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Michael Parzer, Alexander Riss, Fabian Garmroudi, Johannes de Boor, [Takao Mori](https://orcid.org/0000-0003-2682-1846), Ernst Bauer

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SeeBand: a highly efficient, interactive tool for analyzing electronic transport datanpj | computationalmaterials ReviewPublished in partnership with the Shanghai Institute of Ceramics of the Chinese Academy of Scienceshttps://doi.org/10.1038/s41524-025-01645-ySeeBand: a highly efficient, interactivetool foranalyzingelectronic transportdataCheck for updatesMichael Parzer1 , Alexander Riss1 , Fabian Garmroudi1 , Johannes de Boor2,3, Takao Mori4,5 &Ernst Bauer1SeeBand is an interactive tool for extracting microscopic material parameters by fitting temperature-dependent thermoelectric transport properties using Boltzmann transport theory. With real-timecomparison between electronic band structures and transport data, it analyzes the Seebeckcoefficient, resistivity, and Hall coefficient. Neural-network-assisted guesses and efficient fittingroutines enable high-throughput processing of large datasets. SeeBand accelerates material designby allowing electronic band structure models to be derived directly from a single sample’s transportmeasurements.Rapidly increasing numbers of experimental data call for enhanced tools ofdata analysis. In the field of thermoelectrics, tens of thousands oftemperature-dependent transport data have been accumulated. The Star-rydata2 openweb database1 alone aggregates TE property data from 51,985samples and 8,956 papers as of May 2024. The analysis of these immenseamounts of data was often forsaken or performed within a simple single-parabolic-band model in the unipolar regime with a single dominant scat-tering mechanism in the original publications. Since most high-performance TE materials are, however, often far more nuanced, e.g.,showcasing bipolar transport2–4, convergence ofmultiple electronic bands5–8or complex carrier scattering9–11, the lack of adequate and accurate analysesleads to limited or even erroneous interpretations and researchers revertingto time-consuming empirical studies.Tuning and manipulating electronic transport is crucial for realizingnext-generation technologies, such as quantum computing or energy har-vesting, which are expected to play an integral role in our future society12–15.In this context, it is imperative to understand the behavior of charge carriersto enable informed and targeted research and accelerate material’s devel-opment. This is of utter importance across various branches of the physicalsciences, encompassing, for instance, the pursuit of high-temperaturesuperconductivity in fundamental condensed matter physics, or the opti-mization of TE transport in materials science. By comparing experimentaldata with theoretical predictions, profound insights into electronic corre-lations, microstructure, and structure-property relationships can beobtained. Furthermore, the synergy between transport data analysis andtheoretical calculations validates respective results, establishing a robustfoundation for understanding material properties. While novel theoreticalframeworks and codes capable of directly calculating transport propertiesfrom ab initio electronic structure calculations are constantly beingdeveloped16–20, there is currently no common tool that can derive micro-scopic EBSparameters in reverse fromexperimental transport data. Instead,various groups have developed similar frameworks for data analysis,demonstrating the demand for such measurement data analysis, whichcomplements theoretical calculations21–26. The inaccessibility of these tools,however, has led to the common usage of much simpler formalisms formaterial characterization like theweighted mobility, primarily derived fromapproximations to the single-parabolic-band model27–30.Here, we introduce SeeBand, a software package for electronic trans-port data analysis, based on the Boltzmann transport formalism andparabolic band approximation. SeeBand constitutes a highly efficient fittingtool that allows visualizing and linking the interdependence of fundamental,microscopic features such as the EBS around the Fermi energy EF withmacroscopic quantities such as measured temperature dependencies ofelectronic transport. Our code features a unique, neural-network-assistedleast-squares fitting algorithm, which is able to concurrently handle thetemperature-dependent electrical conductivity, Seebeck coefficient, andHall coefficient, different but complementary electronic transport proper-ties that are most ubiquitously and easily measured in a laboratory. Thehighly efficient numerical framework of SeeBand allows extremely rapidprocessing of data, which can be leveraged in the high-throughput analysesof big datasets, as we demonstrate in this paper on the basis of around 1000datasets of half-Heusler compounds.While SeeBand is designed to enhance the comprehension of TEtransport in narrow-gap semiconductors, fostering more targeted andefficient experimental research, its utility extends beyond thermoelectrics toother domains within materials science and solid-state physics, where a1Institute of Solid State Physics, TU Wien, Vienna, Austria. 2Institute of Materials Research, German Aerospace Center (DLR), Cologne, Germany. 3University ofDuisburg-Essen, Faculty of Engineering, Institute of Technology for Nanostructures (NST) and CENIDE, Duisburg, Germany. 4International Center for MaterialsNanoarchitectonics (WPI-MANA), National Institute forMaterials Science, Tsukuba, Japan. 5GraduateSchool of Pure andAppliedSciences, University of Tsukuba,Tsukuba, Japan. e-mail: michael.parzer@tuwien.ac.at; alexander.riss@tuwien.ac.at; f.garmroudi@gmx.atnpj Computational Materials |          (2025) 11:171 11234567890():,;1234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41524-025-01645-y&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41524-025-01645-y&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41524-025-01645-y&domain=pdfmailto:michael.parzer@tuwien.ac.atmailto:alexander.riss@tuwien.ac.atmailto:f.garmroudi@gmx.atwww.nature.com/npjcompumatsprofound understanding of charge transport, especially in semimetallic andsemiconducting systems, is crucial.This paper is organized as follows: First, we provide a brief overview ofthe theoretical foundations underlying charge transport in TE materials,along with their implementation in the SeeBand code. In the subsequentsection, a practical illustration and demonstration of usage (see also theSupplementaryMovie as an additional user guide) is offered. Following this,section “Test examples” delves into real-world test examples, highlightingthe applicability across differentmaterial classes. Finally, we underscore thatthe high efficiency and capability for automated fitting in the SeeBand codeenables high-throughput data analysis, exemplified through the examina-tion of 1000 TE property datasets of half-Heusler compounds within theStarryData2 database1.Theoretical backgroundThe SeeBand code is constructed on a framework based on the Boltzmanntransport theory and the parabolic band approximation. In the semiclassicalBoltzmann theory, the generalized transport coefficients Lα(μ, T), whichdepends on both temperature and the chemical potential μ(T). Note that theFermi level, EF, denotes the temperature-dependent chemical potential atT = 0K. Lα(μ, T) can be written in terms of a general transport distributionfunction Σ(E):Lαðμ;TÞ ¼ q2Z 1�1ΣðEÞðE � μÞα � ∂f∂E� �dE: ð1ÞΣ(E) depends on the EBS and dominant scattering processes:ΣðEÞ ¼XNi¼11ð2πÞ3Zτi;kvi;kvi;kδðE � Ei;kÞ d3k: ð2ÞThe summation is performed over N bands and integration is doneover the first Brillouin zone, with vi,k and τi,k being the band velocities andelectron relaxation times, respectively. For parabolic bands in 3Dmaterials,the spherical symmetry of the Fermi surface in k space allows simplificationof Equation (2) toΣðEÞ ¼ DðEÞv2ðEÞτðEÞ; ð3ÞwithD(E) being the density of states. SinceD(E)∝E1/2, v2(E)∝ E and τ(E)∝E−1/2 for acoustic-phonon and alloy-disorder scattering, it follows that Σ(E)∝E. Thus, Equation (1) canbewritten in termsof Fermi integrals for a singlebandFjðηðTÞÞ ¼Z 10ξjeξ�η þ 1dξ: ð4ÞHere, ξ = E/(kBT) and η = μ/(kBT) represent the reduced energy andchemical potential, respectively. Hence, the number of variables is reducedand fitting routines become reasonable. Utilizing Fermi integrals sig-nificantly enhances the efficiency of the fitting process, as robust solutionsfor these integrals are readily available. Since these integrals are at the heartof the SeeBand framework and enter all the expressions of the transportequations, we especially focused on optimizing the numerical efficiency oftheir computation. This enables a rapid processing of the data, where theFermi integrals can be evaluated up to 106 times per second.Equations used for the fitting processBelow, we briefly list the equations and fit parameters (highlighted in bold)which are used in the refinement process. For a single parabolic band, theequations for the transport properties are given by31SðTÞ ¼ kBeη� 2F1ðηÞF0ðηÞ� �; ð5ÞρðTÞ ¼ 2ffiffiffi2pe2kBTπ2_3eτm� �TγF0 η� �� ��1; ð6ÞRHðTÞ ¼π2_32e 2mkBT� �3=2F�1=2 η� �F20 η� � : ð7ÞHere, kB, e and ℏ are natural constants (Boltzmann constant, electroncharge and reduced Planckian constant, respectively), while η, ~τ andm arethe reduced chemical potential, the scattering prefactor and the effectivemass, respectively, representing the refinement parameters. For two para-bolic bands (2PB), the single-band contributions are weighted and summedup appropriately31,32 (a general derivation of the transport fitting equationsfor N bands is given in Supplementary Note 3), yielding for the Seebeckcoefficient, the resistivity and the Hall coefficient.S2PBðTÞ ¼F0ðηÞS1ðη;TÞ þ ετ=εm F0ðη� Eg=kBTÞS2ðη� Eg=kBT;TÞF0ðηÞ þ ετ=εm F0ðη� Eg=kBTÞ;ð8Þρ2PBðTÞ ¼2ffiffiffi2pe2kBTγþ1N1π2_3m1eτ1 F0ðηÞ þ ετ=εmF0ðη� Eg=kBTÞ  � ��1;ð9ÞRH;2PBðTÞ ¼3π2_32e 2kBT� �3=2 m�3=21F�1=2ðηÞ þ N1=N2� �5=2ε2τ=ε7=2m F�1=2ðη� Eg=kBTÞF0ðηÞ þ ετ=εmF0ðη� Eg=kBTÞ  2264375:ð10ÞThe scope of SeeBand is to extract relevant microscopic EBS para-meters from experiments, by fitting data using above equations. Figure 1sketches the refinement technique implemented in the SeeBand software. Inthe case of two parabolic bands, modeling the Seebeck coefficient viaEquation (8) requires three independent fitting parameters: the position ofthe Fermi level EF, which determines the temperature-dependent chemicalpotential η, the band gapEg, i.e., the energy difference between the two bandedges, as well as a weighting parameter ετ/εm. Here, ετ ¼ eτ2=eτ1 denotes theratio of the reduced electron relaxation times of the two bands andεm = (N1m2)/(N2m1) is the band mass ratiom2/m1, weighted with the banddegeneraciesN1 andN2,whichare inputparameters that have tobe specifieda priori. Note that during thefitting procedure, the chemical potential has tobe continuously recalculated in each step when the EBS parameters EF, Egand εm are changed (seeFig. 1a).Most importantly, eachof these parametershas aunique effect on the temperature-dependentbehavior of S(T), enablingrobust and unambiguous conclusions to be drawn when fitting experi-mental data of S(T).Other transport properties like the temperature-dependent resistivityρ(T) and Hall coefficient RH(T) should in principle be describable by thevery same EBSmodel. To further enhance the robustness of the obtained fitparameters, simultaneous analyses of all these temperature-dependenttransport properties (S(T), ρ(T), RH(T)) can be performed in theSeeBandframework.Since ρ(T) depends on the relaxation times τi of the charge carriersassociated with their respective electronic bands, additional fit parametershave to be introduced. In the case of two parabolic bands, the minimalnumber of additional free parameters is given by (i) the electron relaxationtime eτ1 of the first band and (ii) the ratio of the electronic relaxation timesbetween the twobands ετ ¼ eτ2=eτ1. Forparabolic bands andhence sphericalhttps://doi.org/10.1038/s41524-025-01645-y Reviewnpj Computational Materials |          (2025) 11:171 2www.nature.com/npjcompumatsFermi surfaces, τ can be written in a general formτðE;TÞ ¼ C � TγDðEÞ ¼ eτ TγE�1=2m�3=2: ð11ÞHere,C is a constant, γ is a scattering-specific parameter andeτ denotesan energy- and temperature-independent prefactor. The most importantcharge carrier scattering mechanisms in thermoelectric materials are (i)acoustic phonon and (ii) alloy disorder scattering, for which γ =−1 and 0,respectively. In themetallic limit, this yields thewell-known linear resistivitybehavior expected for electron-phonon scattering ρ(T)∝ τ−1∝ T at elevatedtemperatures and a temperature-independent residual resistivity ρ0 foralloy-disorder scattering. The electron relaxation time is further determinedby material-specific parameters31,33,34, which can be subsumed in a constantprefactoreτph ¼ π_4v2l ρmffiffiffi2pkBΞ2ph; ð12Þeτdis ¼ffiffiffi2pπ_4nAxAð1� xAÞU2dis: ð13ÞWhileSeeBand currently is limited to these two scatteringmechanisms,often relevant at medium and high temperatures, in future versions, addi-tional mechanisms can be included to the framework.Other important scattering mechanisms include ionized-impurityscattering, which typically is important at low temperatures and yieldsdifferent energy-dependence (∝E−2 andγ =−1)35,36 or polar-optical phononscattering, which is especially relevant for polarmaterials36–38 anddominatesthe electronic transport in the same. In case the of dominance of scatteringmechanisms other than acoustic-phonon and alloy-impurity scattering,SeeBand should not be employed. If in doubt, at least the underlyingassumptions with respect to the scattering mechanisms should be statedclearly togetherwith theobtained results on theband structureparameters37.For dominant acoustic-phonon scattering, the relevant physicalparameters defining eτ are the electron-phonon interaction, which may berepresented by an acoustic deformation potential Ξph, the longitudinalsound velocity vl and the material density ρm. In the case of alloy-disorderscattering, eτ depends on the atomic fraction of alloyed atoms xA, theirparticle density nA and on a scattering potentialUdis, which accounts for therandom fluctuations of the periodic lattice potential caused by the randomsubstitution of alloyed atoms. In the SeeBandframework, the user candecidewhether one or both of these scattering processes are taken into account.When both scattering processes are considered at the same time, Mathies-sen’s rule is employed to calculate the total relaxation time τtot = τphτdis/(τph+ τdis).As sketched in Fig. 1a, b the SeeBand code uses an iterative loop,wherein the parameters obtained from the Seebeck modeling are directlytransferred to evaluate ρ(T). Since ετ changes when ρ(T) is fitted, S(T) ismodeled again using the relaxation times from the resistivity fit. Both stepsare reiterated until convergence is achieved. Finally, when data of thetemperature-dependentHall coefficient are also available, the absolute valueof the effective masses can be evaluated by performing a final single-parameter (m1) fit of RH(T) (see Fig. 1c). This procedure allows for aminimal number of free parameters to be used, ensuring more robust andunambiguous results. Simultaneously, the user is able to extract a variety ofrelevant physical parameters concerning the EBS and scattering timesdirectly from synergistic temperature-dependent measurements on a singlesample. The ambiguity of the refinement results can easily be checked by theuser when running the fit with different sets of starting parameters.Demonstration of useTo enhance the user experience and aid understanding of how various bandparameters influence transport properties, we developed a user-friendlygraphical interface (GUI) for SeeBandproviding interactive control over theparameters. As the usermodifies the values, the transport properties updatein real-time, along with a visualization of the corresponding effective bandstructure. Additionally, the GUI facilitates direct control over the fittingprocess and allows extraction of further information from the results. Datacan also be exported for external analysis. A concise overview of the GUI isprovided here, with a more comprehensive user guide available in Supple-mentary Note 2.Graphical user interfaceThe user interface for the two-parabolic-bandmodel and all three transportcoefficients is illustrated in Fig. 2. The top toolbar (a) of the GUI providesaccess to different windows to input data, adjust and fit the transportproperties based on available data and switch between the single- and two-parabolic-band model.Relevant band parameters can be entered in the text fields and sliderson the left (b). After successfully fitting the data, the results of the fitting aredisplayed in green. The four panels on the right (c) show the three transportcoefficients (S, ρ, RH) along with the effective band structure derived fromthe band parameters. Gray circles represent measurement data, while thered and green lines show the transport properties derived from the inputparameters and the least-squares fit, respectively.The Adjust to fit button sets the user controlled parameters (and theresulting calculated transport properties) to the values of the prior fitting,enabling the analysis of how modifications of those parameters impact thetransport properties. The buttons at the bottom (d) allow the user to setlimits for thefitting process, access additional information such as electronicthermal conductivity or the temperature dependence of the chemicalpotential μ(T), and export relevant data to well-arranged text files.WorkflowThe workflow proceeds as follows: First, the measurement data are inputusing the input window (the first symbol on the toolbar). Next, theFig. 1 | SeeBand’s fitting algorithm. aModeling the temperature-dependent See-beck coefficient S(T) with two parabolic bands requires three independent fittingparameters, i.e., the Fermi energy EF, the band gap Eg and a weighting parameterεm = (N1m2)/(N2m1), which depends on the band degeneracies and effective masses.During the fitting routine, the chemical potential μ(T) is calculated each time a bandparameter is changed. b The band parameters obtained from the Seebeck fit aredirectly used formodeling the temperature-dependent resistivity ρ(T), forwhich twoadditional parameters have to be taken into consideration, namely the reducedelectron relaxation time of thefirst bandeτ1 and the ratio of the relaxation times of thetwo bands ετ ¼ eτ2=eτ1. These parameters are transferred between the two fit loopsand iteratively optimized. c In a final single-parameter fit of the temperature-dependent Hall coefficient RH(T), the absolute value of the effective masses can bedetermined.https://doi.org/10.1038/s41524-025-01645-y Reviewnpj Computational Materials |          (2025) 11:171 3www.nature.com/npjcompumatsappropriate model is selected, based on the measurement data and thedesiredfittingmodel. Theuser has the optionof onlymodeling S(T) (see Fig.1a), a combined iterative analysis of the Seebeck coefficient and resistivity(see Fig. 1a, b) or, when Hall data are also available, a combined analysis ofS(T), ρ(T) and RH(T) together. Additionally, the user can choose between asingle and two-parabolic bandmodel. The trainedneural network (NN)willprovide an initial guess for the relevant band parameters, usually offering asufficiently good starting point for the fit to converge. Before starting the fit,the relevant scattering mechanisms are chosen, and limits for all fittingparameters can be adjusted, if necessary. Finally, the fitting process can beinitiated to determine the optimal electronic structure that best describes themeasurement data within the chosen model. Subsequent to the fit, theinterface allows for further analysis bymanually adjusting parameters whileobserving the changes in S, ρ, andRH.Moreover, additional information canbe displayed based on the number of bands and providedmeasurement data.Test examplesIn this section, to elucidate the usefulness and validity of the refinementtools, we present practical use-cases of well-studied materials, wherereliable comparison of our modeling results with the electronic structurecan be made. To demonstrate applicability across different materialclasses with distinct electronic structures, we study P-doped silicon (Si:P)as a wide-gap semiconductor, and the full-Heusler compoundFe2VAl0.9Si0.1 as a narrow-gap thermoelectric material. Comprehensivetemperature-dependent transport data for the Seebeck coefficient, theelectrical resistivity, and the Hall coefficient are available in the literaturefor both materials39,40. While we discuss in the following the case of anEBS consisting of a valence and a conduction band it can also beemployed for two bands of the same type.Phosphorous-doped siliconFigure 3a–c show the temperature-dependent Seebeck coefficient S(T),electrical resistivity ρ(T) as well as the Hall coefficient RH(T) of Si:P39 in thetemperature range 300–650 K. The temperature-dependent transportproperties were modeled assuming acoustic-phonon scattering using themulti-stage refinement algorithm implemented in the SeeBand code (cf. Fig.1) and the least-squares fits are depicted as red solid lines in Fig. 3a–c. Verygood agreement between experimental data and the theoretical curves isapparent for all three independent transport properties, highlighting therobustness of the underlying theory.The obtained EBS parameters from the least-squares fits are sum-marized in Table 1 and compared to values from various literature studies,yielding very good overall agreement. This demonstrates the ability ofcombined analyses of various transport properties for extracting robustinformation of the EBS.cabdFig. 2 | User interface of SeeBand. Exemplary snapshot of the user interface of theSeeBand software. aToolbar allowing to toggle between different windows. The usercan choose between a single- or two-parabolic band analysis of the transportproperties. Additionally, the interface offers to either analyze just the Seebeckcoefficient, or a combination of different transport properties simultaneously,depending on what data are available. b Different fit and input parameters, asexplained in section “Theoretical background” and section “Demonstration of use”.c Imported experimental data and partially adjusted effective electronic structure inthe graphical user interface (GUI). The experimental data are depicted as graycircles, while the calculated transport properties corresponding to the adjustableparameters of the sliders are depicted as the solid red lines. The panel on the top rightshows the effective band structure close to the Fermi level EF (dashed red line),corresponding to the transport properties displayed as a red solid lines. Afterpressing the Fit from manipulate button, the mathematically best fit of band para-meters for the respective data and the resulting transport coefficients are displayed ingreen. d Buttons for further features including additional graphs, like the Seebeckcoefficients and conductivities of the individual bands, the temperature dependenceof the chemical potential, as well as the electronic thermal conductivity includingbipolar contributions.https://doi.org/10.1038/s41524-025-01645-y Reviewnpj Computational Materials |          (2025) 11:171 4www.nature.com/npjcompumatsIn lightly doped semiconductors like Si:P, the Fermi energy is situatedat the localized electronic states arising from the dopant’s impurity levels,which lie about 0.044 eVbelow the conduction band edge41. In SeeBand, thelocalized nature of such localized in-gap impurity states cannot be captured,since only parabolic bands, i.e., free carriers, are assumed. Nonetheless, theobtained position of EF, which lies only about 0.02 meV within the con-duction band, aligns with the notion of EF being pinned just at the con-duction band edge.The band gap, extracted with SeeBand, yields Eg = 1.08 eV, in near-perfect agreementwith the real band gap of silicon, which is 1.12 eVat roomtemperature and decreases down to ≈1.03 eV at 650 K42,43.The conduction band of Si forms six equivalent ellipsoidal constantenergy surfaces, associated with different values of the effective mass inlongitudinal and transverse directions. The valence band complex consistsof a doubly degenerate light and heavy hole band plus a “split-off” holeband43. In our transport modeling, these bands are assumed parabolic andisotropic and the derived values of the effective mass have to be consideredan effective average. Despite these simplifications, the derived values of εmand m1 are in good agreement with those from literature43. It should benoted that phonondrag also contributes to the Seebeck coefficient in silicon,particularly at lower temperatures44,45. Since this effect is not capturedwithinour modeling framework, it represents an additional source of slightdeviation between the experimental data and the fit.Full-Heusler Fe2VAl0.9Si0.1Most state-of-the-art thermoelectric materials are narrow-gap semi-conductors. Todemonstrate the applicability of SeeBand to such systems,westudied thermoelectric transport in Fe2VAl-based full-Heusler compounds,which have emerged as an important class of thermoelectric materials forpotential applications in the temperature range 300–500 K40,46–49. In Fe2VAl,a threefold degenerate valence and conduction band form an almost zero-T (K)300 400 500 600 700S (μV K-1)-1500-1000-5000T (K)300 400 500 600 700ρ (Ω m)0.000.050.100.150.20T (K)300 400 500 600 700RH (10-2 m3  C-1)-1.5-1.0-0.50.0T (K)0 200 400 600 800-150-100-500T (K)0 200 400 600 8000100200300400T (K)0 200 400 600 800-4-3-2-10P-doped Si P-doped Si P-doped SiFe2VAl0.9Si0.1 Fe2VAl0.9Si0.1 Fe2VAl0.9Si0.1Fe2VAl0.9Si0.1 (Exp.) 2PB fit SeeBandSi:P (Experiment)2PB fit SeeBandρ (μΩ cm)S (μV K-1)RH (10-3 cm3  C-1)a b cd e fFig. 3 | Two test examples, a wide- and a narrow-gap semiconductor system, foraccurate electronic transport analysis with SeeBand. Temperature-dependentSeebeck coefficient, electrical resistivity and Hall coefficient of, a–c phosphorus-doped silicon39 and, d–f full-Heusler-type Fe2VAl0.9Si0.140. Symbols representexperimental data, and solid lines least-squares fits from the simultaneous analysesof all transport properties as sketched in Fig. 1.Table 1 | Comparison of band structure parameters for twotest examplesPhosphorus-doped SiParameter Literature RefinementEF 1.076 eV41 1.08 eV+ 0.02 meVEg 1.03–1.12 eV42,43 1.08 eVεm 0.643 0.74eτ1 - 3.44 ⋅ 105 kg2 m Kετ - 1m1 0.55 me43 0.49 mem2 0.33 me43 0.36 men (300 K) 5.7 ⋅ 1014 cm−3 39 1.5 ⋅ 1015 cm−3Heusler-type Fe2VAl0.9Si0.1Parameter Literature RefinementEF - 0.110 eVEg 0.03–0.1 eV50,60,61 0.055 eVεm 2.7150 1.34eτ1 - 3.00 ⋅ 105 kg2 m Kετ - 0.96m1 2.3 me50 5.6 mem2 6.1 me50 7.5 men (10 K) 1.4 ⋅ 1021 cm−3 40 1.22 ⋅ 1021 cm−3Microscopic EBS parameters of phosphorus-doped silicon and Heusler-type Fe2VAl0.9Si0.1,extracted from simultaneous least-squares fits of temperature-dependent transport data shown inFig. 3, utilizing SeeBand. The obtained band parameters are compared to existing values in theliterature41–43,50,60,61, yielding good agreement. All energies are givenwith respect to the valencebandedge (Ei− EV BM).https://doi.org/10.1038/s41524-025-01645-y Reviewnpj Computational Materials |          (2025) 11:171 5www.nature.com/npjcompumatsgap electronic structure50–52. Depending on the doping concentration,bipolar conduction sets in already below room temperature, requiring a two-band framework for properly describing the electronic transport properties.Devising suitable EBSmodels in such materials is crucial, since it allows, forexample, estimating the bipolar contribution to the thermal conductivity,which severely limits the performance and zT and cannot be differentiatedexperimentally bymaking use of theWiedemann-Franz law. Neglecting thepivotal role of minority carrier contributions on the transport properties orthe chemical potential leads to erroneous interpretations and predictions ofthe maximum achievable figure of merit.Figure 3 d–f shows experimental data of S(T), ρ(T), andRH(T) togetherwith least-squaresfits in a 2PBmodel framework forn-dopedFe2VAl0.9Si0.1.The onset of bipolar conduction manifests itself in a distinct maximum ofS(T) at T ≈ 400 K, and also ρ(T) at slightly higher temperatures, as well as akink in RH(T) at around 200–300K.Theoretical curves yield excellent alignment not only with the experi-mentalS(T) andρ(T), but also for theHall coefficientRH(T), considering thatonly a single free fitting parameter has been used for the latter. The obtainedEBSparameters are once again compared toprevious information existing inthe literature (see Table 1), for which we find very good quantitativeagreement. Crucially, while previous modeling attempts are very time-consuming or require the synthesis of samples with different doping con-centrations, SeeBand enables a highly efficient treatment of temperature-dependentdata of a single sample.Thewholemulti-stage refinementprocessusually takes only a couple of seconds (≈1.5 s for Fe2VAl0.9Si0.1), whichmakes SeeBand a potent tool for high-throughput analyses of large data sets.High-throughput data analysisThe sheer volume of experimental TE property data available in the lit-erature is immense, with much of it remaining unanalyzed and under-utilized. The advent of comprehensive materials databases such asStarryData2 and MaterialsProject, which are progressively incorporatingexperimental data, underscores the need for advanced tools to handle largedatasets. These databases enable the extraction of myriad data, makingautomated fitting essential. SeeBand addresses this need by offering auto-mated fitting capabilities, though it still requires robust start parameters tonavigate local minima and avoid divergences from the correct results. Toovercome this issue, we trained a neural network (for details see Supple-mentary Note 1) to provide initial parameter guesses, which can be lever-aged to facilitate the high-throughput analysis of thermoelectric transportproperties across thousands of compounds.In this section, we demonstrate this on the example of XNiSn-basedhalf-Heusler compounds, where X =Ti, Zr, Hf. Owing to their exceptionalmechanical properties and high zT, n-type XNiSn half-Heuslers representone of the hottest candidates for realizing thermoelectric applications atelevated temperatures13,53,54. To this aim, we downloaded hundreds oftransport datasets of different classes of half-Heusler compounds (see Fig. 4a)and fitted the temperature-dependent Seebeck coefficient employing the AI-assisted framework of SeeBand.In the high-throughput analysis of these data, we focus on XNiSn-based half-Heuslers, comparing samples with and without Hf. Figure 4c, dshows zTmax as a function of the extracted EBS parameters such as theweighted effective mass ratio εm and the band gap Eg, both of which can bedirectly derived from fitting S(T) (cf. Fig. 1a). Only for less than 5 % of thedatasets the NN guess was insufficiently accurate for the fit to converge.These datasets were subsequently discarded from the analysis. Despitesubstantial scatter of the obtained EBS parameters, most likely arising formvarying sample and data quality, there exists a clear accumulation, as can beseen from the smoothed statistical frequencies plotted in Fig. 4b, c. Thecorresponding modes D1,2(εm, Eg) reveal that both Hf-free and Hf-containing XNiSn half-Heuslers can be described with similar EBS para-meters. This aligns well with reports in the literature55,56 and ascertains thatthe main enhancement of the figure of merit zTmax stems from a reductionof the lattice thermal conductivity and not from optimizations of the EBS.The mode for the band gap is D1 Eg � D2ðEgÞ ¼ 0:22� 0:23 eV.This value is in very good agreement with what has been reported in theliterature up until now. For instance, a simple estimation of the thermal bandgap via the Goldschmid-Sharp relation Eg ¼ 2e jSmaxjTmax yields 0.27 eV57.For ZrNiSn, Aliev et al. derived band gaps of 0.25 eV using infrared opticalspectroscopy and 0.18 eV from Arrhenius-type activation of ρ(T), whileSchmitt et al. derived smaller values of around 0.13 eV, using diffuse opticalreflectance measurements57. It should be noted that the value of Eg obtainedexperimentally is significantly smaller than that obtained from DFT calcu-lations, since Eg in real samples does not correspond to the energy gapbetween valence and conduction bands but to the gap between intrinsic in-gap impurity bands and the conduction band55,58,59. Using SeeBand, we expectto assess the smaller gap between impurity band and conduction band, ascharge carriers contribute by being excited from less mobile impurity statesinto the conductive conductionband states. Thepivotal role of these impuritybands is also reflected in the effective mass ratio εm = (N1m2)/(N2m1) = 0.28–0.29. By taking into account the band degeneracies58 and theeffective mass of the conduction band carriers m2 ≈ 2.9me59, we derive anT (K)S (μV K-1)-400-200020040000210 009006003n-type HHp-type HH(N1m2) / (N2m1)zTmaxEg (eV)a bAI-assisted high-throughput0.1 1 100.00.51.00.0 0.5 1.0XNiSnHf-freeΣ020035D1 ≈ 0.22 eVD2 ≈ 0.23 eVD1 ≈ 0.29 D2 ≈ 0.28 dce1.5Fig. 4 | Neural-network-assisted high-throughput analysis of hundreds of half-Heusler compounds. a Temperature-dependent Seebeck coefficient S(T) for hun-dreds of p- and n-type half-Heusler compounds from the Starrydata2 open webdatabase1. Neural-network-assisted initial guesses and a fast and automated sub-sequent fitting routine enable high-throughput modeling of S(T). b, c The statisticalfrequency Σ andmodesD1(εm, Eg) andD2(εm, Eg) reveal an almost identical effectivemass ratio εm = (N1m2)/(N2m1) = 0.28–0.29 between conduction and valence bands,as well as an almost identical band gap Eg = 0.22–0.23 eV for Hf-free and Hf-containingXNiSn. This confirms that the enhancement of zT forHf-containing half-Heuslers primarily arises from a reduction of the lattice thermal conductivity andthat Hf has a negligible effect on the electronic structure. d, e Maximum figure ofmerit zTmax versus electronic band structure parameters for XNiSn half-Heuslers(X = Ti, Zr, Hf), with and without hafnium.https://doi.org/10.1038/s41524-025-01645-y Reviewnpj Computational Materials |          (2025) 11:171 6www.nature.com/npjcompumatseffective hole mass of m1≈ 30.5me, which implies that hole-type chargecarriers, associatedwith the impuritybandsbehaveasmassive charge carriers,in agreement with the flat dispersion expected from such impurity bands58.ConclusionSummarizing, we developed a highly efficient, easy-to-use yet powerfulrefinement tool, SeeBand, capable of deriving important physical band struc-ture parameters and relevant information regarding charge carrier scattering,directly from temperature-dependent transport data. The Boltzmann trans-port theory and parabolic band framework make SeeBandespecially suitablefor the study of thermoelectric materials, which are typically doped semi-conductors, where the conducting Fermi surface pockets can often be descri-bed by a parabolic band dispersion. SeeBand possesses the ability toconcurrently analyze temperature-dependent Seebeck, resistivity, and Halleffectdata,whichguarantees robust andclear results, asdemonstratedwith twowell-studiedmaterial classes: P-doped silicon and Fe2VAl-based full-Heuslers.Moreover, a trained neural network embedded in the SeeBand frameworkprovides initial parameter guesses for thefitting procedure, enabling extremelyfast processing of the data and consequently high-throughput data analyses,which we exemplified for about 1000 TE property data sets of half-Heuslercompounds.Adequatelyandefficientlymodelingelectroniccharge transport iscrucial for rationally designing functional electronic materials such as ther-moelectrics, especially considering the vast quantity of data which has beenaccumulated over the recent decades of research. SeeBand in its current formalready constitutes a powerful tool for transport data analysis andhas potentialfor even broader range of applications. By addressing next steps such asexpanding the range of selectable scattering mechanisms, incorporating non-parabolic band models, and including formulas for phononic heat transport,SeeBand canevolve into a comprehensive tool for analyzingmeasurementdataacross various fields of solid-state physics.Data availabilityThe datasets used and/or analyzed during the current study available fromthe corresponding authors on reasonable request. The underlying code andtraining/validation datasets for this study are available from the Git repo-sitory for SeeBand and can be accessed via this link [https://github.com/xAngelswordx/SeeBand]. Also an executable version of the software See-Band is available as a GitHub release in the same repository via [https://github.com/xAngelswordx/SeeBand/releases/tag/Executable].Received: 13 November 2024; Accepted: 2 May 2025;References1. Katsura, Y. et al. Data-driven analysis of electron relaxation times inPbTe-type thermoelectric materials. Sci. Technol. Adv. 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We thank the anonymous reviewers for theirconstructive feedback, which significantly improved both the manuscriptand theSeeBandsoftware.TheauthorsacknowledgeTUWienBibliothek forfinancial support through its Open Access Funding Programme.Author contributionsM.P. and A.R. developed the software SeeBand. M.P., F.G., and A.R.developed the theoretical framework behind the parabolic-band transportmodel. J.d.B. substantially contributed to both the development of thetheoretical framework and the software. A.R. and M.P. developed the neu-ronal network, implemented in the software for initial parameters guesses.M.P., A.R., and F.G. devised and performed the high-throughput analysis.J.d.B. and E.B. supervised the work. T.M. organized the funding. F.G., M.P.,and A.R. wrote the initial draft. All authors discussed the results and editedthe final manuscript.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41524-025-01645-y.Correspondence and requests for materials should be addressed toMichael Parzer, Alexander Riss or Fabian Garmroudi.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution-NonCommercial-NoDerivatives 4.0 International License,which permits any non-commercial use, sharing, distribution andreproduction in any medium or format, as long as you give appropriatecredit to the original author(s) and the source, provide a link to the CreativeCommons licence, and indicate if you modified the licensed material. Youdo not have permission under this licence to share adapted materialderived from this article or parts of it. The images or other third partymaterial in this article are included in the article’s Creative Commonslicence, unless indicated otherwise in a credit line to thematerial. If materialis not included in thearticle’sCreativeCommons licenceandyour intendeduse is not permitted by statutory regulation or exceeds the permitted use,you will need to obtain permission directly from the copyright holder. Toview a copy of this licence, visit http://creativecommons.org/licenses/by-nc-nd/4.0/.© The Author(s) 2025https://doi.org/10.1038/s41524-025-01645-y Reviewnpj Computational Materials |          (2025) 11:171 8https://doi.org/10.1038/s41524-025-01645-yhttp://www.nature.com/reprintshttp://creativecommons.org/licenses/by-nc-nd/4.0/http://creativecommons.org/licenses/by-nc-nd/4.0/www.nature.com/npjcompumats SeeBand: a highly efficient, interactive tool for analyzing electronic transport data Theoretical background Equations used for the fitting process Demonstration of use Graphical user interface Workflow Test examples Phosphorous-doped silicon Full-Heusler Fe2VAl0.9Si0.1 High-throughput data analysis Conclusion Data availability References Acknowledgements Author contributions Competing interests Additional information