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Fabian Garmroudi, Michael Parzer, [Takao Mori](https://orcid.org/0000-0003-2682-1846), Andrej Pustogow, Ernst Bauer

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[Thermoelectric Transport in <math display="inline">  <msub>    <mi>Ru</mi>    <mn>2</mn>  </msub>  <mrow>    <mi>Ti</mi>    <mi>Si</mi>  </mrow></math> Full-Heusler Compounds](https://mdr.nims.go.jp/datasets/ba0a4b6c-f58a-4da9-a31a-fbbd73a89866)

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Thermoelectric Transport in Ru2TiSi Full-Heusler CompoundsPRX ENERGY 4, 013010 (2025)Thermoelectric Transport in Ru2TiSi Full-Heusler CompoundsFabian Garmroudi ,1,* Michael Parzer ,1 Takao Mori ,2,3 Andrej Pustogow ,1 and Ernst Bauer11Institute of Solid State Physics, TU Wien, 1040 Vienna, Austria2International Center for Materials Nanoarchitectonics (WPI-MANA),National Institute for Materials Science, Tsukuba 305-0044, Japan3University of Tsukuba, Tsukuba 305-8577, Japan (Received 3 December 2024; accepted 29 January 2025; published 3 March 2025)Heusler compounds with six valence electrons per atom have attracted interest as thermoelectric materi-als owing to their semimetallic and semiconducting properties. Here, we theoretically and experimentallyinvestigate electronic transport in Ru2TiSi-based full-Heuslers. We show that electronic transport in thissystem can be well captured by a two-parabolic-band model. The larger band gap of Ru2TiSi promisesa higher thermoelectric performance, compared to its isovalent family member Fe2VAl, which has beenstudied as a thermoelectric material for over two decades. Additionally, we identify p-type Ru2TiSi as farmore efficient than previously studied n-type compounds and demonstrate that this can be traced back tomuch lighter and more mobile holes originating from dispersive valence bands. Our findings suggest thatan exceptionally high dimensionless figure of merit zT > 1 can be realized in these p-type compoundsaround 700 K upon proper reduction of the lattice thermal conductivity, e.g., by substituting Zr or Hffor Ti.DOI: 10.1103/PRXEnergy.4.013010I. INTRODUCTIONHeusler compounds represent a highly tunable mate-rial platform encompassing over one thousand differentmembers that exhibit a variety of interesting electronicphases, ranging from half-metallicity to semiconductingstates and nontrivial topological band structures [1–3].Heusler compounds are typically subcategorized into half-Heuslers (hHs) and full-Heuslers (fHs) with XYZ and X2YZstoichiometries, respectively. In hHs, crystallizing in thenoncentrosymmetric C1b structure (space group no. 216,F43m), one of the X sublattices is vacant, whereas the fHstructure with L21 ordering (space group no. 225, Fm3m)can be interpreted as four interpenetrating fcc sublattices,two of which are comprised of X atoms, while the othersare built up from the Y and Z atoms. The X and Y atoms areusually transition metals, whereas the Z atom is typically amain-group-III, -IV, or even -V element [1].In general, research on Heusler materials is guided bysimple electron-counting rules, such as the Slater-Paulingrule [1,4–6], which states that Heusler compounds with*Contact author: f.garmroudi@gmx.atPublished by the American Physical Society under the terms ofthe Creative Commons Attribution 4.0 International license. Fur-ther distribution of this work must maintain attribution to theauthor(s) and the published article’s title, journal citation, andDOI.an average valence electron concentration (VEC) of sixvalence electrons per atom are nonmagnetic semiconduc-tors or semimetals, depending on the strength of hybridiza-tion. However, as the VEC increases or decreases, mag-netic metals emerge. Although there are few exceptions tothis [7,8], hundreds of hH and fH compounds align withthis concept and researchers have adhered to these straight-forward electron-counting rules, as they enable an accurateand quick estimate of the electronic- and magnetic-ground-state properties and serve as fundamental principles [1],particularly in the quest for thermoelectric semiconductors[9–12].The performance of thermoelectric materials is evalu-ated by the dimensionless figure of merit zT = S2σκ−1T,which depends on the Seebeck coefficient S, the electricalconductivity σ , the thermal conductivity κ , and the abso-lute temperature T, and directly determines the efficiencyof a thermoelectric conversion device. For practical pur-poses, zT > 1 is commonly considered as a threshold [13].While numerous hH compounds, e.g., (Ti,Zr,Hf)NiSn,(Nb,Ta)FeSb, etc., are already widely recognized as effi-cient thermoelectric materials [14–16], their fH relativeshave not yet reached zT values that would make themcompetitive with state-of-the-art materials [17].Fe2VAl is unarguably the most prominent fH thermo-electric material, with attractive near-room-temperaturethermoelectric properties [18–20]. Despite consisting onlyof metallic elements, Fe2VAl is a semimetal with a tiny2768-5608/25/4(1)/013010(11) 013010-1 Published by the American Physical Societyhttps://orcid.org/0000-0002-0088-1755https://orcid.org/0000-0003-3509-7474https://orcid.org/0000-0003-2682-1846https://orcid.org/0000-0001-9428-5083https://ror.org/04d836q62https://ror.org/026v1ze26https://ror.org/02956yf07https://crossmark.crossref.org/dialog/?doi=10.1103/PRXEnergy.4.013010&domain=pdf&date_stamp=2025-03-03http://dx.doi.org/10.1103/PRXEnergy.4.013010https://creativecommons.org/licenses/by/4.0/FABIAN GARMROUDI et al. PRX ENERGY 4, 013010 (2025)T (K)0 200 400 600 800 1000 1200 S (µV K–1)10100Ru2TiSi, Ref. [29]Ru2TiGe, Ref. [28]Ru2NbAl, Ref. [27]Ru2NbGa, Ref. [25]Fe2VAl, Ref. [26]Fe2VAl, Ref. [24]Fe2VGa, Ref. [18]FIG. 1. A comparison of the temperature-dependent Seebeckcoefficients of various semimetallic and semiconducting fHs withsix valence electrons per atom [18,24–29].band overlap or an almost gapless semiconductor [21–23].There are two primary factors limiting the performance ofFe2VAl thermoelectrics: (i) their intrinsically large latticethermal conductivity and (ii) the small band gap, whichalready results in bipolar conduction at T � 300 K. InFig. 1, we compare the temperature-dependent Seebeckcoefficient S(T) of various fH compounds with an effectivevalence electron concentration of six valence electrons peratom, VEC = 6, reported and experimentally studied inthe literature previously [18,24–29]. Note the pronouncedmaximum in S(T) of Fe2VAl at around 200 K, arisingfrom the aforementioned bipolar conduction, i.e., the acti-vation of minority carriers across the narrow band gap.On the other hand, isovalent Ru2TiSi, a novel fH com-pound recently studied by Fujimoto et al. [29], displaysa much larger Seebeck coefficient and a broad maximum,vastly surpassing all other known fH compounds over theentire temperature range. This has motivated us to exper-imentally and theoretically study electronic transport inRu2TiSi in detail and assess its potential thermoelectricperformance at optimized doping.II. EXPERIMENTAL METHODS AND MODELINGPolycrystalline Ru2TiSi1−xAlx materials have beensynthesized by melting raw elements with high-purity(99.99% Ru, 99.95% Ti, 99.9999% Si, and 99.999% Al)using a high-frequency induction-heating furnace. Eventhough powder x-ray diffraction investigations displayeda single Heusler phase directly after the melting proce-dure, the samples have been further annealed at 1273 Kfor 2 days in vacuum (10−5 mbar) to optimize homo-geneity. The samples have then been cut using a high-speed cutting device (Accutom by Struers) equipped witha diamond cutting wheel. The electrical resistivity andSeebeck coefficient at high temperatures were measuredin the temperature range 300–860 K in an inert Heatmosphere, using a commercially available setup (ZEM3by ULVAC-RIKO). To analyze the temperature- andcarrier-concentration-dependent Seebeck coefficient, weemployed a least-squares-fit model based on Boltzmann-transport theory and the parabolic band approximation,as implemented in the SeeBand software [30]. Withinthis framework, thermoelectric transport is modeled bynumerically solving the respective Fermi-transport inte-grals and summing up the contributions of the individualbands, assuming parallel conduction through two transportchannels, i.e., one valence band and one conduction band.III. RESULTS AND DISCUSSIONA. Ru2TiSi versus Fe2VAlIn order to understand the thermoelectric propertiesand enhanced performance of Ru2TiSi, a natural ques-tion would be what distinguishes it from the archetypalthermoelectric fH compound Fe2VAl. In Fig. 2(a), weshow the electronic density of states (DOS) of Ru2TiSiand Fe2VAl [31]. Both compounds display a deep well-pronounced pseudogap at the Fermi energy EF , althoughthe gap is significantly wider for Ru2TiSi. Note that thispseudogap arises from the hybridization between the tran-sition metal and the main-group-II or -IV elements, alongwith the absence of significant d-d hybridization [32]—amechanism different from the ones frequently discussed inthe context of high-temperature cuprate superconductors.Additionally, similar to Fe2VAl, the DOS of Ru2TiSi ischaracterized by sharp peaks (owing to the rather localizedRu-4d states) rising next to both edges of the pseudogap,which results in a large differential DOS. Additionally,there are many more dispersive states (with a small DOS)dangling into the gap region, hereafter referred to as pseu-dogap states. It has been shown in previous studies thatin Fe2VAl, electronic transport is almost exclusively gov-erned by these pseudogap states. In Fig. 2(b), we showthe DOS around EF and a close-up of these pseudogapstates. One can immediately note that the pseudogap statesare much broader and the DOS much smaller for Ru2TiSithan for Fe2VAl, especially at E < EF . This implies muchlighter more mobile charge carriers when EF is placedin the vicinity of these states. Moreover, the lower-DOSeffective mass m∗DOS suggests a much more efficient dop-ing scenario for Ru2TiSi. We illustrate this by drawingand comparing the Fermi level of hole-doped Fe2VAl(gray dashed line) and Ru2TiSi (purple dashed line) for ahole doping concentration of 0.05 holes per formula unit,assuming rigid-band doping. This corresponds approx-imately to the concentration for which p-type Fe2VAldisplays its optimal thermoelectric performance. It can beseen that while for Fe2VAl, this would place EF about0.2 eV below the valence-band edge, EF is shifted by013010-2THERMOELECTRIC TRANSPORT IN Ru2TiSi. . . PRX ENERGY 4, 013010 (2025)–4 –2 0 2 4DOS (states per eV per atom)01234 Ru2TiSiFe2VAlE − EF (eV) –1.0 –0.5 0.0 0.5 1.001234E − EF (eV)DOS (states per eV per atom)(a) (b)–1.0 –0.5 0.0 0.5 1.020222426Integrated DOS FIG. 2. The electronic density of states (DOS) of the fH compounds Ru2TiSi and Fe2VAl. (a) A deep pseudogap with an almostnegligibly small DOS at the Fermi level is present in both compounds, which is framed by sharp features in the DOS, originatingfrom rather localized Fe-3d and Ru-4d states. (b) A close-up around the Fermi energy emphasizes that the pseudogap is much broaderin Ru2TiSi. Additionally, it can be seen that the dispersive pseudogap states, reaching into the gap region (small DOS), especiallythe valence-band states, have a much larger bandwidth and are therefore even more dispersive for Ru2TiSi compared to Fe2VAl. Theblack dashed line indicates the Fermi energy EF and the gray and purple dashed lines show EF for a rigid-band doping scenario with0.05 holes per formula unit. The inset shows the integrated DOS of both compounds.almost 0.6 eV into the valence band for Ru2TiSi. Hence,a significantly smaller number of holes has to be dopedto reach optimal performance [compare also with the inte-grated DOS in the inset of Fig. 2(b)]. Interestingly, thedifference in m∗DOS appears less sizeable for the conductionband at E > EF .In Figs. 3(a) and 3(b), we display the temperature-dependent and doping-concentration-dependent Seebeckcoefficient (S(T) and S(p , n)), respectively. In Fig. 3(a),we model experimental S(T) data of Fe2VAl and Ru2TiSiavailable in the literature [26,29], by employing a two-parabolic-band (2PB) least-squares-fit model as imple-mented in the SeeBand software package [30]. To fitthe data, three independent fit parameters are adjusted:(i) the position of the Fermi level; (ii) the band gapor overlap Eg; and (iii) a weighting parameter εm =(NVBm∗CB)/(NCBm∗VB), which includes the degeneracies Niand effective masses mi of the two bands (i = {VB, CB}).EF determines the slope of S(T) at low temperatures, Egthe maximum Seebeck coefficient Smax as well as the tem-perature of the maximum Tmax, and ε dictates the sharpnessof the maximum, i.e., the tail of S(T) in the bipolar regimeat temperatures above Tmax. Details regarding the modelingframework, which we have previously applied successfullyto a number of fH [19,46] and skutterudite thermoelectricmaterials [48], are described in Ref. [30]. As for Fe2VAl,excellent agreement with the experimental data is alsofound for Ru2TiSi. The somewhat similar slope of S(T)at low temperatures for Fe2VAl and the extrapolated curveof the 2PB model for Ru2TiSi indicate a comparable posi-tion of EF with respect to the valence-band edge. The mostnotable difference, however, is that Smax is shifted towardmuch higher temperatures in Ru2TiSi and reaches a verybroad maximum of almost 200 µV K−1, as opposed to only70–80 µV K−1 in Fe2VAl.In Fig. 3(b), we show the room-temperature Seebeckcoefficient as a function of the hole (p) and the elec-tron (n) doping concentration, calculated from the VECvia p , n = 16 nv/a3, where a3 denotes the cubic-unit-cellvolume of Ru2TiSi or Fe2VAl and 16 nv represents thenumber of valence electrons in the unit cell. Each pointrefers to a different sample with a different VEC, which hasbeen varied through aliovalent-element substitution. Thedata points have been taken from numerous doping stud-ies in the literature and p-type Ru2TiSi1−xAlx compoundshave been synthesized and investigated in the course ofthis work. The solid lines are theoretical calculationsemploying a 2PB model. Anand et al. have previouslyshown that the S(p , n) dependence of Fe2VAl can be ade-quately described employing a 2PB model, assuming asmall positive band gap of Eg ≈ 0.02 eV and valence- andconduction-band effective masses of m∗VB ≈ 4.7 me andm∗CB ≈ 12.8 me, respectively [22].Contrary to Fe2VAl, we find from our temperature- anddoping-dependent analysis of the Seebeck coefficient ofRu2TiSi a much larger band gap of Eg ≈ 0.22–0.24 eV andmuch lighter effective masses m∗VB ≈ 1.0 me and m∗CB ≈3.3 me. The effect of the latter is directly visible in the muchmore rapid decrease of S as p and n increase and is con-sistent with the smaller DOS and broader bandwidth ofthe pseudogap states of Ru2TiSi shown in Fig. 2(b). Forp-type Ru2TiSi1−xAlx, for instance, S(300 K) decreases013010-3FABIAN GARMROUDI et al. PRX ENERGY 4, 013010 (2025)p,n (1021 cm–3)10 5 0 5 10–200–1000100T (K)0 200 400 600 800 1000S (µV K–1)S (µV K–1)050100150200Ru2TiSi, Ref. [29]Fe2VAl, Ref. [26]n-Ru2(Ti,Ta)Si, Ref. [29]Fe2VAl, Refs. [18,19,33–47]p-Ru2Ti(Si,Al),this work 2PB model(a) (b)n-typep-type300 KFIG. 3. Modeling of the thermoelectric transport in Fe2VAl and Ru2TiSi fHs. (a) A comparison of the temperature-dependent See-beck coefficients of Fe2VAl and Ru2TiSi. The experimental data have been taken from Refs. [26] and [29], respectively. The solidlines are least-squares fits employing a two-parabolic-band model, which captures the temperature-dependent S(T) of both compoundsin the entire temperature range excellently. (b) The doping-concentration-dependent Seebeck coefficient of p- and n-doped Fe2VAland Ru2TiSi at room temperature. The Seebeck coefficient decreases much more rapidly as a function of the carrier concentrationfor the latter, indicating that doping is far more efficient in Ru2TiSi compared to Fe2VAl, owing to the much lighter and more dis-persive bands; p-type Ru2TiSi1−xAlx-based compounds (filled symbols) have been synthesized and investigated in this work. Theunfilled purple symbols have been taken from Ref. [29] and the filled gray symbols have been taken from various literature studies[18,19,33–47].from 130 µV K−1 in pristine Ru2TiSi down to only22 µV K−1 for x = 0.05. In this sense, the valence-bandelectronic structure of Ru2TiSi appears much more similarto those of chalcogenide semiconductors such as Bi2Te3and PbTe, where the underlying DOS is composed of s-and p-like states with a larger bandwidth (low m∗DOS),rather than hH and fH compounds, where usually a muchlarger doping concentration is required due to the morelocalized nature of the d orbitals building up the DOS(high m∗DOS). At this point, we also note that, interestingly,S(n) first decreases and then increases with n for n-typeRu2Ti1−xTaxSi reported in Ref. [29], instead of monoton-ically decreasing. While this might be within the marginof experimental uncertainty, it could also be an indicationof the contribution of a second conduction band as EF isshifted further toward higher energies with increasing Tasubstitution. Indeed, this would be consistent from a purelyrigid-band-like shift of EF , considering the DOS presentedin Fig. 2(b), which changes its slope around 0.4 − 0.5 eVabove EF .B. Detailed analysis of p- and n-doped Ru2TiSiNext, we present a detailed analysis of the temperature-dependent transport properties of n-type Ru2Ti1−xTaxSifrom Ref. [29] and p-type Ru2TiSi1−xAlx from this work.In Fig. 4(a), we display the S(T) data of the former,collected by Fujimoto et al. in the temperature range300–1000 K. Least-squares fits using the 2PB model areagain able to reproduce all the measured curves reasonablywell. Particularly striking is the distinct maximum in S(T)at 700 K, followed by a sign reversal of S(T) at around900 K for Ru2Ti0.97Ta0.03Si. This is a direct signature ofthe strong electron-hole asymmetry and the much highermobility of holes compared to the conduction-band elec-trons. This is directly reflected in the weighting parameterεm derived from our fit. A large εm implies that either amuch larger degenerate set of hole pockets (compared tothe electron pockets) contributes to the transport propertiesand/or that conduction-band electrons are much heaviercompared to the hole-type carriers. An extraordinarilylarge value of εm ≈ 60 is found for Ru2Ti0.97Ta0.03Si (andeven larger values are found for higher Ta concentrations;see Appendix C), which is especially remarkable consid-ering that NVB = NCB = 3 is derived from band-structurecalculations provided in the Materials Project open webdatabase [49]. We attribute this extreme electron-holeasymmetry to the very dispersive valence-band and themuch more localized and heavy conduction-band states,approximately 0.5 eV above EF [see Fig. 2(b)]. Thesestates likely become important at elevated temperaturesand higher doping concentrations, which explains why amuch smaller band asymmetry εm ≈ 3.3 is derived fromthe analysis of the doping-dependent Seebeck coefficientof Ru2TiSi-based compounds at 300 K, shown in Fig. 3(b).Nonetheless, this band asymmetry is likely critical for thethermoelectric performance of Ru2TiSi, which peaks atmuch higher temperatures T > 300 K.The notion of much lighter holes compared to theconduction electrons is also confirmed when examining013010-4THERMOELECTRIC TRANSPORT IN Ru2TiSi. . . PRX ENERGY 4, 013010 (2025)S(µV K–1)S(µV K–1)Ru2TiSi1–xAlx (µcm)200 400 600 800 1000 1200–200–150–100–50050100T (K)T (K) T (K)200 400 600 800 1000 12000100200300400500600xx0.00 0.05 0.10 0.15 0.200100200300400500600200 400 600 800 1000 1200–100–500501001502002PB modelx = 0.001x = 0.003x = 0.005x = 0.052PB modelx = 0.03x = 0.06x = 0.12x = 0.2Ru2Ti1–xTaxSi, Ref. [29] Ru2TiSi1–xAlxn-doped Ru2Ti0.8Ta0.2Sip-doped Ru2TiSi0.95Al0.05Ru2TiSiRu2Ti1–xTaxSi, Ref. [29]T = 300 K(a) (b)(c) (d)0.00 0.05 0.10 0.15 0.200200400600800Second CBFIG. 4. The electron-hole band asymmetry in Ru2TiSi. The temperature-dependent Seebeck coefficient S(T) of (a) n-typeRu2Ti1−xTaxSi from Ref. [29] and (b) p-type Ru2TiSi1−xAlx from this work. The solid lines are least-squares fits employing a two-parabolic-band model. Notably, the maximum in |S(T)| is much sharper for n-type Ru2Ti1−xTaxSi and even a sign reversal of theSeebeck coefficient takes place, once minority carriers from the valence band are activated. This implies that holes are much moremobile than carriers occupying the conduction-band states in Ru2TiSi. For p-type Ru2TiSi1−xAlx, the maximum in |S(T)| is muchbroader and doping is much more efficient, i.e., a substitution of only a few at.% Al for Si results in a strong decrease of S(T) as EFmoves rapidly away from the band edge. (c) The weighted-mobility comparison of heavily n-doped Ru2Ti0.8Ta0.2Si, pristine Ru2TiSi,and heavily p-doped Ru2TiSi0.95Al0.05. (d) The composition-dependent weighted mobility evaluated at 300 K. The solid and dashedlines are a guide to the eye, with the yellow area corresponding to an expected gain in μW arising from the contribution of a secondconduction band. Most importantly, μW is much greater for p-type Ru2TiSi, confirming its superiority over n-type Ru2TiSi. The insetshows the corresponding composition-dependent resistivity.the temperature-dependent Seebeck coefficient of p-type Ru2TiSi1−xAlx in Fig. 4(b). Both the experimen-tal data collected in this study and the 2PB modelextrapolating toward higher temperatures show a muchbroader maximum of S(T), almost saturating at hightemperatures. This demonstrates that the weighted con-tribution of the minority carriers, activated at high tem-peratures, is small. In Figs. 4(c) and 4(d), we display thetemperature- and composition-dependent weighted mobil-ity μW. The weighted mobility, μW ≈ μD(m∗DOS/me)3/2, isa temperature-dependent material property, which, unlikethe drift mobility μD and the Hall mobility μH, is inde-pendent of the carrier concentration—at least within theparabolic band approximation. From a thermoelectric per-spective, μW represents a unique relationship between Sand σ and determines the maximum figure of merit zthat can be achieved at optimized doping. We have calcu-lated μW via the scheme presented in Ref. [50]. Despitethe limitations of such an analysis owing to (i) bipolartransport and (ii) the rather complex band structure, partic-ularly concerning the conduction-band states, it is nonethe-less obvious that μW is several times larger for p-doped013010-5FABIAN GARMROUDI et al. PRX ENERGY 4, 013010 (2025)Ru2TiSi, as opposed to the n-doped compounds reportedpreviously [29]. The increase of μW with increasing Alconcentration x in Ru2TiSi1−xAlx is anomalous and hintsat either (i) contributions from additional valence bandsas EF is lowered deeper into the valence-band states,(ii) deviation of parabolicity of the contributing band val-leys, or (iii) a change in the dominant carrier-scatteringmechanism. For instance, it is well known that in intrin-sic semiconductors, such as pristine Ru2TiSi, tiny amountsof ionized impurities can limit carrier transport, while withan increase in the carrier concentration, these impuritiesbecome screened, enhancing the mobility. In the highlydoped regime, μW is most likely primarily limited bycharge carriers scattering off phonons and random potentialfluctuations caused by the alloy disorder at the Si site.C. Assessing optimal thermoelectric performanceTo evaluate the optimal performance of p- and n-typeRu2TiSi, we have modeled the composition-dependentpower factor PF = S2σ—the product of the Seebeck coef-ficient squared, S2, and the electrical conductivity σ—andthe thermal conductivity. The maximum zT has been esti-mated based on these theoretical results. In Fig. 5(a),we show the carrier doping-dependent PF at 300 K and700 K. Despite some limitations, our 2PB model tracesthe trend of the experimental data fairly well and correctlyreproduces the enhanced performance of the p-type com-pounds. The electronic part of the thermal conductivity hasbeen calculated via the Wiedemann-Franz law, followingthe same procedure as described in Ref. [22]. The lat-tice thermal conductivity κL, shown in Fig. 5(b), can bedescribed by a simple alloy-scattering model [22], whichconsiders two primary scattering contributions, namely,(i) Umklapp phonon-phonon scattering and (ii) scatteringof high-frequency phonons with point defects introducedby the random substitution of other atoms in the com-pound. The latter depends on mass and volume fluctuationsand is especially significant when substituting heavy 5delements, e.g., Ta instead of Ti. Following this argu-ment, one may assume that other 5d elements with similaratomic mass and size, such as Hf, would have a similareffect and composition dependence of κL, which is indeedwhat is observed in Fe2VAl fH compounds as well [51].According to the Materials Project database, Ru2HfSi isalso a theoretically predicted stable Heusler compound,as is Ru2ZrSi, both with a very similar electronic struc-ture to Ru2TiSi. This suggests that alloying of Ru2TiSiwith Ru2ZrSi and Ru2HfSi could be possible, consequentlyleading to reduced κL and hence increased zT. Since thePF (mW m–1 K–2)(a) (b) (c)n-Ru2Ti1–xTaxSiFIG. 5. Modeling and estimation of the thermoelectric performance in Ru2TiSi-based fH compounds. (a) The doping-concentration-dependent power factor PF of Ru2TiSi at 300 K and 700 K. The open and filled symbols are experimental data from this work and fromRef. [29], respectively. The solid lines are theoretical calculations using a two-parabolic-band model. A more than 2 times larger PF ispredicted (and experimentally achieved) for p-type Ru2TiSi due to the superior quality of the valence-band structure. At a fixed geom-etry and for a given temperature difference, the power factor determines the power output that can be generated within a thermoelectricdevice. (b) The composition-dependent trend of the lattice thermal conductivity for Ru2Ti1−xTaxSi from Ref. [29]. The dashed-dottedlines have been calculated employing a simple alloy-scattering model [22]. The large mass and volume fluctuations imposed by thesubstitution with heavy 5d elements effectively scatter high-frequency heat-carrying phonons. (c) The doping-concentration-dependentzT of Ru2TiSi at 300 K and 700 K. The power factor PF and the electronic contribution to the thermal conductivity have been calculatedemploying a two-parabolic-band model. The lattice thermal conductivity has been estimated from (b) under the reasonable assumptionthat the substitution with other 5d elements, such as Hf/Ti, results in a similar suppression of κL. Hf substitution does not change thetotal number of valence electrons and, therefore, it remains near the predicted maximum zT, exceeding zT = 1 at 700 K, especiallyin the case in which a full solid solution between Ru2TiSi and Ru2HfSi is possible. The inset shows the temperature-dependent zT atoptimal doping.013010-6THERMOELECTRIC TRANSPORT IN Ru2TiSi. . . PRX ENERGY 4, 013010 (2025)valence-band states of fH compounds, such as Fe2VAl andRu2TiSi, are almost exclusively governed by the X atoms,introducing disorder at the Y site hardly affects the elec-tronic transport properties and retains high values of theweighted mobility [22]. On the other hand, the pseudogapstates of the conduction band have a Y-eg orbital charac-ter, which results in a strong trade-off between κL and μW.Thus, substituting heavy 5d elements at the Ti site wouldbe especially promising for p-type Ru2TiSi, where EF islocated in the Ru-t2g valence bands.In Fig. 5(c), we show the theoretical prediction of thedoping-dependent zT for an alloy of Ru2TiSi and Ru2HfSifrom our 2PB model, assuming similar electronic trans-port. The darker colors refer to a 20% Ti-Hf substitution,while the lighter colors refer to a 50% alloy, minimizingthe lattice thermal conductivity of the compound. For theRu2Ti0.5Hf0.5Si alloy, our calculations predict a high max-imum zT = 1 − 1.2 at 700 K for optimal doping, whichmotivates experimental exploration of this virginal mate-rial platform and showcases that not only hH but also fHcompounds bear the potential for competitive thermoelec-tric performance.IV. CONCLUSIONSTo summarize, we have investigated thermoelectrictransport of Ru2TiSi-based fH compounds and comparedtheir transport properties to those of Fe2VAl. A two-parabolic-band model accurately captures the temperature-and doping-dependent thermoelectric transport propertiesof Ru2TiSi. The resulting effective band structure underliesthat the valence-band electronic structure displays muchgreater potential for realizing high thermoelectric perfor-mance compared to p-type Fe2VAl. Moreover, we predictthat p-type Ru2TiSi would outperform n-type compounds,studied previously, by a factor of 2–3, potentially realizingzT > 1 at 700 K upon appropriate reduction of the latticethermal conductivity, e.g., by isovalent substitution with Zror Hf at the Ti site. Our work encourages further investi-gation of the vast phase space of fH next to hH compoundsas thermoelectric materials.ACKNOWLEDGMENTSF.G., M.P., E.B., and T.M. were financially supported bythe Japan Science and Technology Agency (JST) programMIRAI, JPMJMI19A1. A.P. acknowledges support fromOeAD WTZ (Projects CZ 08/2023 and HR 05/2024).DATA AVAILABILITYThe experimental data are available at TU Wien’sresearch data respository [52].APPENDIX A: STRUCTURAL PROPERTIESThe structural properties and phase purity of the p-type Ru2TiSi1−xAlx Heusler compounds synthesized andinvestigated in this work have been studied using x-raypowder diffraction (XRPD), making use of a commer-cially available diffractometer (AERIS by PANalytical).Cu Kα radiation has been used and measurements havebeen conducted in a Bragg-Brentano geometry. An exem-plary XRPD pattern for the sample with the largest amountof Al substitution, i.e., Ru2TiSi0.95Al0.05, is shown in Fig. 6together with Rietveld refinements, which have been per-formed using the program PowderCell. The powder patterndisplays a single fH (L21 structure) phase with a lat-tice parameter a ≈ 0.5967 nm derived from the Rietveldrefinement. The room-temperature lattice parameter is onlyabout 0.14% larger than that of pristine Ru2TiSi [29],which aligns with expectations, since the atomic radii ofsilicon (rSi ≈ 111 pm) and aluminum (rAl ≈ 125 pm) arequite similar.APPENDIX B: ELECTRICAL RESISTIVITYIn Fig. 7, we summarize the temperature-dependentelectrical resistivity ρ(T) of the Ru2TiSi system. First,ρ(T) of pristine Ru2TiSi from Ref. [29] is compared withρ(T) of pristine Fe2VAl from Ref. [26] in Fig. 7(a). Despitethe larger band gap, evident from the larger Seebeck coef-ficient of Ru2TiSi, the resistivity at low temperatures isactually lower compared to Fe2VAl. We attribute this tothe fact that Ru2TiSi is intrinsically doped, with EF locatedabout 0.06 eV below the valence band edge. Thus, atlow temperatures ρ(T) should show metalliclike behav-ior, as is indeed observed. Fujimoto et al. have reporteda carrier mobility of around 100 cm2 V−1 s−1 for pristineRu2TiSi [29], about an order of magnitude larger than2  (degrees)20 40 60 80 100Intensity (105  counts)0.00.51.01.5YobsYcalcYobs − YcalcBragg position Ru2TiSi0.95Al0.05TiRuSi/AlFIG. 6. The x-ray powder diffraction pattern of phase-pure p-type Ru2TiSi0.95Al0.05 alongside Rietveld refinement. The insetshows the fH crystal structure.013010-7FABIAN GARMROUDI et al. PRX ENERGY 4, 013010 (2025)T (K)0 200 400 600 800 1000(µΩ cm)(µΩ cm)050010001500200025003000T (K)200 400 600 800 100010100100010 000Ru2TiSi, Ref. [29]Fe2VAl, Ref. [26]x = 0.005x = 0.05x = 0.03x = 0.06x = 0.2Ru2Ti1–xTaxSi, Ref. [29]Ru2TiSi1–x Alx, this work(a) (b)FIG. 7. The temperature-dependent electrical resistivity of Ru2TiSi Heusler compounds. (a) The temperature-dependent electricalresistivity of pristine Ru2TiSi [29] compared to that of pristine Fe2VAl [26]. Although Ru2TiSi has a larger band gap, the electricalresistivity, especially at low temperatures, is lower than that of Fe2VAl, highlighting that EF is intrinsically doped in the valenceband, which, due to its lower effective mass, enables a higher carrier mobility and lower resistivity for Ru2TiSi. (b) The temperature-dependent resistivity for different n- and p-doped Ru2TiSi samples from Ref. [29] and this work, respectively. It is evident that forp-type Ru2TiSi1−xAlx, ρ(T) decreases much more rapidly as a function of the doping concentration.that of Fe2VAl. The carrier mobility in undoped Fe2VAlis not only smaller due to less dispersive bands at EF butalso hampered by pivotal carrier scattering off localizedin-gap impurity states arising from intrinsic Fe-V and Fe-Al exchange antisite defects [53]. It is possible that theformation of such antisite defects, involving the Ru sublat-tice, is suppressed in Ru2TiSi due to the larger atomic sizemismatch between Ru and Ti-Si, as opposed to Fe versusV-Al.In Fig. 7(b), we show ρ(T) for various n- and p-doped Ru2TiSi-based fHs from Ref. [29] and this work,respectively. It is evident that for p-type compounds, ρ(T)decreases extremely quickly, again reflecting the disper-sive nature of the valence-band electronic structure. ForRu2TiSi1−xAlx, an Al substitution of only x = 0.05 yieldsa very strong decrease of the room-temperature resistivitydown to only 56 µ� cm, which is comparable to that ofordinary metals.APPENDIX C: FIT PARAMETERSIn Table I, we list the obtained fit parameters fromour analysis of the temperature-dependent Seebeck coef-ficient of n-type Ru2Ti1−xTaxSi from Ref. [29] andp-type Ru2TiSi1−xAlx from this work. There are threeindependent fit parameters that can be extracted to developan effective band-structure model. The first parameter,εm, serves as a weighting parameter, which representsthe weighted contribution between the two bands. Thelarger εm is, the smaller is the weighted contributionof the conduction-band carriers. Since mVB ≈ 1 me couldbe derived from the carrier-concentration dependence ofS and because NVB = NCB ≈ 3, εm can be consideredthe effective mass of the conduction-band electrons εm ∼mCB. Taking a look at the values in Table I, it is clearthat mCB increases dramatically with Ta substitution andsort of saturates at very large values of several hundredtimes the free electron mass [left axis in Fig. 8(a)]. Thisaligns with the notion of a second, much heavier, con-duction band, further above EF , which is also observedin Fe2VAl and similar Heusler compounds and the ori-gin of which has been extensively discussed by Bilc et al.[54]. Similarly, the band gap derived from our 2PB modelTABLE I. The fit parameters obtained from modeling thetemperature-dependent Seebeck coefficient of Ru2Ti1−xTaxSiand Ru2TiSi1−xAlx. The values of the Fermi energy are givenwith respect to the valence-band/conduction-band edge for p-and n-type samples respectively.Sample x εm Eg (eV) EF (eV)n-type Ru2Ti1−xTaxSi 0 1.2 0.24 −0.060.03 62 0.42 0.090.06 392 0.67 0.110.12 385 0.78 0.110.20 239 0.74 0.12p-type Ru2TiSi1−xAlx 0 1.2 0.24 −0.060.001 3.4 0.18 −0.050.003 2.1 0.16 −0.080.005 1.5 0.11 −0.090.05 · · · · · · −0.26013010-8THERMOELECTRIC TRANSPORT IN Ru2TiSi. . . PRX ENERGY 4, 013010 (2025)x0.20m ~ mCB (me)1101001000Eg (eV)x0.00 0.05 0.10 0.15 0.00 0.05 0.10 0.15 0.20EF (eV)–0.10–0.050.000.050.100.15EF − ECBMEF − EVBMRu2Ti1–xTaxSi Ru2Ti1–xTaxSiX(a) (b)FIG. 8. The evolution of the electronic structure in n-doped Ru2Ti1−xTaxSi, extracted from fits of the temperature-dependent Seebeckcoefficient. (a) The concentration-dependent weighting parameter εm (left axis) and band gap Eg (right axis) of n-type Ru2Ti1−xTaxSi.The former corresponds to the effective mass of the conduction band, given that mVB ≈ 1 me and NVB = NCB. Both εm ∼ mCB and Egextracted from our 2PB fits increase with x as EF is shifted further toward the flat band, located 0.75 eV above the top of the valencebands. (b) The doping level of n-type Ru2Ti1−xTaxSi. EF jumps abruptly from the valence toward the conduction band but saturateswith increasing x due to the high DOS of the flat-band states.seemingly increases with x and saturates at around 0.7–0.8eV [right axis in Fig. 8(a)]. We interpret this as the posi-tion of the second conduction-band minimum with respectto the valence-band top. The Fermi level EF , which isgiven with respect to the top of the valence band for p-type and with respect to the bottom of the conductionband for n-type samples, rapidly jumps from the top ofthe valence band and is shifted into the conduction-bandstates. As x increases further, however, EF almost satu-rates and is seemingly pinned in the conduction band [seeFig. 8(b)], which is another indirect proof of the flat andheavy band and its associated high DOS that prevents effi-cient doping (shifts of EF ). A schematic of the effectiveband structure expected for Ru2TiSi, the respective energygaps, and the position of the Fermi level is presented inFig. 8(b).To summarize, there is conclusive evidence from vari-ous fit parameters obtained by modeling the temperature-dependent Seebeck coefficient that Ru2TiSi is a narrow-gapsemiconductor with dispersive valence- and conduction-band states and significant electron-hole asymmetryarising from another conduction band hosting charge car-riers that are orders of magnitude heavier. This becomesparticularly important when EF is shifted deep into theconduction bands via n-type doping, or at high tem-peratures, where states further away from EF becomeexcited.[1] T. Graf, C. Felser, and S. S. Parkin, Simple rules for theunderstanding of Heusler compounds, Prog. Solid StateChem. 39, 1 (2011).[2] F. Casper, T. Graf, S. Chadov, B. Balke, and C. Felser, Half-Heusler compounds: Novel materials for energy and spin-tronic applications, Semicond. Sci. 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Lett. 114, 136601 (2015).[55] A. Jain, S. P. Ong, G. Hautier, W. Chen, W. D. Richards,S. Dacek, S. Cholia, D. Gunter, D. Skinner, and G. Cederet al., Commentary: The Materials Project: A materi-als genome approach to accelerating materials innovation,APL Mater. 1, 011002 (2013).013010-11https://doi.org/10.1016/j.jallcom.2009.05.032https://doi.org/10.1103/PhysRevB.102.075117https://doi.org/10.1103/PhysRevB.103.085202https://doi.org/10.1103/PhysRevB.106.235138https://doi.org/10.1016/j.intermet.2022.107567https://doi.org/10.1002/adma.202001537https://doi.org/10.3390/met8110864https://doi.org/10.48436/04ts4-pp738https://doi.org/10.1103/PhysRevB.107.014108https://doi.org/10.1103/PhysRevLett.114.136601https://doi.org/10.1063/1.4812323 I.. INTRODUCTION II.. EXPERIMENTAL METHODS AND MODELING III.. RESULTS AND DISCUSSION A.. Ru2TiSi versus Fe2VAl B.. Detailed analysis of p- and n-doped Ru2TiSi C.. Assessing optimal thermoelectric performance IV.. CONCLUSIONS . ACKNOWLEDGMENTS . DATA AVAILABILITY . APPENDIX A: STRUCTURAL PROPERTIES . APPENDIX B: ELECTRICAL RESISTIVITY . APPENDIX C: FIT PARAMETERS . 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