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[Byungki Ryu](https://orcid.org/0000-0002-0867-6457), Jaywan Chung, Masaya Kumagai, [Tomoya Mato](https://orcid.org/0000-0002-0918-6468), Yuki Ando, Sakiko Gunji, Atsumi Tanaka, Dewi Yana, Masayuki Fujimoto, Yoji Imai, [Yukari Katsura](https://orcid.org/0000-0002-8905-2995), SuDong Park

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[Best thermoelectric efficiency of ever-explored materials](https://mdr.nims.go.jp/datasets/173cb96d-15a6-4d8d-a7fd-efdbf8177f3f)

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Best thermoelectric efficiency of ever-explored materialsllOPEN ACCESSiScienceArticleBest thermoelectric efficiency of ever-exploredmaterialsByungki Ryu,Jaywan Chung,Masaya Kumagai,..., Yoji Imai, YukariKatsura, SuDongParkbyungkiryu@keri.re.krHighlightsReport of bestthermoelectric efficiencyof ever-explored materialsusing big data100 million deviceefficiencies calculatedover 13 thousandmaterialsBest single-stageefficiency can reach17.1%, surpassing 13.3%recordMultistage devices showpotential for over 25%efficiencyRyu et al., iScience 26, 106494April 21, 2023 ª 2023 TheAuthor(s).https://doi.org/10.1016/j.isci.2023.106494mailto:byungkiryu@keri.re.krhttps://doi.org/10.1016/j.isci.2023.106494https://doi.org/10.1016/j.isci.2023.106494http://crossmark.crossref.org/dialog/?doi=10.1016/j.isci.2023.106494&domain=pdfllOPEN ACCESSiScienceArticleBest thermoelectric efficiencyof ever-explored materialsByungki Ryu,1,5,* Jaywan Chung,1 Masaya Kumagai,2,3 Tomoya Mato,4 Yuki Ando,4 Sakiko Gunji,4Atsumi Tanaka,4 Dewi Yana,4 Masayuki Fujimoto,4 Yoji Imai,4 Yukari Katsura,2,4 and SuDong Park11Energy Conversion ResearchCenter, KoreaElectrotechnology ResearchInstitute (KERI), Changwon51543, Republic of Korea2Center for AdvancedIntelligence Project, RIKEN,Tokyo 103-0027, Japan3SAKURA Internet ResearchCenter, SAKURA internet Inc.,Osaka 530-0001, Japan4Research and ServicesDivision of Materials Dataand Integrated System(MaDIS), National Institute forMaterials Science (NIMS),Ibaraki 305-0044, Japan5Lead contact*Correspondence:byungkiryu@keri.re.krhttps://doi.org/10.1016/j.isci.2023.106494SUMMARYA thermoelectric device is a heat engine that directly converts heat into elec-tricity. Many materials with a high figure of merit ZT have been discovered inthe anticipation of a high thermoelectric efficiency. However, there has been alack of investigations on efficiency-based material evaluation, and little is knownabout the achievable limit of thermoelectric efficiency. Here, we report the high-est thermoelectric efficiency using 12,645 published materials. The 97,841,810thermoelectric efficiencies are calculated using 808,610 device configurations un-der various heat-source temperatures (Th) when the cold-side temperature is300 K, solving one-dimensional thermoelectric integral equations with tempera-ture-dependent thermoelectric properties. For infinite-cascade devices, a ther-moelectric efficiency larger than 33% (z1/3) is achievable when Th exceeds1400 K. For single-stage devices, the best efficiency of 17.1% (z1/6) is possiblewhen Th is 860 K. Leg segmentation can overcome this limit, delivering a very highefficiency of 24% (z1/4) when Th is 1100 K.INTRODUCTIONA thermoelectric device composed of P- and N-type thermoelectric material legs placed between hot- andcold-side substrates can directly convert thermal energy into electrical energy via thermoelectric effects.1,2As thermoelectric devices can generate electricity regardless of the amount of the temperature differenceand heat sources, the thermoelectric technology has been considered a promising solution for power sour-ces in space and waste heat recovery systems in the industry and transportation sectors.1–7One of themost successful applications of thermoelectric technology has been in space exploration, whereit has been used in Radioisotope Thermoelectric Generators (RTGs) to generate over a 100W scale power.2Currently, the technology is being investigated for use in industrial and transportation sectors. Commer-cially available devices are based on Bi2Te3 alloys, and each device produces nearly 10 W with a conversionefficiency of approximately 5–7% under a temperature difference of 200–250 K.3 Using these devices, KELKreported a kW-scale power generation from waste heat in steel works.3 Additionally, flexible and organicthermoelectrics have been proposed for room temperature energy harvesting and power generation fromhuman body heat, with the potential of producing energy in the order of mW to mW from a temperaturedifference of approximately 10 K.6,7To put thermoelectric technology into wide and practical use, it is essential to increase the efficiency (h) ofthe devices. Because a thermoelectric device is a heat engine, the thermoelectric efficiency increases withtemperature difference and is bounded by the Carnot efficiency. Moreover, as discovered by Ioffe in 1957,8in some cases, the efficiency is determined by a single material parameter, called the dimensionless ther-moelectric material figure of merit ZT = a2T=rk, which is defined as the ratio between absolute temper-ature T and three thermoelectric transport properties (TEPs), namely, the Seebeck coefficient a, electricalresistivity r, and thermal conductivity k. Precisely, for an ideal one-dimensional thermoelectric leg havingtemperature-independent TEPs, the maximum efficiency of thermoelectric conversion (hmax) under a givenoperating temperature range from the cold-side temperature Tc to the heat-source temperature Th isexactly determined by ZT :hmax =Th � TcThffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1+ ZTmp � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1+ ZTmp+ Tc=Th; (Equation 1)iScience 26, 106494, April 21, 2023 ª 2023 The Author(s).This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).1mailto:byungkiryu@keri.re.krhttps://doi.org/10.1016/j.isci.2023.106494https://doi.org/10.1016/j.isci.2023.106494http://crossmark.crossref.org/dialog/?doi=10.1016/j.isci.2023.106494&domain=pdfhttp://creativecommons.org/licenses/by/4.0/llOPEN ACCESSiScienceArticlewhere Tm is ðTh +TcÞ=2. From the observation that the higher the Z, the higher the efficiency, many high ZTmaterials have been discovered and developed, from the traditional Bi2Te3-, PbTe-, GeTe-, and Si1-xGex-based alloys9–12 to the recently developed SnSe- and Mg3Sb2-based alloys.13,14However, little is known about the current status of device efficiency. The efficiency estimation using ZT isnot very accurate for wide-temperature-range applications because TEPs are temperature-dependent.15,16Additionally, the estimation ignores several factors introduced in device fabrication processes such asparasitic resistances. Although the measurement of device efficiency is demanded, the number of suchexperimental studies is less than a few dozen,11 likely because of more complex processes in device fabri-cation than in materials synthesis. Therefore, the efficiency-based evaluation of materials and devices iscrucial for understanding the current status. Furthermore, determining the achievable limit will guide futuredirections and accelerate research regarding the design of high-performance devices.RESULTSThermoelectric efficiency explorationHere, we report the theoretical best thermoelectric efficiency of devices resulting from the ever-exploredthermoelectric material data collected in the Starrydata2.org thermoelectric database (DB). Starrydata2 isthe world’s largest thermoelectric property DB, which contains 43,601 material samples for thermoelectricproperties from 7,994 publications as of 2021-November-24.17 After data filtering, we obtain high-qualitythermoelectric big data composed of 13,338 samples from 3,120 publications. Then, the material samplesin the big data DB are theoretically evaluated by the computed maximum thermoelectric conversion effi-ciency under various temperature differences of DT = Th � Tc when Tc = 300 K (12,645 samples from 2,919publications are available for T> 300 K): see STARMethods (data preparation, filtering, and cleansing, ther-moelectric device model, and thermoelectric performance calculation). The thermoelectric efficiency iscomputed by solving one-dimensional thermoelectric integral equations for the temperature distributionTðxÞ and heat currents at the hot and cold sides15,16: see STAR Methods (temperature distribution inside aleg). Using the searched high-performance P- and N-leg samples, of which the efficiency is larger than orequal to 85% of best P- and N-material efficiencies for a given heat source temperature, single-stage P-Nleg-pair devices with various leg geometries and interfacial resistances are constructed: see STARMethods(thermoelectric device model and computation of the best efficiency). The interfacial resistances allow us toinclude efficiency loss from device fabrication. Finally, the best thermoelectric efficiency for various electri-cal and thermal operating conditions is theoretically explored over 7,650,225 material efficiency data and97,841,810 device efficiency data points; overall, 105,492,035 efficiency data spaces are explored; see STARMethods (computation of the best efficiency).Best thermoelectric efficiencyFigure 1 shows the achievable best thermoelectric device efficiency (hðdevÞ) among 97,841,810 device effi-ciency data from 808,610 P-N leg-pair thermoelectric device configurations made of 12,645 ever-exploredmaterials, for a given heat-source temperature Th and fixed Tc = 300 K. The infinite-cascade device, wherethe electric current at each temperature point is optimised, attains the theoretical maximum efficiency for agiven ZT curve. We obtain the maximally achievable ZT curve (ZTbestðTÞ) from the DB (see key resourcestable (data500). Hence, the best efficiency of the infinite-cascade device (hðN�CascadeÞmax ) is simply calcu-lated18 byhðN�CascadeÞmax = 1 � exp0B@�ZThTcdTTffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1+ ZTbestðTÞp � 1ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1+ ZTbestðTÞp+ 11CA; (Equation 2)which increased strictly with Th. The theoretical maximum efficiency is 25% (z1/4) at Th = 880 K (DT = 580 K).This high efficiency corresponds to a device ZT value of 2. Furthermore, an even higher theoreticalmaximum efficiency of 33% (z1/3) is possible at Th = 1400 K (DT = 1100 K). For single-stage P-N leg devices,the best efficiency increases with Th and has the highest value of 17.1% (z1/6) at Th = 860 K (DT = 560 K).However, the best efficiency no longer increases but significantly drops when Th > 940 K. This efficiencydrop at very high temperatures is due to the absence of a stable material at high temperatures andmaterialself-compatibility issues18 arising from the strong temperature dependence of thermoelectric propertiesand low average ZT . If the radiation and convection losses are considered, the decrease in efficiency athigh temperatures will be more significant.192 iScience 26, 106494, April 21, 2023Figure 1. Best thermoelectric efficiencyThermoelectric efficiencies are explored using the thermoelectric property big data from Starrydata2 thermoelectricdatabase. After data filtering, 12,645 materials that can be used for power generation above 300 K, are obtained from2,919 publications. Using high-performance P- and N-leg materials, 808,610 single-stage P-N leg-pair devices wereconstructed. Then, 97,841,810 device efficiencies were computed over various heat source temperature, electricalcurrents, geometries, and interfacial resistances, when Tc = 300 K. Finally, best thermoelectric device efficiency istheoretically explored at every heat source temperature. For the clarity, we have only showed the achievable theoreticalboundaries for device efficiencies. Solid, dotted-dashed, and dashed lines are the best efficiency curves for single-stagedevices with a perfect interface (no thermal and electrical resistances), a normal interface (rðnormalÞc = 10� 8 Um2 andkðnormalÞc = 104 W m� 2 K� 1), and an inferior interface (rðinferiorÞc = 10� 7 Um2 and kðinferiorÞc = 103 W m� 2 K� 1), respectively.The dotted line is the best efficiency curve for an infinite-cascade device. Filled squares are efficiencies for calculated(Calc.) multistage segmented P-N leg-pair devices working at Th = 850 K and Th = 1100 K. Additional data pointsrepresent the experimental (Expt.) efficiencies of fabricated devices: single-stage devices (unfilled black circle),3,20–38segmented devices (Seg., unfilled black square),11,22,29,39–43 and a cascaded device (Cascd., unfilled black triangle).20 TheZT values of the grey-guide lines are inversely calculated using the maximum efficiency Equation 1 for a given Th whenTc = 300 K: ZT =�Th �Tc ð1�hÞThð1�hÞ�Tc�2� 1.44 See also key resources table (data500) for the best efficiency curves andexperimental reports, key resources table (data300) for the considered P-N leg-pair configurations, and key resourcestable (data400) for the 100 million P-N leg-pair device efficiencies data.llOPEN ACCESSiScienceArticleA multistage structure in devices such as segmented legs may overcome the self-compatibility issue, astheoretically reported by Ouyang and Li in 2016 (h = 21.0% at DT = 700 K).19 Higher theoretical efficiencieshave been reported in P-type single-leg device: Ryu et al. in 2021 (h = 21.9–24.5% at DT = 600–800 K)15 andWabi et al. in 2022 (h= 22.0% atDT = 600 K).45 In addition, in this study, we find amultiple-stage P-N leg pairdevice with very high efficiency: h= 20.0–23.9% atDT = 550–800 K; see key resources table (data700) for thedevice configuration generated.However, it should be noted that there is a large difference between the theoretical and measured best ef-ficiencies; see Figure 1 and Table 1. For the single-stage devices in Table 1, the measured best efficienciesrange from approximately 7–10%: the single-stage Bi2Te3-based device by KELK (7.2% at Th = 553 K)3 andtheGeTe(P)-Mg3Sb2(N)-baseddevice by Tongji University (10.0%atTh =600K).46 For themultistagedevicesin Table 1, the measured best efficiencies are approximately 12–13%: the GeTe/BiTe(P)-PbSe/Bi2Te3(N)segmented device by SUSTECH (13.3% at Th = 873 K),11 the half-Heusler (HH)/Bi2Te3 segmented deviceby PSU (12.0% at Th = 873 K),39 the skutterudite (SKD)/BiTe segmented device by SICCAS (12.0% at Th =849 K),40 and the PbTe/Bi2Te3 cascaded device by AIST (12.0% at Th = 873 K).20 However, their efficiencyvalues are much lower than the theoretically best device efficiency of 17.1% found in this work. Such aloss in efficiencymay be due to a suboptimal choice of materials and significant interfacial resistances. Ther-mal radiation and convection might also cause a nonnegligible loss in efficiency when Th is high.19iScience 26, 106494, April 21, 2023 3Table 1. Selected values of the best thermoelectric efficiency (h)Device Th [K] Tc [K] DT [K] h hCarnot hTE ReferenceCalc. Infinite cascade 1400 300 1100 33.0%z1/378.6% 42.0% This workSegmented (P-N) 1100 300 800 23.9%z1/472.7% 32.8% This workSingle-stage (P-N) 860 300 560 17.1%z1/665.1% 26.2% This workExpt. Segmented (P-N) 800 294 506 13.3%z1/863.3% 21.0% Ref.11Single-stage (P-N) 600 280 320 10.0%z1/1053.3% 18.8% Ref.46553 303 250 7.2%z1/1445.2% 15.9% Ref.3The efficiencies of multistage and single-stage thermoelectric devices are described with working temperatures, Carnot ef-ficiency (hCarnot = DT=Th), and reduced thermoelectric efficiency (hTE = hCarnot=h). See also Figure 1 and key resources table(data500).llOPEN ACCESSiScienceArticleDistribution of thermoelectric propertiesFigure 2 shows the distribution of thermoelectric property values in the filtered thermoelectric big data of13,338 published material samples. Figure 2A shows the available temperature ranges that are defined asthe range from the minimum measured temperature to the maximum measured temperature for a givenmaterial’s thermoelectric property. There are two distinct measurement patterns: (1) cryogenic-type ther-moelectric property measurements below room temperature (<300 K) and (2) power generation-type mea-surements above room temperature (>300 K). For the latter, 80% of the samples are measured from 300 Kup to 800 K, which are suitable for mid-temperature thermoelectric power generation applications. Thehighest measured temperature is 1500 K. Figure 2B shows the distribution of 215,526 Seebeck coefficientvalues (aðTÞÞ. The distribution is somewhat symmetric between the P- and N-type materials. When aðTÞ issmall, it increases with T, showing metallic behavior. When aðTÞ is large, it increases with T, while T < 300 K,showing bipolar transport behavior of the narrow-gap semiconductors. Figure 2C shows the distribution of223,404 electric resistivity values. The resistivity varies exponentially with temperature. For temperaturesbelow 300 K, most of the explored samples have very small resistivity, similar to semimetallic or heavilydoped semiconductors. Some have very high resistivity, which might be related to charge carrier quench-ing in doped semiconductors. For temperatures larger than 300 K, the distribution is symmetrical about theaxis of the critical value rcrit = 10� 4 Um. Furthermore, the resistivity value seems to converge to rcrit as thetemperature increases. Figure 2D shows the distribution of the 187,244 thermal conductivity values. Whenthe temperature is below 100 K, there are unusually small and large lattice thermal conductivities. The smallvalue is due to the small number of phonon activation modes, while the large value is due to less phononscattering and ballistic phonon transport. With increasing temperature, the number of large thermal con-ductivity samples decreases, which may be due to the activation of the Umklapp three-phonon process. Onthe whole, the samples can have small thermal conductivities below 10 W m� 1 K� 1 above 300 K.History of thermoelectric performances: ZT and efficiencyFor decades, the thermoelectric performance of materials has been developed to search for high ZTvalues. Figure 3A shows how the best ZT value has been improved over time. In 2000, the ZT exceeded1 for the first time due to nanostructuring and low thermal conductivity.47,48 Then, the ZT finally reached2 due to the synergetic effect of electron and phonon transport10,49; the largest improvement was achievedin the mid-temperature range of approximately 600–950 K. Recent studies report very high peak ZT valuesexceeding 2.5 in GeTe11 and SnSe.13Figure 3B shows how the best thermoelectric material efficiency (hðmatÞ) of single-leg P- or N-type materialshas been improved over time for various Th’s and Tc = 300 K. Between 2000 and 2009, the material effi-ciency was highly enhanced for all temperature ranges, similar to the improvement of ZT values. The4 iScience 26, 106494, April 21, 2023A BC DFigure 2. Distribution of thermoelectric properties(A–D) (A) Hexa-bin plot of the maximum and minimum available temperatures from the samples’ thermoelectricproperties. Hexa-bin plots of the (B) Seebeck coefficient distribution, (C) electrical resistivity distribution, and (D) thermalconductivity distribution over measured temperatures. In each panel, color represents the number of data points. Notethat Starrydata2 contains 43,601 material samples for thermoelectric properties from 7,994 publications as of2021-November-24. After data filtering, high-quality thermoelectric big data composed of 13,338 samples from 3,120publications is obtained and displaced. See also key resources table (data010) and STAR Methods (data preparation,filtering, and cleansing).llOPEN ACCESSiScienceArticleimprovement was the largest at Thz900 K, similar to the improvement of the peak ZT . However, after 2010,the increase in the best efficiency for low- and high-temperature applications has been rather small. This ismainly because, even if the peak ZT is high, the average ZT can be low. Additionally, for wide-temperatureapplications, the self-compatibility problem18 of materials may also responsible for the limited efficiencyincrease. Our observation of the large discrepancy between ZT and the material efficiency implies theimportance of studying device efficiency.Performance by material compositionFigure 4 shows the ideal thermoelectric material efficiency over 17 material groups based on materialcomposition; see the key resources table (data150 and data261). In Figure 4A, the theoretical material per-formances are represented for well-known telluride alloys (Te-alloys): Bi2Te3-, AgSbTe2-, GeTe-, and(Pb,Sn)Te-based alloys. For Te alloys, P-type materials perform better than N-type materials. Bi2Te3 alloysare the best materials for low-temperature heat sources (Th < 600 K); their best efficiency can reach 10–12%.For mid-temperature heat sources (600 K < Th < 950 K), P-type AgSbTe2-, GeTe-, and (Pb,Sn)Te-based al-loys show superior thermoelectric conversion performance compared to other alloys.iScience 26, 106494, April 21, 2023 5A BFigure 3. History of thermoelectric performances: ZT and efficiency(A and B) (A) The best material figure of merit ZT and (B) best thermoelectric material efficiency hðmatÞ for the given year ranges were drawn over ever-explored materials. The achievable best ZT and hðmatÞ were explored over 13,338 samples from 3,120 publications (see also key resources table (data070))and 12,645 samples from 2,919 publications (see also key resources table (data261)), respectively. Note that the number of samples for the latter is smallerthan the former owing to the temperature range restriction: T > 300 K for efficiency calculations.llOPEN ACCESSiScienceArticleIn Figure 4B, the thermoelectric material performances of oxide-, sulfide-, and selenide-related alloys (X-O,X-S, X-Se) are provided. Compared to tellurides, these alloys can operate at high temperatures. In the caseof La2S3, although DT is very large (1100 K), its best material efficiency is smaller than that of Bi2Te3 for DT =300 K. In Figure 4C, the performances of Mg and Si alloys are shown. Note that the P-type MgAgSb-based and N-type Mg3Sb2-based materials have comparable efficiency to Bi2Te3-based devices forlow-temperature heat sources, which explains the experimental device efficiency.21 For N-type materials,Mg2Si-based alloys exhibit good thermoelectric performance at mid-temperatures. Alternatively, Si1-xGexalloys demonstrate good performance at high temperatures (Th > 950 K), as reported.12 Figure 4D showshigh-temperature thermoelectric materials such as clathrate (Clath.), HH, and SKD thermoelectric materialsand other antimonide compounds (X-Sb). Although their performance is relatively poor at low and mid-temperatures, they show the best efficiency with regard to high-temperature heat sources.Representative P-N leg-pair devicesFigure 5 and Table 2 show the theoretical device efficiency curves of nine representative single-stage P-Nleg-pair devices; see key resources table (data400 and data600). Within available temperature ranges, thedevice efficiencies increase with temperature, indicating that thermoelectric efficiencies are limited by theavailable temperature ranges, which might be determined by the material thermal stability. For low-tem-perature heat sources, the fully Bi2Te3-based P-N leg-pair device shows the best device efficiency, which ishigher than 10%. For 600 K < Th < 950 K, the devices based on P-type chalcogenides and N-type Mg3Sb2,Mg2(Si,Sn), HH, or SKD show the highest efficiency. A limit efficiency of 17% is found in the single-stagePbTe-SKD and SnSe-SKD devices. For Th > 950 K, however, the device efficiency is smaller than that ofmid-temperature devices. Notably, there is a distinct pattern in the curves; that is, the curves for low-tem-perature devices are concave, but the other curves are convex at low temperatures and become linear athigh temperatures. The convexity of the curves is related to the poor ZT value at low temperatures in addi-tion to the temperature-dependent nature of thermoelectric transport properties. This suggests that thetemperature gradient in thermoelectric properties should be controlled using segmented or cascaded de-vice structures. On the other hand, the linearity of the curves at high temperatures implies that linearextrapolation of the curves can be used to estimate the efficiency at higher temperatures.6 iScience 26, 106494, April 21, 2023A BC DFigure 4. Performance by material composition(A–D) Achievable thermoelectric material efficiencies for 17 material groups are drawn for (A) Te alloys, (B) O/S/Se-related alloys, (C) Mg- or Si-based alloys,and (D) other alloys. See also key resources table (data150 and data261) and STAR Methods (material group classification of compositions) for samplematerial group classification.llOPEN ACCESSiScienceArticleEfficiency loss due to interfacial resistancesFigure 6 shows the degradation of efficiency of P-N leg-pair devices under electrical and thermal contact resis-tances compared to the best efficiency of perfect devices; see key resources table (data500). While the P andN legs generate electrical power, some of the power is lost via internal resistances.19 The P- andN-type thermo-electric semiconductors are metallized and connected to the electrodes on the substrates, and the device sub-strates are in contact with the external heat source and sink. Such a complex layered structure causes parasiticelectrical and thermal interfacial resistances (rc and k� 1c ), resulting in net power reduction. The net temperaturedifference decreases with interfacial thermal resistance, and additional Joule heating occurs at the ends of thelegs via additional interfacial electrical resistance. When contact resistances are extremely small (rc =10� 10 Um2 and kc = 105 W m� 2 K� 1), the relative efficiency loss is only 1.5% less than the best device efficiencywith perfect contact. For rc = 10� 9 Um2 and kc = 104 W m� 2 K� 1, the drop in the best efficiency is approxi-mately 12% on average. For the normal interface condition with rc = 10� 8 Um2 and kc = 104 W m� 2 K� 1,the enhanced interfacial Joule heating causes a significant drop in relative efficiency by 22% on average. Foran inferior interface condition (rc = 10� 7 Um2 and kc = 103 W m� 2 K� 1), a large efficiency loss occurs (67%loss on average), and the best efficiency is highly reduced to 6.5% (z1/15).iScience 26, 106494, April 21, 2023 7Figure 5. Representative P-N leg-pair devicesThermoelectric device efficiency curves are drawn for 9 representative devices with respect to various heat-sourcetemperatures when Tc is 300 K. Low-T (Th < 600 K), mid-T (600 K < Th < 950 K), and high-T (Th > 950 K) range applicationsare separated with vertical gray lines. The dotted line indicates the achievable best device efficiencies using single-stageP-N leg devices. The dot on each curve means that the device has realized the achievable best efficiency. The bestefficiency is computed with leg-area ratio optimization. See also Table 2 and the key resources table (data400 anddata600) for detailed information on the designed P-N leg-pair devices.llOPEN ACCESSiScienceArticleFor a given device, the maximum generated power Pmax can be approximately determined from the gener-ated voltage Vdev and total internal resistance Rdev,15 as follows:P = IðVdev � IRÞ % Pmax yV 2dev4Rdev=a2 DTeff4ðRmat +RcÞ ; (Equation 3)where a is the average Seebeck coefficient over the effective working temperature difference DTeff , Rmat isthe electrical resistance of thermoelectric material, and Rc is the parasitic interfacial/contact resistance.Thermal interfacial resistance causes a temperature drop at the interface (DTIF > 0).15 In addition, the Peltierheat flow causes higher DTIF and lower DTeff = DT � DTIF <DTideal = Th � Tc , compared to the open-cir-cuit condition.15,66 As a result, the device power and efficiency can be significantly decreased by the inter-facial thermal and electrical resistances.Table 2. Representative thermoelectric P-N leg-pair devicesP-leg composition (Ref., sample ID) N-leg composition Th [K] DT [K] hðdevÞ Ap/AnSi0.8Ge0.2/P (ref.50, sampleid = 21962) Si80Ge20 (ref.51, sampleid = 21190) 1285 985 13.02% 0.939(Nb0.60Ta0.40)0.8Ti0.2FeSb(ref.52, sampleid = 31566)Si80Ge20 (ref.53, sampleid = 21211) 1185 885 16.15% 0.426Pb0.9Na0.02Mg0.08Te (ref.54, sampleid = 300) (Hf0.6Zr0.4)NiSn0.99Sb0.01 - W0.087(ref.55, sampleid = 38585)940 640 16.70% 3.615Sn0.97Na0.03Se0.9S0.1 (ref.56, sampleid = 41016) Yb0.3Co4Sb14.4 (ref.57, sampleid = 41697) 885 585 17.05% 3.181Na0.035Eu0.03Mn0.03Pb0.905Te (ref.58,sampleid = 35891)xCo/Ba0.3In0.3Co4Sb12, x = 0.2% (ref.59,sampleid = 31358)860 560 17.01% 3.391Ge0.89Sb0.1In0.01Te (ref.60, sampleid = 31973) Mg2(Si0.4Sn0.6)Sb0.018(ref.61, sampleid = 9777)785 485 16.45% 1.912AgSbTe2 (ref.62, sampleid = 16668) Mg3.15Mn0.05Sb1.5Bi0.49Se0.01 (ref.63,sampleid = 27133)655 355 13.68% 0.961Bi0.4Sb1.6Te3Ag0.003 (ref.64, sampleid = 38722) Bi0.24Sb0.05Te0.61Se0.10 (ref.30, sampleid = 38264) 585 285 11.67% 1.106Bi0.5Sb1.5Te3 (ref.9, sampleid = 42662) Y0.2Bi1.8Se0.3Te2.7 (ref.65, sampleid = 16900) 435 135 7.75% 1.034Thermoelectric device efficiencies of nine representative thermoelectric P-N leg-pair devices and corresponding P- andN-leg compositions with sample ID (sam-pleid) numbers in Starrydata2. See Figure 5 and the key resources table (data400 and data600).8 iScience 26, 106494, April 21, 2023A BFigure 6. Efficiency loss due to interfacial resistances(A) The achievable best thermoelectric device efficiency under interfacial electrical and thermal resistances.(B) Relative efficiency of the best device efficiency under interfacial electrical and thermal resistances compared to the perfect interface devices. See also keyresources table (data500) for detailed information.llOPEN ACCESSiScienceArticleThe interfacial thermal resistance between the thermoelectric devices and outside heat source/sink can becharacterized using impedance spectroscopy67 and I-V measurement methods.66 These methods yieldedthe thermal contact conductivities of 5,800–19,000 W m� 2 K� 1 and 2,800 W m� 2 K� 1, respectively, whichare comparable to the good/bad interfaces, as shown in Figure 6. However, the exact interface responsiblefor the significant increase in resistance has not yet been identified, and could be situated between the ma-terials, electrodes, substrates, or heat sources. A spatially resolved method such as the time-domain ther-moreflectancemethod68may play an important role in characterizing the position-dependent local thermalconductivity inside a leg and device. On the other hand, the electrical interfacial/contact resistance hasbeen characterized by the spatial resolution measurement of the electrical potential along a singleleg.69 Recently, low contact resistivity of rc % 10� 9 Um2 has been developed in contacted thermoelectriclegs for several alloy systems.70,71 A 14% single-leg hðmatÞ was reportedly achieved for contact-developedGeTe alloys under DT = 440 K and Tc = 300 K.70ConclusionThe achievable best thermoelectric device efficiencies are theoretically investigated over ever-exploredmaterials. Theoretically, an efficiency of 1/3 is possible in an infinite-cascade device. An efficiency of 1/4is possible in a segmented device, while an efficiency of 1/6 is possible in a single-stage device. However,these theoretical limits are much higher than the measured device efficiencies of 1/8 and 1/10 of the multi-stage and single-stage devices. A poor interface quality may yield a low conversion efficiency of 1/15 to1/14. The discrepancy between theoretical and experimental efficiency can be mitigated in multistage de-vices by reducing interfacial resistance and selecting optimal thermoelectric materials. It ultimately sug-gests that collaboration between materials and energy-related fields could accelerate the industrializationof thermoelectric power generation technology.Limitations of the studyAlthough the thermoelectric property data from Starrydata2 are filtered, the thermoelectric property data-set used in this study may contain errors. This study is based on theoretical efficiency calculations usingone-dimensional thermoelectric equations, assuming that heat and current follows a one-dimensionalpath. In addition, the calculations neglect the thermal energy loss by radiation and convection processes.Therefore, the efficiencies in this study are overestimated compared to three-dimensional calculations.Thus, the best efficiency reported in this study is the upper bound of the best thermoelectric efficiencyof ever-explored materials.iScience 26, 106494, April 21, 2023 9llOPEN ACCESSiScienceArticleSTAR+METHODSDetailed methods are provided in the online version of this paper and include the following:d KEY RESOURCES TABLEd RESOURCE AVAILABILITYB Lead contactB Materials availabilityB Data and code availabilityd EXPERIMENTAL MODEL AND SUBJECT DETAILSd METHOD DETAILSB Overview of efficiency calculation processB Data preparation, filtering, and cleansingB Thermoelectric device modelB Thermoelectric performance calculationB Temperature distribution inside a legB Computation of the best efficiencyB Dimensional effect and radiation lossB Material group classification of compositionsd ADDITIONAL RESOURCESACKNOWLEDGMENTSThis work was supported by the Korea Electrotechnology Research Institute (KERI) Primary research pro-gram through the National Research Council of Science & Technology (NST) funded by the Ministry ofScience and ICT (MSIT) (23A01002), by the Korea Institute of Energy Technology Evaluation and Planning(KETEP) grant funded by the Ministry of Trade, Industry and Energy (MOTIE) (2021202080023D), and bythe National Research Foundation of Korea (NRF) grant funded by the Korea Government (MSIT)(2022M3C1C8093916), the Republic of Korea. Y.K. and M.K. were supported by the Japan Science andTechnology Agency (JST) CREST grant number (JPMJCR19J1), Japan.AUTHOR CONTRIBUTIONSB.R. conceptualized the study, led the design of thework, filtered the data, performed the efficiency calculations,analyzed the efficiency data, created the figures, andwrote the article. J.C. developed the software, analyzed theefficiency data, discussed the results, andwrote the article. S.P. supervised the project and discussed the results.M.K.andT.M.developedtheStarrydatawebsystemandgenerated thedatasetfile.Y.A., S.G.,A.T.,D.Y., andM.F.collected the thermoelectric property data from publications. S.G., Y.I., A.T., and Y.K. collected target publica-tions. M.K. and Y.K. served as an advisor on the project and discussed the results.DECLARATION OF INTERESTSThe authors declare no competing interests.Received: December 15, 2022Revised: March 14, 2023Accepted: March 22, 2023Published: March 27, 2023REFERENCES1. Goldsmid, H.J. (2016). 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World Appl. 68, 103649.https://doi.org/10.1016/j.nonrwa.2022.103649.iScience 26, 106494, April 21, 2023 13https://doi.org/10.1103/PhysRevB.73.144301https://doi.org/10.1109/ICT.2005.1519875https://doi.org/10.1126/sciadv.abf2738https://doi.org/10.3390/ma14226774https://doi.org/10.3390/ma14226774https://doi.org/10.1038/s41598-020-70320-7https://doi.org/10.1038/s41598-020-70320-7https://doi.org/10.1016/j.nonrwa.2022.103649https://doi.org/10.1016/j.nonrwa.2022.103649llOPEN ACCESSiScienceArticleSTAR+METHODSKEY RESOURCES TABLEREAGENT or RESOURCE SOURCE IDENTIFIERDeposited dataThermoelectric property database: Starrydata2.org,rawdata 20211124.zipStarrydata2.org; GitHub https://github.com/starrydata/starrydata_datasets/tree/master/datasets(data010) Sample’s formatted thermoelectric properties. This paper; Mendeley Data https://doi.org/10.17632/r9bhpv6vx9.1(data030) Sample information and metadata table This paper; Mendeley Data https://doi.org/10.17632/r9bhpv6vx9.1(data040) Sample’s filtering table This paper; Mendeley Data https://doi.org/10.17632/r9bhpv6vx9.1(data070) Sample’s interpolated TEPs This paper; Mendeley Data https://doi.org/10.17632/r9bhpv6vx9.1(data150) Sample’s composition classification into material group This paper; Mendeley Data https://doi.org/10.17632/r9bhpv6vx9.1(data234) Material efficiency raw data for different interfaces This paper; Mendeley Data https://doi.org/10.17632/r9bhpv6vx9.1(data261) Material efficiency raw data (only for 25 K interpolated) This paper; Mendeley Data https://doi.org/10.17632/r9bhpv6vx9.1(data300) Device configuration data generated usinghigh-efficiency P- and N-samplesThis paper; Mendeley Data https://doi.org/10.17632/r9bhpv6vx9.1(data400) Device efficiency data (only for optimalcurrents, small-size version)This paper; Mendeley Data https://doi.org/10.17632/r9bhpv6vx9.1(data500) Best ZTs, best material and device efficiencies fromcalculations, and experimentally reported efficienciesThis paper; Mendeley Data https://doi.org/10.17632/r9bhpv6vx9.1(data600) 9 representative best efficiency P-N leg-pair devices This paper; Mendeley Data https://doi.org/10.17632/r9bhpv6vx9.1(data700) A multiple-stage P-N leg pair device with avery high efficiency of 23.9%This paper; Mendeley Data https://doi.org/10.17632/r9bhpv6vx9.1Software and algorithmsPython 3.8. 5 Python software foundation https://www.python.orgNumPy 1.19.2 NumPy project and community http://www.numpy.org/SciPy 1.5.2 SciPy developers https://scipy.org/Pandas 1.1.3 Pandas developers https://pandas.pydata.org/Barycentric polynomial interpolation at Chebyshev nodes Chung et al.72 https://doi.org/10.1038/s41598-020-70320-7One-dimensional temperature-solving integral algorithmfor a thermoelectric legRyu et al.15 https://doi.org/10.1016/j.isci.2021.102934RESOURCE AVAILABILITYLead contact� Further information and requests for resources should be directed to and will be fulfilled by the leadcontact, Byungki Ryu (byungkiru@keri.re.kr).Materials availability� This study did not generate new unique reagents.Data and code availabilityd Data for the thermoelectric properties, related sample and publication information, and calculated effi-ciencies have been deposited at Mendeley and are publicly available as of the date of publication (Men-deley Data: https://doi.org/10.17632/r9bhpv6vx9.1), as summarised in the key resources table. TheDATAIDs and related calculation processes are described in the STAR Methods (overview of efficiencycalculation process).d This paper does not report original code. Instead, all detailed algorithms are explained in the STARMethods.14 iScience 26, 106494, April 21, 2023mailto:byungkiru@keri.re.krhttps://doi.org/10.17632/r9bhpv6vx9.1http://Starrydata2.orghttps://www.starrydata2.org/https://github.com/starrydata/starrydata_datasets/tree/master/datasetshttps://github.com/starrydata/starrydata_datasets/tree/master/datasetshttps://doi.org/10.17632/r9bhpv6vx9.1https://doi.org/10.17632/r9bhpv6vx9.1https://doi.org/10.17632/r9bhpv6vx9.1https://doi.org/10.17632/r9bhpv6vx9.1https://doi.org/10.17632/r9bhpv6vx9.1https://doi.org/10.17632/r9bhpv6vx9.1https://doi.org/10.17632/r9bhpv6vx9.1https://doi.org/10.17632/r9bhpv6vx9.1https://doi.org/10.17632/r9bhpv6vx9.1https://doi.org/10.17632/r9bhpv6vx9.1https://doi.org/10.17632/r9bhpv6vx9.1https://doi.org/10.17632/r9bhpv6vx9.1https://www.python.org/http://www.numpy.org/https://scipy.org/https://pandas.pydata.org/https://doi.org/10.1038/s41598-020-70320-7https://doi.org/10.1016/%20j.isci.2021.102934llOPEN ACCESSiScienceArticled Any additional information required to reanalyse the data reported in this paper is available from thelead contact upon request.EXPERIMENTAL MODEL AND SUBJECT DETAILSThis study did not perform experimental works. This study is theoretical.METHOD DETAILSOverview of efficiency calculation processThe best thermoelectric efficiency is calculated and explored using the following procedure. The values inparentheses at the end of the following sentences are the corresponding data identification codes(DATAIDs) for the analysis and data. First, a formatted thermoelectric property file (data010) is generatedby extracting the raw data from Starrydata2. A samplemetadata file (data030) is also generated; it containsthermoelectric data length, data reference, composition information, available temperature range, pub-lished date, etc. A data filter is used to test that eachmaterial sample’s dataset of thermoelectric propertiesis complete, valid, or errorless, and the result is recorded as sample lists (data040). The thermoelectricproperties are interpolated (data070) for the ZT distribution analysis. Composition data are transformedinto a composition vector, and the samples are classified into given material groups (data150). The ther-moelectric material efficiency of the samples in the list is computed for available temperature ranges(data234), and material efficiency data are interpolated (data261). Using high-efficiency samples, P-Nleg pair device configurations are generated (data300), and the device efficiencies are computed(data400). Finally, the best ZT , best material efficiency, and best device efficiency curves are obtained(data500). The best efficiencies are compared to the experimentally reported device efficiencies(data500). Among them, 9 representative best efficiency devices are analysed (data600). A multiple-stageP-N leg pair device with a very high efficiency of 23.9% is found (data700). Related data of correspondingDATAIDs have been deposited at Mendeley and are publicly available as of the date of publication (Men-deley Data: https://doi.org/10.17632/r9bhpv6vx9.1) as summarized in the key resources table.Data preparation, filtering, and cleansingHigh-quality thermoelectric property data are obtained from the Starrydata2 thermoelectric web-DB,17which is growing with time. For the investigations, we use the thermoelectric property data of version2021-November-24, consisting of 43,601 material samples from 7,994 published papers (‘‘20211124.zip’’in https://github.com/starrydata/starrydata_datasets). Among them, 16,420 samples are chosen from3,585 published papers that contain a full set of three thermoelectric properties. Note that a completeset of thermoelectric properties is mandatory for the calculation of thermoelectric performance. Then,high-quality thermoelectric property data are obtained after filtering out unphysical and erroneous data.A total of 33 data filters are used, that is, physics filters to remove unphysical values of thermoelectric prop-erties and temperature ranges and error filters to remove insufficiently/incorrectly labelled data that arehardly readable by a computer. See key resources table (data040) for the thermoelectric sample filteringtable. For high-efficiency samples, the correctness of the thermoelectric properties from Starrydata2 isconfirmed by visual inspection. Consequently, high-quality thermoelectric property data are obtained,which consist of 13,338 complete and valid samples from 3,120 published papers. The correspondingdata size is 215,526 ðT ;aðTÞÞ pairs, 223,404 ðT ; rðTÞÞ pairs, and 187,244 ðT ; kðTÞÞ pairs. For efficiency calcu-lations, thermoelectric properties are interpolated in a piecewise linear manner and extrapolated in a con-stant value manner so that the resulting properties are continuous functions of temperature. Among thecomplete and valid samples, 12,645 samples from 2,919 publications have thermoelectric properties atTh > 300 K. Using these samples, thermoelectric efficiencies at Th > 300 K are evaluated for various devicemodels: see key resources table (data261). Related data on the correspondingDATAIDs have been depos-ited at Mendeley and are publicly available as of the date of publication (Mendeley Data: https://doi.org/10.17632/r9bhpv6vx9.1) as summarized in the key resources table.Thermoelectric device modelIn the thermoelectric device, P- and/or N-type thermoelectric materials, called thermoelectric legs, areplaced between hot- and cold-side substrates. The thermal boundary conditions are assumed to be theDirichlet condition, which means that Th and Tc are fixed during device operation. We compute the theo-retical maximum conversion efficiency, ignoring radiation and convection losses. The length of a thermo-electric leg is assumed to be 3 mm. Since the leg is connected to substrates, we consider the interfacialiScience 26, 106494, April 21, 2023 15https://doi.org/10.17632/r9bhpv6vx9.1https://github.com/starrydata/starrydata_datasetshttps://doi.org/10.17632/r9bhpv6vx9.1https://doi.org/10.17632/r9bhpv6vx9.1llOPEN ACCESSiScienceArticleelectrical and thermal resistances. The electrical and thermal resistances are simultaneously imposed usingtwo additional segments of 0:1 mm attached at the ends of the leg unless interfacial resistances are zero; intotal, the leg length is 3 mm for a perfect interface and 3:2 mm if there are interfacial resistances. We as-sume that the electrical and thermal currents flow perpendicular to the substrate, which implies a one-dimensional flow. For a single-leg device, the leg cross-sectional area of A = 3 mm33 mm is adopted.For a P-N leg-pair device, the cross-sectional areas of P- and N-type legs (Ap; An) are set to Ap =ð1 � xÞ,A and An = x,A, for a number x between 0.02 and 0.98. For x, we consider 11 values at the Cheby-shev nodes of the second kind72 between 0.02 and 0.98.Thermoelectric performance calculationIn a one-dimensional thermoelectric leg, the heat currents at the hot and cold sides (Qh; QcÞ and the powerP are given as follows:Qh;c = � Akh;c�dTdx�h;c+ I aðTh;cÞTh;c ; (Equation 4)P = Qh � Qc ; (Equation 5)where h and c denote the hot and cold sides, A is the leg cross-sectional area, and I is the electric currentflowing through the leg. Once the temperature distribution inside the leg is known, the thermoelectric ma-terial efficiency can be easily computed for given electrical and thermal conditions:hðmatÞ = hðI;Th; TcÞ =Qh � QcQh: (Equation 6)The P-N leg-pair device efficiency can be computed as follows:hðdevÞ =PðtotÞQðtotÞh=PðpÞ +PðnÞQðpÞh +QðnÞh: (Equation 7)Temperature distribution inside a legIn a steady-state one-dimensional leg, thermoelectric effects are governed by the following thermoelectricdifferential equation15,73:ddx�kdTdx�� T�dadT��dTdx�J + rJ2 = 0: (Equation 8)This equation can be transformed into a thermoelectric integral equation for temperature distribution TðxÞvia double integration on fT ðxÞd � T dadTdTdx J+ rJ2 as follows15:TðxÞ =0@Th � KDTAZx01kðsÞds1A+0@�Zx0FT ðsÞkðsÞ ds +KdTAZx01kðsÞds1A; (Equation 9)where FT ðxÞdZx0fT ðsÞds, dTdZL0FT ðxÞkðxÞ dx, and L is the leg length. By iteratively computing (9), we find thetemperature distribution inside the legs.15Computation of the best efficiencyA thermoelectric sample is evaluated using the maximum thermoelectric efficiency calculation undervarious temperature differences DT = Th � Tc , where the cold-side temperature is 300 K and Th is chosenbetween 301 K and the maximum available temperature (Tmax) plus 15 K, including two ends. For Th, weconsider 11 values at the Chebyshev nodes of the second kind72 between 301 K and Th:The generated electrical power and input heat current are computed using Equations 4, 5, 6, 7, 8, and 9 15under a given electrical current and thermal boundary conditions. Then, the maximum efficiencies aresearched by varying electrical currents for 12,645 samples, which are available for T > 300 K. Using thesearched high-performance P- and N-samples, of which the efficiency is larger than or equal to 85% ofthe best efficiency, single-stage P-N leg-pair configurations of 14,702 are constructed; see key resources16 iScience 26, 106494, April 21, 2023llOPEN ACCESSiScienceArticletable (data300). We also consider 11 leg-pair geometries and 5 interfacial resistance conditions for the P-Nleg-pair devices. In total, 63; 225 ð= 12; 6453135Þ single-leg devices and 808; 610 ð= 14; 70231135Þ P-Nleg-pair devices are considered. For each device configuration, 11 electrical current points and 11 thermalboundary conditions are considered. During this process, the leg geometry parameter x, current I, andtemperature Th are sampled at the Chebyshev node of the second kind, which is suited for polynomialinterpolation.72 In total, we calculate hðmatÞ’s for 7,650,225 configurations (over 12645materials, 5 interfaces,11 I points, 11 Th points) and hðdevÞ’s for 97,841,810 configurations (over 14702 material leg pairs, 11 geom-etry x values, 5 interfaces, 11 I points, 11 Th points). Finally, we explore the 100 million thermoelectricmaterial and device efficiency space (7650225 material cases + 97841810 device cases = 105,492,035 effi-ciencies) and obtain the best thermoelectric efficiencies.Dimensional effect and radiation lossWe tested the efficiency performance difference between the one- and three-dimensional models. Forthree-dimensional leg-pair calculations, the efficiencies are found to be dependent on leg geometry.The efficiency difference between the 3 mm33 mm33 mm leg and 0:5 mm30:5 mm33 mm was found tobe �3.6%. As the ratio of leg height to leg width increases, the efficiency increases and converges to1-dimensional leg efficiency. Adding radiation (RAD) and convection (CONV) heat transfers from the deviceto the outside may lower thermoelectric efficiency. Thus, we can conclude that the one-dimensional modelgives the upper bound for the best thermoelectric efficiency:hð1�dimÞ R hð3�dimÞ R hð3�dimÞ+ ðRAD and CONVÞ:Depending on the geometry and heat source temperature, the thermoelectric efficiency can be furtherreduced by �20% or more,19 when blackbody-like radiation occurs, that is, the emissivity is 1.Material group classification of compositionsSamples are grouped into 17 material groups based on composition analysis. We manually define the 17material groups with representative compositions. To classify a sample into amaterial group, the fraction ofhost or anionic elements is calculated. If the fraction is greater than a certain value (90% for the host, 30% foranionic analysis, 20% for oxide classification), then the sample is grouped into one of the 17material groupsof which the representative composition is most similar to the sample’s composition. For example,(Bsi0.4Sb1.6)Te3Ag0.03 is similar to Bi2Te3Ag0.03 and is grouped into Bi2Te3. The classification results canbe found in the key resources table (data150 and data261).ADDITIONAL RESOURCESThere are two related preprint versions of this paper by same authors.� At arXiv.org: https://doi.org/10.48550/arXiv.2210.08837.� At Research Square: https://doi.org/10.21203/rs.3.rs-2179853/v1.iScience 26, 106494, April 21, 2023 17https://doi.org/10.48550/arXiv.2210.08837https://doi.org/10.21203/rs.3.rs-2179853/v1 Best thermoelectric efficiency of ever-explored materials Introduction Results Thermoelectric efficiency exploration Best thermoelectric efficiency Distribution of thermoelectric properties History of thermoelectric performances: ZT and efficiency Performance by material composition Representative P-N leg-pair devices Efficiency loss due to interfacial resistances Conclusion Limitations of the study Acknowledgments Author contributions Declaration of interests References STAR★Methods Key resources table Resource availability Lead contact Materials availability Data and code availability Experimental model and subject details Method details Overview of efficiency calculation process Data preparation, filtering, and cleansing Thermoelectric device model Thermoelectric performance calculation Temperature distribution inside a leg Computation of the best efficiency Dimensional effect and radiation loss Material group classification of compositions Additional resources