperatures displayed for (a) n-type materials and (b) p-type materials.

Table 1
State-of-the-art thermoelectric materials displaying the material class, composition, peak zT value, and at which temperature the peak zT value is obtained.