# Fileset

[Physical Review X---Topological Flat-Band-Driven Metallic Thermoelectricity.pdf](https://mdr.nims.go.jp/filesets/ba652bb5-21ba-4431-a028-55bc9c8fd3b8/download)

## Creator

Fabian Garmroudi, Jennifer Coulter, [Illia Serhiienko](https://orcid.org/0000-0002-3072-9412), Simone Di Cataldo, Michael Parzer, Alexander Riss, Matthias Grasser, Simon Stockinger, Sergii Khmelevskyi, Kacper Pryga, Bartlomiej Wiendlocha, Karsten Held, [Takao Mori](https://orcid.org/0000-0003-2682-1846), Ernst Bauer, Antoine Georges, Andrej Pustogow

## Rights

[Creative Commons BY Attribution 4.0 International](https://creativecommons.org/licenses/by/4.0/)

## Other metadata

[Topological Flat-Band-Driven Metallic Thermoelectricity](https://mdr.nims.go.jp/datasets/4ff06013-77d0-4052-aaef-0072bf8b9113)

## Fulltext

Topological Flat-Band-Driven Metallic ThermoelectricityTopological Flat-Band-Driven Metallic ThermoelectricityFabian Garmroudi ,1,2,* Jennifer Coulter,3,† Illia Serhiienko ,4,5 Simone Di Cataldo ,1,6 Michael Parzer ,1Alexander Riss,1 Matthias Grasser ,1 Simon Stockinger ,1 Sergii Khmelevskyi ,7 Kacper Pryga ,8Bartlomiej Wiendlocha ,8 Karsten Held ,1 Takao Mori ,4,5 Ernst Bauer,1Antoine Georges,3,9,10,11 and Andrej Pustogow 1,‡1Institute of Solid State Physics, TU Wien, 1040 Vienna, Austria2Materials Physics Applications—Quantum, Los Alamos National Laboratory,Los Alamos, New Mexico 87545, USA3Center for Computational Quantum Physics, Flatiron Institute, New York 10010, USA4International Center for Materials Nanoarchitectonics (WPI-MANA),National Institute for Materials Science, Tsukuba 305-0044, Japan5University of Tsukuba, Tsukuba 305-8577, Japan6Dipartimento di Fisica, Sapienza University of Rome, Piazzale Aldo Moro 5, 00185 Roma, Italy7Vienna Scientific Cluster Research Center, TU Wien, 1040 Vienna, Austria8AGH University of Krakow, Faculty of Physics and Applied Computer Science,Aleja Mickiewicza 30, 30-059 Krakow, Poland9Collège de France, PSL University, 75005 Paris, France10Department of Quantum Matter Physics, University of Geneva, 1211 Geneva, Switzerland11Centre de Physique Théorique, Ecole Polytechnique, 91128 Palaiseau, France(Received 12 April 2024; revised 13 January 2025; accepted 18 April 2025; published 14 May 2025)Materials where flattened electronic dispersions arise from destructive phase interference, rather thanlocalized orbitals, have emerged as promising platforms for studying emergent quantum phenomena.Crucial next steps involve tuning such flat bands to the Fermi level, where they can be studied at low energyscales, and assessing their potential for practical applications. Here, we show that the interplay of highlydispersive and ultraflat bands inherent to these systems can lead to extreme interband scattering-inducedelectron-hole asymmetry, which can be harnessed in thermoelectrics. Our comprehensive theoretical andexperimental investigation of Ni3In1−xSnx kagome metals supports this concept, showing that it could leadto thermoelectric performance on par with state-of-the-art semiconductors such as Bi2Te3. In Ni3In,scattering-induced electron-hole asymmetry is, however, subdued by an exotic conduction mechanismarising from quantum tunneling of charge carriers between Dirac bands, unrelated to the flat band itself. Weoutline strategies to selectively switch off this tunneling transport through negative chemical pressure orstrain. Our study proposes a new direction to explore in topological flat-band systems and vice versaintroduces a novel tuning knob for thermoelectric materials.DOI: 10.1103/PhysRevX.15.021054 Subject Areas: Condensed Matter PhysicsI. INTRODUCTIONTopological flat-band (TFB) materials are of immensecurrent interest, as they promise a rich tapestry of emergentcorrelation phenomena and novel physics to be discovered[1,2]. Certain frustrated geometries such as the dice, Lieb,or kagome lattices are theoretically predicted to supportcompletely flat electronic bands, that is, a quasi-infinitelydegenerate set of quantum states arising from internalsymmetries or local topology [3–5]. For instance, withina single-orbital tight-binding model for the kagomelattice—a corner-sharing network of triangles—destructiveinterference among electronic hopping pathways leads toperfectly flat bands in two dimensions [Fig. 1(a)]. Thepyrochlore lattice, a corner-sharing network of tetrahedra,extends this notion to the third dimension [Fig. 1(b)]. Suchlattice-borne TFBs have emerged as promising platformsfor designing novel quantum phases of matter on demand,as the electron-electron interaction becomes decisive com-pared to the quenched kinetic energy [6,7]. While variousTFB candidates have been proposed in recent years [8–10],*Contact author: f.garmroudi@gmx.at†Contact author: jcoulter@flatironinstitute.org‡Contact author: pustogow@ifp.tuwien.ac.atPublished by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license.Further distribution of this work must maintain attribution tothe author(s) and the published article’s title, journal citation,and DOI.PHYSICAL REVIEW X 15, 021054 (2025)2160-3308=25=15(2)=021054(11) 021054-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-0088-1755https://orcid.org/0000-0002-3072-9412https://orcid.org/0000-0002-8902-0125https://orcid.org/0000-0003-3509-7474https://orcid.org/0009-0009-1534-9196https://orcid.org/0009-0002-2783-1245https://orcid.org/0000-0001-5630-7835https://orcid.org/0000-0003-3087-7498https://orcid.org/0000-0001-9536-7216https://orcid.org/0000-0001-5984-8549https://orcid.org/0000-0003-2682-1846https://orcid.org/0000-0001-9428-5083https://ror.org/04d836q62https://ror.org/01e41cf67https://ror.org/00sekdz59https://ror.org/026v1ze26https://ror.org/02956yf07https://ror.org/02be6w209https://ror.org/04d836q62https://ror.org/00bas1c41https://ror.org/052bz7812https://ror.org/01swzsf04https://ror.org/05hy3tk52https://crossmark.crossref.org/dialog/?doi=10.1103/PhysRevX.15.021054&domain=pdf&date_stamp=2025-05-14https://doi.org/10.1103/PhysRevX.15.021054https://doi.org/10.1103/PhysRevX.15.021054https://doi.org/10.1103/PhysRevX.15.021054https://doi.org/10.1103/PhysRevX.15.021054https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/and there are now even catalogs and databases listingthousands of such materials [11,12], an outstanding chal-lenge and crucial question is how TFBs manifest in thephysical properties of the hosting materials and whethertheir unique electronic structures can be harnessed forpractical applications [13].Here, we recognize the remarkable prospect of TFBs—given that they can be properly tuned around the Fermilevel—for thermoelectric (TE) materials. Thermoelectricityrefers to the ability of solids to seamlessly interconvertthermal and electrical energy, which is attractive for variouspower generation and cooling applications [15]. An effi-cient TE material requires a large power factor PF ¼ S2σ(S being the Seebeck coefficient and σ the electricalconductivity) and a low thermal conductivity κ, summa-rized in the dimensionless figure of merit zT ¼ S2σκ−1T,where T is the absolute temperature.Traditionally, only selected semiconductors have beenconsidered to be viable TEs ever since Ioffe proposed themin themid-20th century [16]. This preference is rooted in thefact that the bandgap in semiconductors inherently separatesthe contributions of low- and high-energy charge carriers(holes and electrons), prerequisite for achieving large S [17].However, the rather poor σ of semiconductors, even in theirheavily doped forms, results in rather low PF and neces-sitates the use of materials with ultralow lattice thermalconductivity κl. To inhibit heat transport via the lattice, TEsemiconductors are usually complex alloys comprisingmultiple elements and a high degree of structural andchemical bonding intricacy [18]. As a downside, crystallo-graphic and chemical features yielding ultralow κl often gohand in hand with poor mechanical and thermal stability,limiting the applicability of these materials [19]. Thus, it iscrucial to identify materials that display competitive zTarising from large PF rather than from low κl [20].Unfortunately, electronic tuning is hindered by intrinsictrade-offs between σ and S, despite dedicated efforts in thepast aimed at overcoming these challenges [21–23].A. “The best thermoelectric” revisitedWhat electronic structure provides the highest TEperformance? In their seminal work, Mahan and Sofoattempted to answer this simple yet fundamental questionby mathematical derivation, establishing conditions for thebest thermoelectric [24]. They found that the energy-dependent conductivity of charge carriers, characterizedby a unique transport distribution function σðEÞ, must varyrapidly and ideally be shaped like a delta distribution. Inother words, charge carriers should contribute only in aninfinitely narrow energy interval. Recent theoreticalstudies [25–29] suggest that a (rectangular) boxcar-shapedtransport function of finite width would be optimal andsuperior to the delta function. In such a scenario, electronictransport is bounded to a finite energy window with σðEÞinstantaneously dropping to zero outside that interval.In real materials, however, σðEÞ neither is bounded toan infinitely narrow energy interval nor displays aninfinitely steep edge. Instead, σðEÞ increases linearly withKagomePyrochloreΓ Γ W Γ L UWKMEnergy (arb. u.)σ0σσσminFBDBΓ X K L KXDispersionless in 2D Dispersionless in 3DTFB TFBEEFEgE0D(E(()Emeffff3/20σ(E(()Emeffff-10σ(E(()EFBDBSemiconductors MetalsEFμW μW D(E(()E0Weighted mobility(a) (c) (d)(b)FIG. 1. Topological flat bands for thermoelectric materials design. (a) Frustrated kagome and (b) pyrochlore lattices with compactlocalized states (red-yellow) arising from destructive interference (Ø) of nearest-neighbor hopping paths. This yields topological flatbands in two and three dimensions, respectively. (c) Energy-dependent density of states DðEÞ and transport distribution function σðEÞfor a dispersive and for a flat band, separated by an energy gap. The slope of σðEÞ, the weighted mobility μW, determines the electronicquality of TE semiconductors, in the regime near the band edge, where the single-parabolic band approximation holds [14].(d) DðEÞ and σðEÞ for a metal, where dispersive (DB) and flat bands (FBs) overlap and interband scattering creates a verysteep edge in σðEÞ.FABIAN GARMROUDI et al. PHYS. REV. X 15, 021054 (2025)021054-2a finite slope near the band edge [Fig. 1(c)] and becomesweakly energy dependent in the center of a band. In asingle-parabolic band model, σðEÞ ∼DðEÞv2ðEÞτðEÞ, withDðEÞ being the density of states, vðEÞ the velocity, andτðEÞ the relaxation time of charge carriers [17]. In semi-conductors, which can often be reasonably well describedby a single-parabolic band model, S depends on only thedistance of the Fermi level EF from the band edge and is adirect measure of doping. The electrical conductivity σ and,hence, PF, however, depend on the absolute value of σðEÞaround EF. This yields a trade-off, since S decreases as EFis shifted further into the band whereas σ increases. Theslope of σðEÞ, commonly referred to as the weightedmobility μW, determines, in the single-parabolic bandregime, the electronic quality of a TE material, that is,the highest achievable performance at an optimized posi-tion of EF [14]. Enhancing μW is not as straightforward asreducing κl. It is often wrongly assumed that a flat bandwith a steep DðEÞ would also result in a steep σðEÞ.However, while DðEÞ ∝ m3=2eff , σðEÞ scales inverselywith the effective mass m−1eff due to v2ðEÞ ∝ m−1eff andτðEÞ ∝ m−3=2eff . Hence, charge transport via flat bands withexceptionally steep DðEÞ cannot be directly exploited. Infact, the best thermoelectric semiconductors, such asBi2Te3 and PbTe, usually display very dispersive bands(DBs) around EF [30,31]. This inevitably means, however,that semiconducting thermoelectrics with dispersive(broad) bands cannot realize a narrow boxcar-shaped ordelta-distribution-shaped transport distribution.Here, we present a materials design principle in whichthe exceptionally steep band edge of flat bands can actuallybe exploited to generate a steep edge in σðEÞ [Fig. 1(d)]. Inmetals, where FBs overlap with DBs, carriers from the DBcan scatter into the FB, locally suppressing σðEÞ. Theseinterband transitions and, hence, the edge occurring in σðEÞare directly determined by DðEÞ of the FB, as shown later.Through this paradigm, we tuned interband scattering ofmobile s-like carriers into more localized Ni-3d states and,this way, realized in binary NixAu1−x alloys the highest PFever reported in a bulk material above room temperature,PF ¼ 34 mWm−1 K−2 at 560 K, and the highest zT everreported for a metal, zT ≈ 0.3 (0.5) at 300 (1100) K [32].A similar interband scattering scenario is at work also incertain semimetals like CoSi [33,34]—yielding high PF ≈6 mWm−1K−2 but modest zT ≈ 0.1—and has theoreticallybeen proposed for others [35]. Recently, Graziosi et al.derived materials design criteria for metallic thermoelec-trics employing a two-parabolic band model [36], and thepotential of extreme band asymmetry has even beenrecognized for narrow-gap semiconductors and topologicalsemimetals [37–39].While the malleable and ductile nature of metals makesthem especially attractive for designing new and innovativeTE device structures, e.g., for flexible and wearableelectronics [40,41], in the case of NixAu1−x, the high costof Au poses constraints for practical use and the obviousquestion arises whether the TE performance could beimproved even further up to zT ≈ 1 near room temperature,competitive with commercial Bi2Te3 systems [18,42]. Toachieve such a leap of TE performance, an important issueto address is how to further flatten the bands compared tothe situation in binary NixCu1−x and NixAu1−x, wherealloying and disorder substantially smear the band edges.We argue that metallic TFB systems appear to be near-perfect candidates in this regard, not only because theirdispersion is significantly more flattened by geometricalfrustration, but also because their band flatness is topo-logically robust and immune to alloying and chemicaldisorder.Additionally, we discover a quantum tunneling mecha-nism for charge carriers between dispersive bands, unrelatedto the flat band, that needs to be taken into considerationwhen designing scattering-tuned metallic thermoelectrics.Given the importance of this tunneling mechanism, inparticular, for linear band crossings near the Fermi level,which are essential band structure features for realizing anoptimal boxcar-shaped transport distribution [28], we stressthe crucial need for theoretical work reconsideringwhat kindof electronic structure produces the best thermoelectric.II. RESULTSA. Extreme interband scattering in Ni3InWe implemented the above described concept in acomprehensive experimental and theoretical investigationof a particular promising candidate: the kagome metalNi3In. This system has recently been predicted to featurealmost dispersionless TFBs in conjunction with two highlydispersive Dirac-like bands in its electronic structure,exactly at EF [43]. Figure 2(a) shows the crystal unit cellalongside a top view of the AB stacked kagome layersformed by Ni atoms. The alternating sign pattern of the Ni-3dxz orbital texture results in destructive phase interferenceamong inter- and intralayer hopping paths and spatiallylocalized electronic states around the triangular Ni pla-quettes [red shaded area in Fig. 2(a)]. This yields ultraflatbands exactly at EF as shown in Fig. 2(b). These FBsproduce an extremely steep edge inDðEÞ [Fig. 2(c)], whoseposition with respect to EF can be tuned exceptionally wellvia aliovalent Sn substitution at the In site in Ni3In1−xSnx[Fig. 2(d)]. Since Ni3Sn exists as an isostructural variant, afull solid solution is possible, as confirmed experimentally[inset in Fig. 2(c)].For a quantitative comparison, we compare in Fig. 3(a)DðEÞ of Ni3In1−xSnx with those of binary transition metalalloys like NixCu1−x and NixAu1−x. It is evident that, owingto the hopping frustration-induced flattening of the Ni-3dxzbands, the DðEÞ edge is much steeper for the former.Because of the rapid variation ofDðEÞ, the scattering phaseTOPOLOGICAL FLAT-BAND-DRIVEN METALLIC … PHYS. REV. X 15, 021054 (2025)021054-3space for charge carriers from the underlying DBs exhibitsa strong energy dependence as well. By direct applicationof Fermi’s golden rule, it follows that τ−1ðEÞ ∼DðEÞ. Inthe following, we refer to it as the D−1ðEÞ model. At thispoint, we also note that the analytic formula for electron-phonon scattering in deformation potential theory yieldsτðEÞ ∼D−1ðEÞ, which, for parabolic bands, results in thewell-known τðEÞ ∼ E−1=2 dependence of the relaxationtime. Figure 3(b) displays σðEÞ, normalized to its valuesat E − EF ¼ 0.2 eV, to highlight and compare the drop inσðEÞ around EF. It can be seen that while in the binaryalloys, with mundane Ni-3d bands, σðEÞ reduces to about50% of its value 0.2 eVabove EF, the kagome metal showsa remarkable drop to less than 10%, while the conductivityremains high at E > EF in either case. This scattering-induced energy filtering creates pronounced electron-holeasymmetry and results in a very large Seebeck coefficientS ≈ −136 μVK−1 at 300 K, well above those predictedtheoretically and achieved experimentally in both NixCu1−xand NixAu1−x [Fig. 3(c)] or any other metal known to date.As shown in Fig. 3(d), the high S coupled with the goodmetallic nature of Ni3In1−xSnx would result in an out-standing zT ≈ S2=L ≈ 1.1 at 300 K, rivaling or evensurpassing those of today’s best semiconductors [42].We note that zT ≈ S2=L is a very good approximationin highly metallic systems, such as Ni3In1−xSnx andNixAu1−x, especially at high temperatures [32]. Indeed,thermal conductivity measurements (Fig. S1 [44]) confirmthat electrons dominate thermal transport and that lattice-driven heat transport is insignificant in Ni3In1−xSnx,resulting in zT ¼ S2=L. Thus, only the Seebeck coefficientS and the Lorenz number L determine the performance.Energy filtering of charge carriers right next to either sideof EF enhances electron-hole asymmetry and thereby S. Onthe other hand, selectively filtering charge carriers at higherenergies (approximately 3kBT above EF) to create aboxcar-shaped transport distribution function σðEÞ canstrongly reduce L relative to the Sommerfeld value, furtherboosting zT in metallic TEs [45]. Achieving such ascenario via flat-band-induced energy filtering wouldrequire different flat bands, one right next to EF andanother slightly above or below EF, depending on theposition of the first, which could be achieved, e.g., bysubstituting transition metal atoms with localized non-hybridizing impurity bands [46].The theoretical estimates by the BTE-D−1ðEÞ model,predicting a large Seebeck effect in metallic Ni3In1−xSnx,motivated us to experimentally study the TE properties ofthese compounds. Since large samples are an essentialrequirement for conducting high-temperature TE measure-ments, we prepared phase-pure polycrystalline samples inthe entire composition range from x ¼ 0 to 1 [experimentalNic ab(a)048-10 -8 -6 -4 -2 0 2 4 60816 Ni3In1-x Snx(c)(d)(b)FBBD BDFB1-10Γ M K Γ A L H ANi3InWave vector E − EF (eV)-0.4 -0.2 0.0 0.2 0.4300 KA HLMΓKx = 0x = 0.1x = 0.2x = 0.8x = 0.9x = 1Ni3Sn Ni3InFB FBx0 1a,b  (Å)5.285.34c (Å)4.234.255cabInD(E)  (eV-1 f.u.-1) D(E) (eV-1 f.u.-1) E − E F (eV)FIG. 2. A kagome system with flat bands at the Fermi level and exceptional tunability. (a) Unit cell of Ni3In with triangular motifsformed by Ni atoms, showing AB stacked breathing kagome lattice planes. Yellow and blue lobes indicate opposite phases of the dxzwave functions, causing destructive interference (Ø) of inter- and intralayer hopping paths (arrows) and localizing electronic states at thetriangular plaquettes (red area). (b) Electronic band structure (without spin orbit coupling) of Ni3In, featuring flat bands (red) anddispersive Dirac-like bands (blue) at the Fermi energy EF. (c),(d) Alloy-averaged densities of states of Ni3In1−xSnx across various alloyconcentrations, showing tunability of the flat band’s position relative to EF via In/Sn substitution. A solid solution is confirmedthroughout the entire composition range by x-ray diffraction. Inset: experimental lattice parameters as a function of x at roomtemperature. The gray shaded area in (d) represents the derivative of the Fermi-Dirac distribution at 300 K, marking the relevant energyrange for electronic transport.FABIAN GARMROUDI et al. PHYS. REV. X 15, 021054 (2025)021054-4lattice parameters in the inset in Fig. 2(c)]. Figure S12 [44]shows SðTÞ of three representative samples (x ¼ 0, 0.5, 1)measured in a very broad temperature range 4–860 K. It canbe immediately seen that there is a large disparity betweenexperimental results and theoretical calculations using theBTE-D−1ðEÞmodel [see also Fig. 3(c)], which is surprisingconsidering that usually very consistent agreement isobtained, e.g., in binary transition metal alloys [32] or insemimetallic CoSi [47]. It should also be pointed out that,while Ni3Sn reveals near-perfect agreement, the disparitybecomes greater toward x ¼ 0. This raises concernswhether the D−1ðEÞ model fails to accurately estimatethe energy-dependent scattering in Ni3In.Therefore, in a second step of more advanced theoreticalassessment, we performed ab initio calculations of theelectron-phonon scattering rates in the relaxation timeapproximation (RTA) using the PHOEBE code [48].Figure 3(e) displays the energy-dependent carrier lifetimesfor electron-phonon scattering in Ni3In from RTA calcu-lations at 300 K and the D−1ðEÞ model. It is evident thatboth frameworks predict an abrupt decrease of τðEÞ,starting about 0.1 eVabove EF, which is exactly the energyrange where the TFB states of Ni3In overlap with the moredispersive bands; the inset shows a projection of the RTAscattering rates onto the band structure in the relevantenergy range. These puzzling findings pose questions as towhy the steep edge in τðEÞ does not manifest itselfexperimentally in a large Seebeck coefficient.B. Zener quantum tunneling in Ni3InTo probe the unusual charge transport behavior of Ni3In,we leveraged the PHOEBE code to perform calculationswithin the Wigner transport formalism, a general theoreti-cal framework that extends beyond semiclassicalBoltzmann transport theory, capturing additional quantummechanical transport effects by considering the spectralnature of electronic structure. Through these calculations,we demonstrate that extreme interband scattering is indeedat play in this material but is shrouded by an exoticconduction mechanism due to quantum tunneling betweenDirac-like bands, unrelated to the FB itself. Finally, weoutline strategies to overcome this challenge in Ni3In andways to avoid it in other TFB compounds.The quantum nature of electrons and their interactionswith other quasiparticles such as phonons results in spectralbroadening of the electronic band structure. This spectralquality allows them to undergo spontaneous verticaltransitions from one band into another, creating anE − EF (eV)-0.2 -0.1 0.0 0.1 0.2σ/σ 0(E) (%)0100T (K)0 200 400 600 800 1000S (μV K-1)-140-120-100-80-60-40-200D(E) (eV-1)012345650Orbital flatteningTopological flatteningNixCu1-xNixAu1-xNi3In1-xSnx ExperimentBTE-D-1(E) modelBi2Te 3Mg 3Bi20.70.81.101Ni-Au0.3zT 40−60%<10%(a)(b)(c) (d)NixAu1-xNi3In1-xSnxNi-Cu Ni 3(In,Sn) NixCu1-x-0.4 -0.2 0.0 0.2 0.4020406080100120E − μ (eV)� e-p (fs) Γ M ΓNi3InRTA�e-p 300 KExperimental(e)300 K�e-p ~ D-1(E)BTE-D-1(E) Metals SemiconductorsKFIG. 3. Comparison to binary transition metal alloys and state-of-the-art semiconductors. (a) Density of states and (b) normalizedtransport distribution function of Ni3In1−xSnx kagome metals with x ¼ 0.25 compared to NixCu1−x and NixAu1−x with x ¼ 0.4;calculations were performed within Boltzmann transport theory and using a D−1ðEÞ interband scattering model for the relaxation time[BTE-D−1ðEÞ]. (c) Calculated temperature-dependent Seebeck coefficient. A much larger Seebeck coefficient is predicted forNi3In1−xSnx due to its frustration-enhanced flat bands and scattering phase space. Surprisingly, the experimental SðTÞ of Ni3In issignificantly lower, raising questions about the accuracy of interband scattering estimates from theD−1ðEÞmodel. (d) Comparison of zTat room temperature. An outstanding zT > 1 is predicted for Ni3In1−xSnx in the BTE-D−1ðEÞ framework, competitive with the bestsemiconductors [42]. Here, zT was calculated from zT ≈ S2=L, which is a good approximation for these metallic systems due toelectronic-dominated heat transport, κe ¼ LσT ≫ κl, at high temperatures (see Fig. S1 [44]). The disagreement with experimentsquestions the validity of the D−1ðEÞ. (e) Carrier lifetimes for electron-phonon scattering from expensive ab initio calculations in therelaxation time approximation, however, exhibit good agreement with the D−1ðEÞ model, suggesting that additional mechanismsbeyond Boltzmann transport theory influence SðTÞ. Inset: variation of RTA carrier lifetimes in Ni3In projected onto the band structurearound the chemical potential.TOPOLOGICAL FLAT-BAND-DRIVEN METALLIC … PHYS. REV. X 15, 021054 (2025)021054-5additional contribution to σðEÞ [Fig. 4(d)] due to the Zenertunneling mechanism [49]. Modern computational theorycan predict this effect by means of the Wigner transportformalism, which has been well established in thermaltransport [50,51], though only a few studies have consid-ered it in a charge transport context [52–54]. Utilizing theWigner transport equation (WTE), we calculated SðTÞ andobtain near-perfect agreement with our experimental data[Fig. 4(b)]. Here, electron-phonon scattering is calculatedfrom first principles in the RTA, and Zener tunneling isaccounted for as an additional additive transport contribu-tion by the Wigner formalism, from now on referred toas WTE-RTA. The additional conductivity from Zenertunneling derived in the framework of the WTE can bewritten asσWigner ¼gse2VNkXk;ν;ν0;ν≠ν0f0ν0 ðkÞ − f0νðkÞϵν0 ðkÞ − ϵνðkÞ×viν;ν0 ðkÞvj�ν0;νðkÞ½ΓνðkÞ þ Γν0 ðkÞ�4½ϵν0 ðkÞ − ϵνðkÞ�2 þ ½ΓνðkÞ þ Γν0 ðkÞ�2; ð1Þwhere gs ¼ 2 stands for the spin degeneracy and V for thecrystal unit cell volume and Nk represents the number ofwave vectors used to integrate the Brillouin zone.Additionally, ν and ν0 represent band indices and v andϵ the electronic band velocities and energies, respectively,and f0ðkÞ is the Fermi-Dirac distribution function. Theelectron-phonon scattering rates Γ are obtained at the RTAlevel. It can be seen that σWigner becomes large when(a) (b)(e) (f)Ni3In0 200 400 600 800-150-100-500BTE-D-1(E(( )E BTE-RTATTWTE-RTA(c)S (μV K-1)(d)Wave vectorcab21T (K)EF0σ (E(()ECRTAD-1(E)BTE-RTAWTE-RTAAccuracySemiclassical Eσmin BTE-RTA+ ZenerBTE-CRTATT-0.10.10.0E − μ (eV)-0.20.0-0.1E − μ (eV)c < 0εab > 00 10 1K1 K2K1 K2 σWigner 0MaxNi3InpristineStrainedand doped600 K600 KFBFBMost transitionsε  = 5 %, n  = 0.8 f.u.0MaxσWignerk,�QuantumσWignerk,�FIG. 4. Zener quantum tunneling transport in Ni3In. (a) The CRTA framework is the least accurate, as it entirely dismisses any energydependence of τðEÞ. While the latter is approximated within the D−1ðEÞ model and fully included within the RTA framework, theWigner formalism also takes into account “vertical” interband transitions due to Zener quantum tunneling, as sketched in (d). (b) Theexperimental data of the Seebeck coefficient SðTÞ of Ni3In are compared to various levels of theoretical calculations: BTE-CRTA clearlyunderestimates S at high T, whereas the BTE-D−1ðEÞ and BTE-RTA results overestimate it. The WTE-RTA formalism yields excellentagreement with experiment. (c) Zener tunneling contribution to electronic transport coming from each high-symmetry band path, as inEq. (1), without the summation over k and ν, showing that only the Dirac-like bands in Ni3In with mixed orbital character and stronglinewidth overlap around their crossing are relevant. On the right, the associated contribution σWigner is plotted (in arbitrary units) fordifferent band fillings. (d) The Wigner term from vertical transitions adds up to the energy-dependent conductivity, yielding asignificantly larger σmin compared to the small (suppressed by interband scattering) RTA value and a reduced S compared to the BTE-RTA result. (e),(f) Shifting apart the Dirac crossing from the FB upon strain tuning of Ni3In enables to switch off the Zener tunnelingcontribution. As sketched in (e), biaxial tensile strain εab > 0 can be realized by compressive strain along the c direction due to anonzero Poisson ratio. (f) This way, the FB can be decoupled from the Dirac bands in biaxially strained Ni3In kagome planes, as theformer is shifted toward higher energies. This allows switching off Zener tunneling while keeping electron-hole asymmetry, when EF isadjusted to the FB and away from the Dirac crossing by doping accordingly. For the unstrained and strained plots, the k point ranges areK1 ¼ ½0.05833; 0.05833; 0�, K2 ¼ ½0.10833; 0.10833; 0� and K1 ¼ ½0.04167; 0.04167; 0�, K2 ¼ ½0.08333; 0.08333; 0� in reciprocallattice coordinates, respectively.FABIAN GARMROUDI et al. PHYS. REV. X 15, 021054 (2025)021054-6spectral broadening increases (large Γ) compared to theseparation of energy bands and when band velocities arehigh (dispersive bands).We observe a significant Zener tunneling transportcontribution, σWigner, which we note does not exhibitpronounced energy dependence around EF [see Fig. 4(c)right]. In the case of Ni3In, the additional conductivitydecreases the electron-hole transport asymmetry, as indi-cated in Fig. 4(d), and, thus, reduces SðTÞ. Typically,σWigner is much smaller than the conductivity of conven-tional transport processes taken into account in theBoltzmann transport equation (BTE). However, whenseveral conditions are met—specifically, for bands withlarge group velocities, small band energy separation, andoverlapping linewidths—these transitions can becomemore pronounced, and in Ni3In, the Dirac-like bandscrossing EF satisfy all these conditions. As shown inFig. 4(c), Zener tunneling is linked to the Dirac bandcrossing around EF in Ni3In.A natural question arises: How can we overcome thischallenge in Ni3In to realize the high SðTÞ predicted by theBTE-D−1ðEÞ and BTE-RTA frameworks, and how can weavoid a parasitic contribution σWigner in other candidatematerials? In Ni3In1−xSnx, this could be realized bydecoupling the FB from the Dirac bands. The Diraccrossing point has to be pushed above or below EF torestrict the phase space for Zener tunneling. However, thiscannot be achieved by mere doping, as the FB needs to beclose to EF to provide the necessary interband scattering fora high S. Figures 4(e) and 4(f) show that biaxial straintuning of the kagome planes can effectively move the FB tohigher energy, while the Dirac bands remain completelyinsensitive, providing the required decoupling. Note thatour results agree well with the recent band-structurecalculations on strain-tuned Ni3Sn [55]. Given that theNi3In structure type exhibits the capability to accommodatea diverse range of elements and compositions, one cananticipate a considerable level of adjustability in the flat-band states without the need to apply mechanical strain. Forexample, we suggest that chemical substitution withheavier Pb/Bi atoms instead of In/Sn or substituting Pd/Pt instead of Ni could expand the kagome planes, recov-ering a high SðTÞ when the flat bands are decoupled fromthe Dirac crossing points.Lastly, we emphasize that, upon evaluating new TEcandidate materials, it is probably advisable to avoid Dirac-or Weyl-like crossing points intersecting with the FB, and aslight offset between the two is recommended. Any futurestudy considering the promise of topological TEs shouldtake care to consider this effect, which has also been foundtheoretically in the topological insulator Bi2Se3 [52].However, Zener tunneling does not necessarily have tobe significant. For instance, charge transport in CoSi, wherea Dirac point also intersects with a slightly flattened bandexactly at EF, can be accurately described using thestandard BTE [34,47]. Thus, multiple specific conditionsmust be met for Zener tunneling to occur, making it acritical, yet ideally rare, obstacle to the design of topo-logical TE materials, in general. In certain band configu-rations, σWigner may even be asymmetric around EF andprovide a beneficial effect.III. CONCLUSIONFrustrated lattice geometries have received enormousattention for hosting exotic correlation phenomena, such asspin-liquid and fractional quantum Hall states. Moreover,destructive interference among electronic hopping paths insuch systems can give rise to topological flat bands, whichare robust to alloying and disorder. In this study, weunveiled the potential of their electronic structures to drivea large TE effect and experimentally achieved high powerfactors up to 5 mWm−1 K−2 at room temperature inNi3In0.5Sn0.5, although an even greater hidden potentialis found. Apart from the obvious choice of engineeringelectronic materials based on the dispersion and bandwidthinherent to the atomic orbitals (s and p orbitals being lesslocalized than d and f orbitals), our work underscoresfrustrated lattice geometry as another pivotal tuning knoband sheds new light on charge transport in kagome andother flat-band hosting materials. A crucial next step forthese intriguing topological and correlated material plat-forms involves tuning their FBs to EF and understandingtheir low-energy excitations. Our work uncovers an impor-tant new transport mechanism (Zener quantum tunneling),which has previously not been considered in the analyses ofthese systems. The study of thermoelectricity in TFBcompounds comes at the perfect time for several reasons.First, relevant databases and catalogs of FB materials[11,12] have recently been compiled, making it easier thanever to identify the most promising candidates for design-ing functional materials, such as TEs. Second, the TE fieldhas made substantial progress in modern theoretical cal-culations of transport properties beyond the constantrelaxation time approximation [48,56,57]. These advance-ments make it more feasible now to study materials withcomplex carrier scattering, as proposed here.IV. METHODSA. Materials synthesis and characterizationHigh-quality polycrystalline Ni3In1−xSnx samples weresynthesized and investigated in two different batches indifferent laboratories at the National Institute for MaterialsScience (NIMS) in Tsukuba, Japan and at the TU Wien inVienna, Austria to confirm the validity and reproducibilityof our results. The first batch of Ni3In1−xSnx with x ¼ 0,0.5, 1 was synthesized in a two-step procedure at NIMS:arc melting and consecutive spark plasma sintering. Thesecond batch of samples with x ¼ 0, 0.2, 0.4, 0.6, 0.8, 1was synthesized at TUWien by first reacting high-purity NiTOPOLOGICAL FLAT-BAND-DRIVEN METALLIC … PHYS. REV. X 15, 021054 (2025)021054-7and In/Sn bulk metals inside an evacuated quartz tube,followed by induction melting in a water-cooled coppercoldboat. Since these compounds do not melt congruently,the resulting ingots were then annealed at 873 K for sevendays to obtain phase-pure Ni3In1−xSnx. The purity of theused raw elements was 99.99% for Ni and 99.999% forIn and Sn. Figure S2 [44] shows that there is excellentreproducibility and agreement between both batchesobtained in different laboratories via different techniques.The structural properties of Ni3In1−xSnx were investi-gated via x-ray powder diffraction using Cu K-α radiation,which confirmed phase-pure samples and a full solidsolution between Ni3In and Ni3Sn. Rietveld refinementson the obtained powder diffraction patterns were performedusing the programs FULLPROF and POWDERCELL.B. Property measurementsThe electrical resistivity and Seebeck coefficient weremeasured in the temperature range 300–860 K, using acommercially available setup (ZEM3 by ULVAC/AdvanceRiko) at TU Wien and at NIMS. Additionally, the low-temperature resistivity of Ni3In was measured from 4 to300 K at TU Wien. For this purpose, thin gold wires werespot-welded onto the sample surface and the sample wasmounted on a homemade sample probe that was thenimmersed in a liquid He bath cryostat. Measurements wereconducted into the longitudinal direction of the sample (thesame direction has been measured at high temperatures)making use of a four-probe method and along the crosssection of the sample making use of the van der Pauwmethod. A comparison between these data is shown inFig. S3 [44], ruling out any anisotropy in our polycrystal-line samples.The thermal conductivity (see Fig. S1 [44]) was mea-sured at NIMS using a laser flash diffusivity setup byNetzsch. Here, the diffusivity is determined experimentally,and the specific heat is measured simultaneously using areference sample. The thermal conductivity is then obtainedfrom κ ¼ αcpρm with α being the thermal diffusivity, cp thespecific heat, and dm the material density. The results ofthese measurements confirm that electrons dominate ther-mal transport and that lattice-driven heat transport isinsignificant in these metals.C. Electronic structure calculationsDensity functional theory (DFT) calculations of theelectronic band structure of pristine Ni3In [Fig. 2(b)] wereperformed using QUANTUM ESPRESSO v. 7.1 [58,59],employing optimized norm-conserving Vanderbilt pseudo-potentials [60] with a cutoff of 80 Ry on the plane waveexpansion and the Perdew-Burke-Ernzerhof exchange-cor-relation functional [61]. To obtain the ground-state chargedensity, a uniform 12 × 12 × 12 mesh was employed forintegration over the Brillouin zone, with a Methfessel-Paxton smearing of 0.02 Ry. Phonon calculations wereperformed using density functional perturbation theory [62]on a uniform 4 × 4 × 4 q mesh, and electron-phononmatrix elements were integrated over a 24 × 24 × 24 gridwith a Gaussian smearing of 0.015 Ry for the double-deltaapproximation. Phonon dispersions and density of stateswere obtained by Fourier interpolation.Alloy-averaged densities of states of Ni3In1−xSnx[Figs. 2(c) and 2(d)] were calculated by using bulkGreen-function methods within the framework of theKohn-Korringa-Rostoker formalism and the coherentpotential approximation (KKR-CPA). This theoreticalframework accounts for the disorder due to the alloyingor substitution of Sn atoms at the In site.D. Electronic transport calculationsFor transport calculations, DFT calculations were per-formed using QUANTUM ESPRESSO v. 7.2 [58,59] with theGarrity-Bennett-Rabe-Vanderbilt pseudopotentials [63] par-ametrized for the Perdew-Burke-Ernzerhof exchange-corre-lation functional [61] and using a 140 Ry plane-wavecoefficient energy cutoff. A 6 × 6 × 8 kmesh and a 3 × 3 ×4 q mesh were used for the initial coarse-grid densityfunctional perturbation theory calculation of the electron-phonon matrix elements. All subsequent transport calcula-tions were performed using the PHOEBE code [48], an open-source package for Boltzmann transport equation solutions.These calculations used the relaxation time approximation(RTA) with the addition of the Wigner transport correction,as well as the constant relaxation time approximation(CRTA) with a 10 fs electron-phonon lifetime as a pointof comparison. The BTE-D−1ðEÞ framework utilized thegroup velocities calculated from the DFT band structures.Instead of multiplication by a constant τ, the energy-dependent relaxation time τðEÞ ∼D−1ðEÞ was multiplied.The Wigner transport formalism for electrons wascalculated as in Ref. [52] using the scattering linewidthsdetermined at the RTA level, where we compute theadditive contributions to the Onsager coefficients relatedto conductivity, σ ¼ LEE and the Seebeck coefficient,S ¼ −½LEE�−1LET :ΔLEE ¼ gse2VNkXk;ν;ν0;ν≠ν0f0ν0 ðkÞ − f0νðkÞϵν0 ðkÞ − ϵνðkÞ×viν;ν0 ðkÞvj�ν0;νðkÞ½ΓνðkÞ þ Γν0 ðkÞ�4½ϵν0 ðkÞ − ϵνðkÞ�2 þ ½ΓνðkÞ þ Γν0 ðkÞ�2; ð2ÞΔLET ¼−gse2VNk×Xk;ν;ν0;ν≠ν0viν;ν0 ðkÞvj�ν0;νðkÞ½dfνdT þ df0νdT �½ΓνðkÞþΓν0 ðkÞ�4½ϵν0 ðkÞ− ϵνðkÞ�2þ½ΓνðkÞþΓν0 ðkÞ�2;ð3ÞFABIAN GARMROUDI et al. PHYS. REV. X 15, 021054 (2025)021054-8where ν is a band index, Γ is an electron-phonon scatteringrate calculated at the RTA level (see Ref. [48] for details), vand ϵ are electronic band velocities and energies, respec-tively, f is the Fermi-Dirac function, gs is a spin degeneracyfactor, and V is a unit cell volume.The RTA and Wigner calculations were performed byinterpolating the electron and phonon band structures, aswell as the electron-phonon matrix elements, to 55 × 55 ×75 k and q meshes. Adaptive Gaussian smearing was usedto broaden the delta functions found in the electron-phononscattering rate expression. To appropriately compare to thepolycrystalline sample used in the experimental work, thetransport coefficients and the images of the Zener tunnelingcontribution to transport on bands were plotted as anaverage over Cartesian directions of the transport tensors.ACKNOWLEDGMENTSResearch in this paper was financially supported bythe Japan Science and Technology Agency (JST) programMIRAI, JPMJMI19A1. The Flatiron Institute is a division ofthe Simons Foundation. F. G. further acknowledges aDirector’s Postdoctoral Fellowship through the Laboratoryand Directed Research & Development (LDRD) program.The authors also acknowledge the TU Wien Bibliothek forfinancial support through its Open Access FundingProgramme. Furthermore, X. Yan fromTUWien is acknowl-edged for assistance with Rietveld refinements. G. J. Snyderfrom Northwestern University, E. S. Toberer from ColoradoSchool of Mines, and J. G. Checkelsky from MassachusettsInstitute of Technology are acknowledged for fruitful andstimulating discussions. F. G. and A. P. conceptualized thestudy. F. G., J. C., A. G., and A. P. analyzed the data andinterpreted the results. F. G., together with J. C. and A. P.,wrote the initial draft. F. G., M. P., M. G., and S. S. syn-thesized samples from batch 1 and performed experiments atTU Wien. I. S. synthesized samples from batch 2 andinvestigated thermoelectric transport at NIMS. F. G. con-ducted low-temperaturemeasurements. J. C. performed elec-tronic and phonon structure calculations and from therecalculated all the electronic transport properties presented inthe text. S. D. C. performed DFT calculations of the elec-tronic and phonon structure of pristine Ni3In and Ni3Sn.K. P. performed DFT calculations of the electronic structurewith and without spin orbit coupling and transport calcu-lations within the CRTA presented in SupplementalMaterial [44]. S. K. calculated the alloy-averaged densitiesof states in the KKR-CPA framework. B.W., K. H., T. M.,E. B., A. G., and A. P. supervised the work.[1] Y. Cao, V. Fatemi, S. Fang, K. Watanabe, T. Taniguchi, E.Kaxiras, and P. Jarillo-Herrero, Unconventional supercon-ductivity in magic-angle graphene superlattices, Nature(London) 556, 43 (2018).[2] J. G. Checkelsky, B. A. Bernevig, P. Coleman, Q. Si, and S.Paschen, Flat bands, strange metals and the Kondo effect,Nat. Rev. Mater. 9, 509 (2024).[3] B. Sutherland, Localization of electronic wave functions dueto local topology, Phys. Rev. B 34, 5208 (1986).[4] D. L. Bergman, C. Wu, and L. Balents, Band touching fromreal-space topology in frustrated hopping models, Phys.Rev. B 78, 125104 (2008).[5] D. Leykam, A. Andreanov, and S. Flach, Artificial flat bandsystems: From lattice models to experiments, Adv. Phys. X3, 1473052 (2018).[6] E. Tang, J.-W. Mei, and X.-G. Wen, High-temperaturefractional quantum Hall states, Phys. Rev. Lett. 106,236802 (2011).[7] Z. Liu, F. Liu, and Y.-S. Wu, Exotic electronic states in theworld of flat bands: From theory to material, Chin. Phys. B23, 077308 (2014).[8] L. Ye, M. Kang, J. Liu, F. Von Cube, C. R. Wicker, T.Suzuki, C. Jozwiak, A. Bostwick, E. Rotenberg, D. C. Bellet al., Massive Dirac fermions in a ferromagnetic kagomemetal, Nature (London) 555, 638 (2018).[9] M. Kang, S. Fang, L. Ye, H. C. Po, J. Denlinger, C. Jozwiak,A. Bostwick, E. Rotenberg, E. Kaxiras, J. G. Checkelskyet al., Topological flat bands in frustrated kagome latticeCoSn, Nat. Commun. 11, 4004 (2020).[10] J. P. Wakefield, M. Kang, P. M. Neves, D. Oh, S. Fang, R.McTigue, S. Frank Zhao, T. N. Lamichhane, A. Chen, S.Lee et al., Three-dimensional flat bands in pyrochlore metalCaNi2, Nature (London) 623, 301 (2023).[11] N. Regnault, Y. Xu, M.-R. Li, D.-S. Ma, M. Jovanovic, A.Yazdani, S. S. Parkin, C. Felser, L. M. Schoop, N. P. Onget al., Catalogue of flat-band stoichiometric materials,Nature (London) 603, 824 (2022).[12] P. M. Neves, J. P. Wakefield, S. Fang, H. Nguyen, L. Ye, andJ. G. Checkelsky, Crystal net catalog of model flat bandmaterials, npj Comput. Mater. 10, 39 (2024).[13] P. F. Rosa and F. Ronning, A quantum collaboration for flatbands, Nat. Phys. 20, 539 (2024).[14] G. J. Snyder, A. H. Snyder, M. Wood, R. Gurunathan, B. H.Snyder, and C. Niu, Weighted mobility, Adv. Mater. 32,2001537 (2020).[15] V. Pecunia, S. R. P. Silva, J. D. Phillips, E. Artegiani, A.Romeo, H. Shim, J. Park, J.-H. Kim, J. S. Yun, and G. C.Welch, Roadmap on energy harvesting materials, JPhysMater. 6, 042501 (2023).[16] A. F. Ioffe, L. Stil’Bans, E. Iordanishvili, T. Stavitskaya, A.Gelbtuch, and G. Vineyard, Semiconductor thermoelementsand thermoelectric cooling, Phys. Today 12, No. 5, 42(1959).[17] A. Zevalkink, D. M. Smiadak, J. L. Blackburn, A. J.Ferguson, M. L. Chabinyc, O. Delaire, J. Wang, K.Kovnir, J. Martin, L. T. Schelhas et al., A practicalfield guide to thermoelectrics: Fundamentals, synthesis,and characterization, Appl. Phys. Rev. 5, 021303(2018).[18] G. J. Snyder and E. S. Toberer, Complex thermoelectricmaterials, Nat. Mater. 7, 105 (2008).[19] Q. Yan and M. G. Kanatzidis, High-performance thermo-electrics and challenges for practical devices, Nat. Mater.21, 503 (2022).TOPOLOGICAL FLAT-BAND-DRIVEN METALLIC … PHYS. REV. X 15, 021054 (2025)021054-9https://doi.org/10.1038/nature26160https://doi.org/10.1038/nature26160https://doi.org/10.1038/s41578-023-00644-zhttps://doi.org/10.1103/PhysRevB.34.5208https://doi.org/10.1103/PhysRevB.78.125104https://doi.org/10.1103/PhysRevB.78.125104https://doi.org/10.1080/23746149.2018.1473052https://doi.org/10.1080/23746149.2018.1473052https://doi.org/10.1103/PhysRevLett.106.236802https://doi.org/10.1103/PhysRevLett.106.236802https://doi.org/10.1088/1674-1056/23/7/077308https://doi.org/10.1088/1674-1056/23/7/077308https://doi.org/10.1038/nature25987https://doi.org/10.1038/s41467-020-17465-1https://doi.org/10.1038/s41586-023-06640-1https://doi.org/10.1038/s41586-022-04519-1https://doi.org/10.1038/s41524-024-01220-xhttps://doi.org/10.1038/s41567-024-02397-0https://doi.org/10.1002/adma.202001537https://doi.org/10.1002/adma.202001537https://doi.org/10.1088/2515-7639/acc550https://doi.org/10.1088/2515-7639/acc550https://doi.org/10.1063/1.3060810https://doi.org/10.1063/1.3060810https://doi.org/10.1063/1.5021094https://doi.org/10.1063/1.5021094https://doi.org/10.1038/nmat2090https://doi.org/10.1038/s41563-021-01109-whttps://doi.org/10.1038/s41563-021-01109-w[20] W. Liu, H. S. Kim, Q. Jie, and Z. Ren, Importance of highpower factor in thermoelectric materials for power gen-eration application: A perspective, Scr. Mater. 111, 3(2016).[21] Y. Pei, X. Shi, A. LaLonde, H. Wang, L. Chen, and G. J.Snyder, Convergence of electronic bands for high perfor-mance bulk thermoelectrics, Nature (London) 473, 66(2011).[22] M. Zebarjadi, G. Joshi, G. Zhu, B. Yu, A. Minnich, Y. Lan,X. Wang, M. Dresselhaus, Z. Ren, and G. Chen, Powerfactor enhancement by modulation doping in bulk nano-composites, Nano Lett. 11, 2225 (2011).[23] T. Mori, Novel principles and nanostructuring methods forenhanced thermoelectrics, Small 13, 1702013 (2017).[24] G. Mahan and J. Sofo, The best thermoelectric, Proc. Natl.Acad. Sci. U.S.A. 93, 7436 (1996).[25] R. S. Whitney, Most efficient quantum thermoelectricat finite power output, Phys. Rev. Lett. 112, 130601(2014).[26] J. Maassen, Limits of thermoelectric performance with abounded transport distribution, Phys. Rev. B 104, 184301(2021).[27] J. Park, Y. Xia, V. Ozoliņš, and A. Jain, Optimal bandstructure for thermoelectrics with realistic scattering andbands, npj Comput. Mater. 7, 43 (2021).[28] J. Park, Globally optimal band structure for thermoelectricsin realistic systems, Phys. Rev. B 106, 235105 (2022).[29] S. Ding, X. Chen, Y. Xu, and W. Duan, The best thermo-electrics revisited in the quantum limit, npj Comput. Mater.9, 189 (2023).[30] I. T. Witting, T. C. Chasapis, F. Ricci, M. Peters, N. A.Heinz, G. Hautier, and G. J. Snyder, The thermoelectricproperties of bismuth telluride, Adv. Electron. Mater. 5,1800904 (2019).[31] Y. Pei, A. D. LaLonde, H. Wang, and G. J. Snyder, Loweffective mass leading to high thermoelectric performance,Energy Environ. Sci. 5, 7963 (2012).[32] F. Garmroudi, M. Parzer, A. Riss, C. Bourgès, S.Khmelevskyi, T. Mori, E. Bauer, and A. Pustogow, Highthermoelectric performance in metallic NiAu alloys viainterband scattering, Sci. Adv. 9, eadj1611 (2023).[33] C. Li, W. Ren, L. Zhang, K. Ito, and J. Wu, Effects of Aldoping on the thermoelectric performance of CoSi singlecrystal, J. Appl. Phys. 98, 063706 (2005).[34] D. Pshenay-Severin, Y. Ivanov, and A. Burkov, The effect ofenergy-dependent electron scattering on thermoelectrictransport in novel topological semimetal CoSi, J. Phys.Condens. Matter 30, 475501 (2018).[35] Y. Xia, J. Park, V. Ozoliņš, and C. Wolverton, Leveragingelectron-phonon interaction to enhance the thermoelectricpower factor in graphene-like semimetals, Phys. Rev. B100, 201401(R) (2019).[36] P. Graziosi, K.-I. Mehnert, R. Dutt, J.-W. G. Bos, and N.Neophytou, Materials design criteria for ultrahigh thermo-electric power factors in metals, PRX Energy 3, 043009(2024).[37] M. Y. Toriyama, A. N. Carranco, G. J. Snyder, and P. Gorai,Material descriptors for thermoelectric performance ofnarrow-gap semiconductors and semimetals, Mater. Horiz.10, 4256 (2023).[38] Z. Li, K. Pal, H. Lee, C. Wolverton, and Y. Xia, Electron-phonon interaction mediated gigantic enhancement ofthermoelectric power factor induced by topological phasetransition, Nano Lett. 24, 5816 (2024).[39] J. Qiu, S. Zhi, P. Zhao, J. Wang, X. Ma, S. Ye, C. Lin, X.Zhang, Z. Wu, S. Duan et al., Revealing the significant roleof band structure asymmetry on thermoelectric bipolarconduction, Phys. Rev. B 111, 045203 (2025).[40] Q. Yang, S. Yang, P. Qiu, L. Peng, T.-R. Wei, Z. Zhang, X.Shi, and L. Chen, Flexible thermoelectrics based on ductilesemiconductors, Science 377, 854 (2022).[41] P. Zhao, W. Xue, Y. Zhang, S. Zhi, X. Ma, J. Qiu, T. Zhang,S. Ye, H. Mu, J. Cheng et al., Plasticity in single-crystallineMg3Bi2 thermoelectric material, Nature (London) 631, 777(2024).[42] J. Mao, G. Chen, and Z. Ren, Thermoelectric coolingmaterials, Nat. Mater. 20, 454 (2021).[43] L. Ye, S. Fang, M. Kang, J. Kaufmann, Y. Lee, C. John,P. M. Neves, S. F. Zhao, J. Denlinger, C. Jozwiak et al.,Hopping frustration-induced flat band and strange metal-licity in a kagome metal, Nat. Phys. 20, 610 (2024).[44] See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevX.15.021054 for additionalthermal transport and reproducibility measurements as wellas additional computational results on the pristine, undopedcompounds.[45] R. W. McKinney, P. Gorai, V. Stevanović, and E. S. Toberer,Search for new thermoelectric materials with low Lorenznumber, J. Mater. Chem. A 5, 17302 (2017).[46] F. Garmroudi, M. Parzer, A. Riss, A. V. Ruban, S.Khmelevskyi, M. Reticcioli, M. Knopf, H. Michor, A.Pustogow, T. Mori et al., Anderson transition in stoichio-metric Fe2VAl: High thermoelectric performance fromimpurity bands, Nat. Commun. 13, 1 (2022).[47] Y. Xia, J. Park, F. Zhou, and V. Ozoliņš,High thermoelectricpower factor in intermetallic CoSi arising from energyfiltering of electrons by phonon scattering, Phys. Rev. Appl.11, 024017 (2019).[48] A. Cepellotti, J. Coulter, A. Johansson, N. S. Fedorova, andB. Kozinsky, PHOEBE: A high-performance framework forsolving phonon and electron Boltzmann transport equa-tions, JPhys Mater. 5, 035003 (2022).[49] C. Zener, A theory of the electrical breakdown of soliddielectrics, Proc. R. Soc. A 145, 523 (1934).[50] M. Simoncelli, N. Marzari, and F. Mauri, Unified theory ofthermal transport in crystals and glasses, Nat. Phys. 15, 809(2019).[51] M. Simoncelli, N. Marzari, and F. Mauri, Wigner formu-lation of thermal transport in solids, Phys. Rev. X 12,041011 (2022).[52] A. Cepellotti and B. Kozinsky, Interband tunneling effectson materials transport properties using the first principlesWigner distribution, Mater. Today Phys. 19, 100412 (2021).[53] G. J. Iafrate, V. N. Sokolov, and J. B. Krieger, Quantumtransport and the Wigner distribution function for Blochelectrons in spatially homogeneous electric and magneticfields, Phys. Rev. B 96, 144303 (2017).[54] G. Kané, M. Lazzeri, and F. Mauri, Zener tunneling in theelectrical transport of quasimetallic carbon nanotubes,Phys. Rev. B 86, 155433 (2012).FABIAN GARMROUDI et al. PHYS. REV. X 15, 021054 (2025)021054-10https://doi.org/10.1016/j.scriptamat.2015.07.045https://doi.org/10.1016/j.scriptamat.2015.07.045https://doi.org/10.1038/nature09996https://doi.org/10.1038/nature09996https://doi.org/10.1021/nl201206dhttps://doi.org/10.1002/smll.201702013https://doi.org/10.1073/pnas.93.15.7436https://doi.org/10.1073/pnas.93.15.7436https://doi.org/10.1103/PhysRevLett.112.130601https://doi.org/10.1103/PhysRevLett.112.130601https://doi.org/10.1103/PhysRevB.104.184301https://doi.org/10.1103/PhysRevB.104.184301https://doi.org/10.1038/s41524-021-00512-whttps://doi.org/10.1103/PhysRevB.106.235105https://doi.org/10.1038/s41524-023-01141-1https://doi.org/10.1038/s41524-023-01141-1https://doi.org/10.1002/aelm.201800904https://doi.org/10.1002/aelm.201800904https://doi.org/10.1039/c2ee21536ehttps://doi.org/10.1126/sciadv.adj1611https://doi.org/10.1063/1.2041843https://doi.org/10.1088/1361-648X/aae6d1https://doi.org/10.1088/1361-648X/aae6d1https://doi.org/10.1103/PhysRevB.100.201401https://doi.org/10.1103/PhysRevB.100.201401https://doi.org/10.1103/PRXEnergy.3.043009https://doi.org/10.1103/PRXEnergy.3.043009https://doi.org/10.1039/D3MH01013Ahttps://doi.org/10.1039/D3MH01013Ahttps://doi.org/10.1021/acs.nanolett.4c01008https://doi.org/10.1103/PhysRevB.111.045203https://doi.org/10.1126/science.abq0682https://doi.org/10.1038/s41586-024-07621-8https://doi.org/10.1038/s41586-024-07621-8https://doi.org/10.1038/s41563-020-00852-whttps://doi.org/10.1038/s41567-023-02360-5http://link.aps.org/supplemental/10.1103/PhysRevX.15.021054http://link.aps.org/supplemental/10.1103/PhysRevX.15.021054http://link.aps.org/supplemental/10.1103/PhysRevX.15.021054http://link.aps.org/supplemental/10.1103/PhysRevX.15.021054http://link.aps.org/supplemental/10.1103/PhysRevX.15.021054http://link.aps.org/supplemental/10.1103/PhysRevX.15.021054http://link.aps.org/supplemental/10.1103/PhysRevX.15.021054https://doi.org/10.1039/C7TA04332Ehttps://doi.org/10.1038/s41467-022-31159-whttps://doi.org/10.1103/PhysRevApplied.11.024017https://doi.org/10.1103/PhysRevApplied.11.024017https://doi.org/10.1088/2515-7639/ac86f6https://doi.org/10.1098/rspa.1934.0116https://doi.org/10.1038/s41567-019-0520-xhttps://doi.org/10.1038/s41567-019-0520-xhttps://doi.org/10.1103/PhysRevX.12.041011https://doi.org/10.1103/PhysRevX.12.041011https://doi.org/10.1016/j.mtphys.2021.100412https://doi.org/10.1103/PhysRevB.96.144303https://doi.org/10.1103/PhysRevB.86.155433[55] H. J. Kim, M. J. Kim, J. Lee, J. M. Ok, and C.-J. Kang,Tuning the flat band with in-plane biaxial strain and theemergence of superconductivity in Ni3Sn, Phys. Rev. B 110,024504 (2024).[56] A. M. Ganose, J. Park, A. Faghaninia, R. Woods-Robinson,K. A. Persson, and A. Jain, Efficient calculation of carrierscattering rates from first principles, Nat. Commun. 12,2222 (2021).[57] P. Graziosi, Z. Li, and N. Neophytou, ElecTra code: Full-band electronic transport properties of materials, Comput.Phys. Commun. 287, 108670 (2023).[58] P. Giannozzi, S. Baroni, N. Bonini, M. Calandra, R. Car, C.Cavazzoni, D. Ceresoli, G. L. Chiarotti, M. Cococcioni, I.Dabo et al., QUANTUM ESPRESSO: A modular and open-source software project for quantum simulations of materi-als, J. Phys. Condens. Matter 21, 395502 (2009).[59] P. Giannozzi, O. Andreussi, T. Brumme, O. Bunau, M. B.Nardelli, M. Calandra, R. Car, C. Cavazzoni, D. Ceresoli,M. Cococcioni et al., Advanced capabilities for materialsmodelling with QUANTUM ESPRESSO, J. Phys. Condens.Matter 29, 465901 (2017).[60] D. R. Hamann, Optimized norm-conserving Vanderbiltpseudopotentials, Phys. Rev. B 88, 085117 (2013).[61] J. P. Perdew, K. Burke, and M. Ernzerhof, Generalizedgradient approximation made simple, Phys. Rev. Lett. 77,3865 (1996).[62] S. Baroni, S. De Gironcoli, A. Dal Corso, and P. Giannozzi,Phonons and related crystal properties fromdensity-functionalperturbation theory, Rev. Mod. Phys. 73, 515 (2001).[63] K. F. Garrity, J. W. Bennett, K. M. Rabe, and D. Vanderbilt,Pseudopotentials for high-throughput DFT calculations,Comput. Mater. Sci. 81, 446 (2014).TOPOLOGICAL FLAT-BAND-DRIVEN METALLIC … PHYS. REV. X 15, 021054 (2025)021054-11https://doi.org/10.1103/PhysRevB.110.024504https://doi.org/10.1103/PhysRevB.110.024504https://doi.org/10.1038/s41467-021-22440-5https://doi.org/10.1038/s41467-021-22440-5https://doi.org/10.1016/j.cpc.2023.108670https://doi.org/10.1016/j.cpc.2023.108670https://doi.org/10.1088/0953-8984/21/39/395502https://doi.org/10.1088/1361-648X/aa8f79https://doi.org/10.1088/1361-648X/aa8f79https://doi.org/10.1103/PhysRevB.88.085117https://doi.org/10.1103/PhysRevLett.77.3865https://doi.org/10.1103/PhysRevLett.77.3865https://doi.org/10.1103/RevModPhys.73.515https://doi.org/10.1016/j.commatsci.2013.08.053 Topological Flat-Band-Driven Metallic Thermoelectricity I. INTRODUCTION A. ``The best thermoelectric'' revisited II. RESULTS A. Extreme interband scattering in Ni3In B. Zener quantum tunneling in Ni3In III. CONCLUSION IV. METHODS A. Materials synthesis and characterization B. Property measurements C. Electronic structure calculations D. Electronic transport calculations ACKNOWLEDGMENTS References