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Michael Parzer, Fabian Garmroudi, Alexander Riss, Michele Reticcioli, Raimund Podloucky, Michael Stöger-Pollach, Evan Constable, Andrej Pustogow, [Takao Mori](https://orcid.org/0000-0003-2682-1846), Ernst Bauer

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PRX ENERGY 3, 033006 (2024)Semiconducting Heusler Compounds beyond the Slater-Pauling RuleMichael Parzer ,1,* Fabian Garmroudi ,1 Alexander Riss,1 Michele Reticcioli ,2Raimund Podloucky ,3 Michael Stöger-Pollach ,4 Evan Constable ,1 Andrej Pustogow ,1Takao Mori ,5,6 and Ernst Bauer11Institute of Solid State Physics, Technische Universität (TU) Wien, Vienna 1040, Austria2Faculty of Physics and Center for Computational Materials Science, University of Vienna, Vienna 1090, Austria3Department of Physical Chemistry, University of Vienna, Vienna 1090, Austria4University Service Center for Transmission Electron Microscopy, TU Wien, Vienna 1040, Austria5International Center for Materials Nanoarchitectonics (WPI-MANA), National Institute for Materials Science,Tsukuba, Ibaraki 305-0044, Japan6Graduate School of Pure and Applied Sciences, University of Tsukuba, Tsukuba, Ibaraki 305-8571, Japan (Received 4 January 2024; revised 20 August 2024; accepted 23 August 2024; published 20 September 2024)Heusler compounds with semiconducting properties represent an important class of functional materials.Usually, research on these systems is guided by simple electron-counting rules, such as the Slater-Paulingprinciple. Here, we report on the discovery of Heusler-type semiconductors, significantly deviating fromthe Slater-Pauling rule. We theoretically predict the occurrence of nonmagnetic semiconducting groundstates in various highly off-stoichiometric full-Heusler alloys, where self-substitution leads to a band-gapopening. This unexpected trend is confirmed experimentally by thermoelectric transport measurements ona multitude of Fe2−2xV1−xAl1+3x samples with up to 20% substitution of Fe and V atoms. The band-gapopening leads to an exceptionally large Seebeck coefficient in p-type Fe2VAl thermoelectrics, previouslylimited by bipolar conduction and low-density-of-states effective mass. Consequently, our work presents aparadigm to tune the band gap of Heusler compounds by self-substitution and introduces a hitherto unex-plored class of semiconductors with exceptional thermoelectric properties, offering significant potentialfor advancements in energy science and sustainable-energy technologies.DOI: 10.1103/PRXEnergy.3.033006I. INTRODUCTIONHeusler compounds in the X2YZ (full-Heusler) and XYZ(half-Heusler) stoichiometry constitute one of the rich-est material classes [1], encompassing over 1000 differentintermetallics with diverse physical and electronic prop-erties, such as half-metallicity [2], nontrivial band topol-ogy [3], superconductivity [4], or semiconducting states.Recently, discoveries of a giant anomalous Nernst effectdue to a large variety of new topological phases havesparked further interest in these compounds [5]. Thus,potential applications of Heusler compounds reach fromthe field of spintronics [6] and magnetism [7] towardphotovoltaics [6] and thermoelectrics [8,9]. For energytechnologies such as photovoltaics and thermoelectrics, the*Contact author: Michael.parzer@tuwien.ac.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.highly tunable band gaps of half-Heusler compounds areparticularly advantageous, positioning them as some of themost promising materials for thermoelectric applications.In contrast, full-Heusler compounds typically have smallerband gaps, limiting their direct application in energytechnologies. Nevertheless, full-Heusler compounds, espe-cially Fe2VAl-based materials, continue to attract sig-nificant research interest due to their cost-effective con-stituents and favorable mechanical properties [10–17]. Thedevelopment of strategies to modify the electronic struc-ture of full-Heuslers is crucial for unlocking their fullpotential.Notably, the electronic structure of Heusler compoundsdepends mainly on the valence electron concentration peratom (VEC), which in turn can be used to effectively andreliably predict and tune the electronic as well as the mag-netic ground state [1]. For instance, Heusler compoundswith exactly six valence electrons per atom are typi-cally semiconductors with vanishing magnetic moment m,whereas m increases linearly for VEC ≶ 6 (see Fig. 1).This is subsumed in the well-known Slater-Pauling (S-P)rule, which was originally developed by Slater and Pauling2768-5608/24/3(3)/033006(17) 033006-1 Published by the American Physical Societyhttps://orcid.org/0000-0003-3509-7474https://orcid.org/0000-0002-0088-1755https://orcid.org/0000-0001-8223-9928https://orcid.org/0000-0003-0118-4238https://orcid.org/0000-0002-5450-4621https://orcid.org/0000-0002-3047-1239https://orcid.org/0000-0001-9428-5083https://orcid.org/0000-0003-2682-1846https://ror.org/04d836q62https://ror.org/03prydq77https://ror.org/03prydq77https://ror.org/04d836q62https://ror.org/026v1ze26https://ror.org/02956yf07https://crossmark.crossref.org/dialog/?doi=10.1103/PRXEnergy.3.033006&domain=pdf&date_stamp=2024-09-20http://dx.doi.org/10.1103/PRXEnergy.3.033006https://creativecommons.org/licenses/by/4.0/MICHAEL PARZER et al. PRX ENERGY 3, 033006 (2024)(a)(b)FIG. 1. Deviation from the Slater-Pauling (S-P) rule. (a) Thereported band gap Eg and (b) magnetic moment m of varioustypes of Heusler alloys as a function of the number of valenceelectrons per atom. The filled symbols depict theoretical val-ues, while the open symbols are values found experimentally.All compounds fall on a single line for the regular Heusler com-pounds in panels (a) and (b), which is commonly described bythe S-P rule. The strongly off-stoichiometric Heusler compoundspresented here are an exception and do not follow this universalrule for a higher concentration of the Z element. This has alsobeen confirmed experimentally for Fe2−2xV1−xAl1+3x, as shownby the open red symbols. The magnetic measurements depictedin (b) are presented in detail in Appendix C 1, while the refer-enced table with the depicted values for Eg and m extracted fromliterature is attached as Supplemental Table [18].to comprehend the electronic and magnetic properties ofsimple transition-metal alloys [19,20] and has later beensummarized for full-Heusler compounds by Galanakiset al. [2]. Exceptions to this rule are rare, with only spe-cial cases such as the eight-electron half-Heuslers (HHs)[21] or the RNiSb compounds with a VEC of ≥ 9 [22],deviating from the conventional S-P behavior. Heuslercompounds with a VEC close to 6 generally fulfill the S-P rule, regarding their electronic and magnetic structure.This fact is illustrated in Fig. 1, which shows the calcu-lated energy band gaps and magnetic moments of hundredsof Heusler-type compounds as a function of their VEC.It can be seen that VEC = 6 marks an insulating non-magnetic singularity in a vast sea of adjacent magneticmetals. A comprehensive list of all members of the Heuslerfamily in Fig. 1 that satisfy the S-P rule is in the Sup-plemental Table [23]. The relevant papers [24–87] havebeen collected employing conventional search engines aswell as natural-language-processing tools as implementedin DIMENSIONS.AI [88].In HHs with VEC = 6, yielding 18 electrons per for-mula unit (f.u.), the most electropositive element donatesits valence electrons, resulting in closed shells on all atoms,a finite band gap, and a zero magnetic moment, in agree-ment with the S-P rule. It follows naturally that changingthe number of electrons by chemical substitution woulddestabilize the compounds and would result in a shift ofthe Fermi level out of the gap. Hence, the majority of sta-ble HH compounds feature a valence electron count of sixper atom [89]. For full-Heusler materials, the outer d shellsof the X and Y elements are not fully filled. The band gapresults from the splitting of their d states into eg and t2gorbitals due to the crystal field of their surroundings. Thisleads to much smaller band gaps in this class of materi-als. In fact, for full-Heusler materials, band gaps have onlybeen reported in unstable or metastable compounds suchas Fe2NbAl [90] or Fe2TiSi [91]. The nature of the chemi-cal bonding in Heusler compounds rationalizes the validityof the S-P rule as a simple chemical electron-countingrule.Moreover, guided by the S-P rule, novel Heusler-typesemiconductors have been discovered in recent years.Some noteworthy examples with promising thermoelec-tric properties include so-called double half-Heuslers(X2YY′Z2), such as Ti2FeNiSb2 [92], three-quarter Heuslers(X1.5YZ), e.g., Ru1.5ZrSb, with 21 valence electrons [93],or off-stoichiometric variations such as Nb0.84CoSb. Inter-estingly, these vacancy-afflicted half-Heusler compoundswere first described as stoichiometric 19-valence electroncompounds [94,95].After thorough investigation, however, it was found thatNbCoSb crystallizes with vacancies as Nb0.8CoSb (VEC= 6) and has excess elemental Nb impurities [32,96,97].Therefore, crucially, all of these examples satisfy the S-Prule, as can be seen in Fig. 1.Here, we report on the discovery of strongly off-stoichiometric full-Heusler compounds, which form a bandgap and therefore show significant deviations from theS-P rule with respect to both their electronic and theirmagnetic properties [see the red symbols in Figs. 1(a)and 1(b)]. The commonly used presentation of the mag-netic moment per atom versus the VEC fails to followthe S-P relation for the X2−2xY1−xZ1+3x alloys, as noneof the compounds are magnetic. Moreover, when increas-ing x up to x = 0.11 (VEC = 5.65), a significant band-gapopening is observed, also contradicting the S-P rule with033006-2SEMICONDUCTING HEUSLER COMPOUNDS... PRX ENERGY 3, 033006 (2024)regard to their electronic structure. Full-Heusler semicon-ductors, although predicted theoretically, have not beenexperimentally realized so far as bulk materials as theyare either semimetallic, such as Fe2VAl and Fe2VGa[98], or metastable, such as Fe2NbAl [90] or Fe2TiSi[91]. Therefore, our findings present a propitious approachtoward achieving highly off-stoichiometric, yet stable, full-Heusler semiconductors. By combining efforts of density-functional-theory (DFT) calculations and thermoelectrictransport as well as magnetic experiments on a large num-ber of samples, we demonstrate that off-stoichiometricfull-Heusler compounds with a X2−2xY1−xZ1+3x stoichiom-etry exhibit semiconducting properties if the additional Zatoms are incorporated as antisite defects on the X and Ysites [see Fig. 2(a)]. In the following, the theoretical andexperimental evidence for this unexpected violation of theS-P rule will be presented.II. MATERIALS AND METHODSA. Computational detailsDFT calculations were conducted using thePerdew-Burke-Enzerhof (PBE) general-gradient approxi-mation as implemented in the Vienna ab initio simulationpackage (VASP) [99–104]. To simulate the disorder, 3 ×3 × 3 supercells with a total of 108 atoms were createdfor the substituted cases. For different substitutions, theappropriate number of X and Y atoms from the respectivelattice site were replaced with the Z atoms. The supercellswere then relaxed regarding their volume as well as atomicpositions, employing a 5 × 5 × 5 k-mesh and an energy-cutoff value of 450 eV. For the self-consistent calculations,a finer k mesh of 8 × 8 × 8 was used, to reduce the noise inthe electronic density of states (DOS). We used the tetra-hedron method with Blöchl corrections to set the partialoccupancies for each orbital.To extract the partial DOS of Al atoms sitting atdifferent lattice sites, i.e., AlFe, AlV, and AlAl, a dis-ambiguation of these atoms was implemented into thePOSCAR structure file. The Bader volumes extracted fromthe regular self-consistent calculation were approximatedto be spherical and the Wigner-Seitz radius of therespective atoms were adjusted accordingly. The element-decomposed DOS is depicted in Fig. 3(b), normalized withthe number of atoms per element in the unit cell.B. Synthesis and characterizationAll samples analyzed in this work were synthe-sized under an argon atmosphere using a high-frequencyinduction-melting furnace. The samples were weighedusing a high-precision scale and elements with a purity of99.99%, 99.95%, and 99.999% for Fe,V and Al, respec-tively. The mass loss was below 0.1% for all samples,yielding well-defined stoichiometry in our samples, whichwas also checked by X-ray fluorescence for selected sam-ples in an earlier work [105]. The XRD patterns were mea-sured using a commercial device from PANalytical andthe lattice parameters were extracted via Rietveld refine-ments as implemented in PowderCell (see Appendix B 1).Electron microscopy was performed using an FEI Quanta250 field-emission gun (FEG) scanning electron micro-scope (SEM) and an FEI TECNAI F20 FEG transmissionelectron microscope, as available at the USTEM researchfacility [106].C. Thermoelectric measurementsTransport measurements above room temperature wereconducted on bar-shaped samples using the ZEM-3 fromULVAC RIKO. Low-temperature Seebeck coefficientswere measured on the same samples using a home-madesetup inside a He-bath cryostat, employing the seesaw-heating technique [107] to cancel out interfering volt-ages, similar to other works [108,109]. Low-temperatureresistivity measurements were conducted using the four-point measurement within a He-bath cryostat. The low-temperature Seebeck and resistivity data were adjusted tothe high-temperature measurement by multiplying by aconstant factor in the range of 0.95 < x < 1.05 to accountfor the slight mismatch in absolute values.D. Analysis of transport dataThe temperature-dependent Seebeck-coefficient datawere analyzed using a three-parabolic band model withinthe framework of the Landauer theory. Assuming acous-tic phonon and alloy-disorder scattering, the dependenceof the energy on the carrier scattering time is given byτ(E, T) ∝ τ0(T)E−1/2, (1)where τ0 depends on the type of scattering process [110].As for the Seebeck coefficient, any energy-independentprefactors cancel out; hence S(T) of a single band can bedescribed by Fermi integrals of the first and zeroth order,as [10,110]S(T) = kBe[η − F1(η, T)F0(η, T)], (2)where η is the chemical potential, T is the temperature, eis the elementary charge, and Fn is the respective Fermiintegral:Fn(η, T) =∫ ∞0ξ ndξ1 + e(ξ−η). (3)Considering multiple bands, the total Seebeck coeffi-cient St can be calculated by adding up the single-band033006-3MICHAEL PARZER et al. PRX ENERGY 3, 033006 (2024)contributions weighted with their conductivity, usingSt = �nSnσn�nσn, (4)where σn ∝ 1/mn is inversely proportional to the respec-tive band mass mn. Assuming that, in total, three bandsare relevant for transport, this leads to five fitting param-eters, including the position of the Fermi level EF withrespect to the first band, the position of the other two bandswith respect to the first band, Eg2 and Eg3, and the rel-ative weighting parameters, σ2 and σ3, of the respectivebands. The position of the first band is set to 0 and therelative weighting parameter σ1 to −1, as the other valuesare relative to these and Eg1 and σ1 can be chosen arbitrar-ily without affecting the results. The closest distance of thebands at the band gap, either Eg1 or Eg2, denotes the effec-tive band gap of the compound extracted from the model,which is plotted in Fig. 5(c). The resulting fitting param-eters of all analyzed samples are given in Table S2 of theSupplemental Material [23].III. ELECTRONIC STRUCTURETo determine the electronic DOS of X1.78Y0.89Z1.33,depicted in Fig. 3, reasonably large supercells contain-ing 108 atoms, with six ZX and three ZY antisites, wereconstructed. For Fe2−2xV1−xAl1+3x, this was shown to bethe most favorable arrangement [111,112]. For comparisonwith experimental data, we put special emphasis on the cal-culation of this system. Calculations on small numbers ofdifferent antisites conducted in the past have found detri-mental effects of the substitutions on the electronic bandgap of full-Heusler compounds. [111–114] Nonetheless,for all considered full-Heusler compounds, we have founda band-gap opening if the Z atoms are distributed evenly onthe X and Y sites as antisites, even for large substitutions(see Fig. 1).For Fe1.78V0.89Al1.33, a total of 15 randomized anti-site configurations were calculated in this way and aver-aged, weighted with their formation energy, employing theBoltzmann average:DOSav = �15n=1DOSne (−Eform,n)/kBT)�15n=1e(−Eform,n)/kBT), (5)where Eform,n is the respective formation energy, DOSn isthe respective electronic DOS, T is the averaging temper-ature, and kB is the Boltzmann constant. The results fordifferent averaging temperatures T are depicted in Fig. 2,together with the formation energy as a function of theelectronic energy gap. The randomization of the antisitepositions was done using true random numbers employ-ing atmospheric noise [115]. The resulting DOSs werethen averaged using these Boltzmann weights to obtaina robust result (for details, see the Supplemental Mate-rial [23]), which is depicted in Fig. 3(a) and revealsstriking changes in the electronic structure with Al self-substitution. Notably, at the highest averaging temperature(11 000 K), the formation energies are largely disregarded,resulting in a more uniform averaging of the DOSs.For the other stoichiometries varying in Al concentra-tions, we started from the most favorable supercell ofFe1.78V0.89Al1.33 and added or removed antisite defects tocreate supercells for different x values. Again, different(c)(b)(a)FIG. 2. The analysis of the different antisite configurations of Fe1.78V0.89Al1.33. (a) A sketch of the full-Heusler crystal structure ofFe2VAl, with Al replacing Fe and V atoms on random lattice sites. (b) The Boltzmann-averaged DOS over all 15 supercells for differentaveraging temperatures. The energetically less favorable antisite configurations are only relevant for the highest averaging temperatureof 11 000 K. (c) The correlation of the extracted band gap with the configuration energy of the different supercells, normalized to themost favorable one: EN = Eform,n − Efav. Evidently, the more favorable antisite configurations exhibit larger band gaps. The linearregression acts as a guide to the eye.033006-4SEMICONDUCTING HEUSLER COMPOUNDS... PRX ENERGY 3, 033006 (2024)antisite configurations were tested to assess the influenceof the antisite positions on the electronic structure fordifferent x. The unfolded band structures for different stoi-chiometries ranging from x = 0 to x = 0.22 are displayedin Appendix A. For stoichiometric Fe2VAl, our calcula-tions reveal a profound pseudogap, aligning with recentliterature observations [9,14,116]. Notably, the introduc-tion of Al into the compound induces substantial changesin the DOS at the valence-band edge, accompanied by aband-gap opening of approximately 0.3 eV. As illustratedin the inset, the number of electrons per atom is reducedfrom 6 to 5.65, in agreement with the diminished numberof electrons in the valence band. One can clearly see thatthe Fermi level EF, however, remains in the energy gapfor the highly off-stoichiometric Fe1.78V0.89Al1.33. More-over, when testing the effect of different values of theon-site Coulomb interactions U − J , we have found onlynegligible changes in the electronic structure for the off-stoichiometric compounds.The partial DOS of Fe1.78V0.89Al1.33 depicted inFig. 3(b) reveals that, remarkably, aluminum atoms stillcontribute very few states to the DOS close to the Fermienergy, even for the compound with x = 0.11, for whichAl atoms make up one third of all atoms in the super-cell. Instead, the sharp rise in the DOS below EF isconstituted by resonant localized states, attributable toall constituent elements of the compound. Hence, thechanges in the electronic structure can be understood aschanges to the hybridization between the constituent ele-ments by the Al substitution, rather than as addition ofAl impurity states to the DOS. Out of the Al atoms,evidently, the AlFe antisites feature the largest contribu-tion to the DOS per atom around EF. To further elu-cidate the changes in the electronic structure caused bythe Z-element off-stoichiometry, we have calculated theunfolded band structure of the most stable supercells forseveral members of the Heusler family using BANDS4VASP[117,118]. The band structures of Fe1.78V0.89Al1.33 and(a)(c)(b)(d)FIG. 3. The electronic structure of strongly off-stochiometric full-Heusler compounds X2−2xY1−xZ1+3x. (a) A schematic of the full-Heusler crystal structure, with additional Z atoms occupying the X and Y lattice sites as antisite defects. (b) A comparison of theelectronic density of states (DOS) of pristine Fe2VAl and averaged Fe1.78V0.89Al1.33. A strong change in the DOS at the Fermi level EFis observable, together with the opening of an energy gap. The inset reports the integrated DOS in states per atom, showing a decreaseof the total number of states per atom in Fe1.78V0.89Al1.33 below EF. (c),(d) The unfolded band structure of two novel semiconductors,(c) Fe1.78V0.89Al1.33 and (d) Ru1.78Ta0.89Al1.33. The color code depicts the spectral function A(k, EF) obtained from the unfoldingcalculations. For better comparison, the band structure of the parent stoichiometric compounds is superimposed as a white solid line.033006-5MICHAEL PARZER et al. PRX ENERGY 3, 033006 (2024)Ru1.78Ta0.89Al1.33 are shown in Figs. 3(c) and 3(d), astwo representative examples. Compared to their stoichio-metric full-Heusler counterpart, a significant broadeningof the dispersive conduction band is observable, effec-tively shifting the edge toward higher energies. For bothcompounds, this leads to the opening of an indirect bandgap of the order of 0.3 eV, while EF remains pinnedat the valence-band edge within the band gap. Notably,this semiconducting ground state constitutes a signifi-cant deviation from the S-P behavior for full-Heuslermaterials.Hybridization-induced band gaps in transition-metalaluminides have been discussed by Weinert and Watson[119]. Besides RuAl2 and FeAl2, they have also inves-tigated Fe2VAl, tracing back the (pseudo) band gap tohybridization between the transition metals (TMs) and Alon the one hand and the lack of Al-TM d-d hybridizationon the other. In our off-stoichiometric Heusler compound,this effect is increased by the additional Al substitution onFe-V sites. As the AlV antisites have their d states at muchhigher energies than V, the V—V bonds are disturbed,leading to the observed broadening of the conduction bandat the X point [as evident from Fig. 3(c)]. Focusing on theFe states that dominate the valence band, the Al antisiteslead to a sizable loss of coordination between Fe atoms,penalizing the orbital hybridization. As a consequence, theFe states become more localized, giving rise to flat Febands, akin to resonant levels, at the � point. Similar con-siderations can be drawn for Ru1.78Ta0.89Al1.33 in Fig. 3(d)and for other full-Heuslers, such as Fe1.78V0.89Ga1.33 andFe1.78Nb0.89Al1.33, where the chemical bonding works in asimilar manner. We note that such flat bands at the Fermienergy are of great interest in fundamental solid-statephysics, as the small kinetic energy of charge carriers com-pared to the Coulomb interaction can give rise to exoticquantum phases of matter [120,121]. Indeed, we haveobserved peculiar anomalies in various low-temperaturetransport properties that will remain a subject for futurestudies.IV. EXPERIMENTAL RESULTSTo confirm the DFT calculations, we synthesized amultitude of Fe2−2xV1−xAl1+3x samples with the amountof Al substitution reaching from x = 0 to x = 0.2(Fe2VAl−Fe1.6V0.8Al1.6). In Fig. 4(a), we display thestructural properties of stoichiometric Fe2VAl in compar-ison to off-stoichiometric Fe1.78V0.89Al1.33. As evidencedby X-ray powder diffraction, both compounds crystallizein the L21 full-Heusler structure, as all peaks are clearlypresent. The evolution of the experimentally determinedlattice parameter with increasing x shows linear behav-ior, indicating full solubility along the transition. Theslight disparity in absolute values of the lattice parameterbetween DFT and experiment is a known anomalyin Fe2VAl, which has been reported in the literatureIntensity (arb. units)(a)(b)(c)FIG. 4. The structural analysis of Fe2−2xV1−xAl1+3x.(a) The X-ray diffraction (XRD) patterns of Fe2VAl andFe1.78V0.89Al1.33; all peaks of the L21 structure are clearlyvisible, without any impurity peaks. The inset on the rightshows the comparison of the experimental lattice parametersof Fe2−2xV1−xAl1+3x, extracted from Rietveld refinements,with the lattice parameters obtained from the DFT calculations.(b) Thr electron-dispersive X-ray spectroscopy (EDX) of thesample Fe1.78V0.89Al1.33. The scanning electron microscopy(SEM) image and the EDX mapping of the three elements Fe,V, and Al on the right exclude any impurity phases. The totalcomposition was determined to be Fe1.75V0.9Al1.35. Panel (c)depicts a TEM image at very large magnifications for the samesample, confirming that the main matrix of the compound issingle phase and homogeneous at the nanoscale.033006-6SEMICONDUCTING HEUSLER COMPOUNDS... PRX ENERGY 3, 033006 (2024)previously [98,111,122,123]. Moreover, the slope of theconcentration-dependent increase of the lattice parame-ter agrees remarkably well with that extracted from DFTcalculations, as depicted in the inset. The high toler-ance toward a large degree of antisite disorder in Fe2VAlis well known and likely results from the similar sizesof the atomic radii and the highly symmetrical cubiccrystal structure. For the Fe2−2xV1−xAl1+3x compounds,the stability is possibly increased further by the band-gap opening, as it separates bonding from antibondingstates. The composition and homogeneity of these sam-ples were confirmed by energy-dispersive X-ray spec-troscopy (EDX) and transmission electron microscopy(TEM), as shown in Figs. 4(b) and 4(c). The EDX mea-surements reveal that samples such as Fe1.78V0.89Al1.33maintain the intended stoichiometry without detectablemicroscale impurities. The TEM measurements furtherconfirm the absence of nanoscale impurities, aside fromminor V4Al4C precipitates accounting for less than 1%of the volume (see Appendix B). This high degree ofcompositional homogeneity, even at the nanoscale, is cru-cial for accurately assessing the thermoelectric propertiesand understanding the electronic structure for materialsoptimization.Narrow-gap semiconductors, such as those predictedtheoretically in Fig. 3, are of great interest for energyapplications such as thermoelectrics, which seamlesslyconvert heat into electricity and vice versa. It is well knownthat S(T) of narrow-gap semiconductors strongly dependson the band gap [128]. Therefore, we have studied thetemperature- and composition-dependent thermoelectricproperties of a large number of Fe2−2xV1−xAl1+3x samples.In Fig. 5, we show the experimental data for the Seebeckcoefficient as a function of the temperature and the VEC.Moreover, we compare the band gap extracted from theexperimental S(T) curves by employing a triple parabolicband model with the prediction from DFT calculations,assuming alloy disorder and acoustic phonon scattering.Inspecting the S(T) of Fe2−2xV1−xAl1+3x in Fig. 5(a), itcan be seen that the maximum Seebeck coefficient Smax isincreased significantly (by approximately 100%) for the Aloff-stoichiometry of x = 0.11.Simultaneously, the temperature of the maximum Tmaxis shifted toward higher values, from about 200 K forFe2VAl to about 400 K for Fe1.78V0.89Al1.33. These sys-tematic changes in S(T) are typical for an increase ofthe band gap. Indeed, by using the triple parabolic bandmodel to fit the experimental S(T) curves, a tiny bandgap, Eg ∼ 0.02 eV, is found for pristine Fe2VAl, but asizable Eg of about 0.3 eV is found for strongly off-stoichiometric Fe1.78V0.89Al1.33. These values are veryconsistent with those predicted theoretically by our ab ini-tio band-structure calculations [see Figs. 3(c) and 3(d) aswell as by Refs. [9,123]]. The main difference between theDFT and the experimentally derived electronic structures(a)(b)(c)FIG. 5. The experimental results for Fe2−2xV1−xAl1+3x. (a)The experimental data of the temperature-dependent Seebeckcoefficient for x = 0 and x = 0.11 in gray and green, respec-tively. The solid lines depict the three parabolic band modelfit, applied to assess the effective band structure close to EF assketched in the insets. The data and the fitting results for all sam-ples are shown in Appendix C 2. (b) The Pisarenko-style plot atT = 300 K reveals a significant deviation from the S-P behaviordemonstrated in doping studies [10], and for other full-Heuslers[124–127], when plotting raw Seebeck data versus the valenceelectron count per atom (VEC). (c) The composition-dependentbehavior of the band gap extracted from fitting experimental S(T)curves and DFT calculations (see Appendix A), respectively. Thetrend follows the raw Seebeck values at T = 300 K depictedin (b). For both, the band gap Eg increases with increasing Alconcentration up to a maximum at around x = 0.11 and VEC =5.65, before decreasing again for higher Al concentrations.033006-7MICHAEL PARZER et al. PRX ENERGY 3, 033006 (2024)is the different position of the Fermi level, with EF slightlydoped into the valence band for the latter. This can mostlikely be explained by a low concentration of VFe antisitedefects present in the samples, as rationalized by additionalcalculations depicted in Fig. S2D of the SupplementalMaterial [23].In Fig. 5(b), we show the room-temperature Seebeckcoefficient of Fe2−2xV1−xAl1+3x as a function of thevalence electron concentration per atom, compared to pre-vious doping studies in Fe2VAl-based compounds as wellas other members of the full-Heusler family. It can beseen that in contrast to the predicted vanishing Seebeckeffect for conventional doping, an anomalous increaseof the Seebeck coefficient is obtained at VEC ≈ 5.65,which corresponds to the off-stoichiometric compositionFe1.78V0.89Al1.33. Indeed, not only does the enhanced See-beck coefficient constitute a record among p-type full-Heusler compounds but it also signifies a severe deviationfrom the rigid-band-doping scenario as well as the S-P rule(red solid line). Notably, for p-type Fe2VAl-based com-pounds, increasing the band gap and therefore the Seebeckcoefficient is one of the key targets to improve thermo-electric performance and bring these compounds closerto application [9,129]. Previous p-type doping studiesin Fe2VAl thermoelectrics have achieved Seebeck coef-ficients up to 110 µV/K, which represents a significantbottleneck for achieving competitive TE performance withtheir n-type counterparts: a simple estimation of the maxi-mum figure of merit zT of Fe2VAl-based compounds witha Seebeck coefficient of 110 µV/K versus compoundswith Smax = 150 µV/K reveals a doubling of zTmax from0.64 to 1.27 [130] when neglecting the lattice thermalconductivity.This has motivated us to theoretically study the unfoldedband structures of off-stoichiometric Fe2−2xV1−xAl1+3x atdifferent Al concentrations to investigate the evolution ofthe band gap and the electronic structure near EF, withvarying stoichiometry. In Fig. 5(c), we show a compari-son of the band gaps from DFT calculations (Appendix A)with those obtained by modeling numerous experimentalS(T) curves of Fe2−2xV1−xAl1+3x, using a triple parabolicband model (for the complete set of temperature-dependentSeebeck-coefficient data, see Appendix C) [9,10,116]. Wefind a remarkable agreement not only in terms of absolutevalues but especially regarding the overall domelike shapeand pronounced peak at x ∼ 0.11. For higher Al substitu-tions, the flat impurity bands are shifted into the band gap,effectively reducing Eg . Moreover, for 0.05 < x < 0.17,the fitting model yields an additional flat valence bandclose to EF, which is reminiscent of the unfolded bandstructures of those compounds, depicted, e.g., in Fig. 3(c)and also in Appendix A. The band gaps extracted indepen-dently from the temperature-dependent resistivity applyingthe Arrhenius law also confirm this behavior, as shown inAppendix C 3.V. CONCLUSIONSIn summary, we have found an unexpected band-gap opening in off-stoichiometric Heusler compoundsX2−2xY1−xZ1+3x with x > 0, in contradiction to theS-P rule. The similarity of the electronic structuredetermined by independent experimental methods forFe2−2xV1−xAl1+3x corroborates our theoretical predictionsand constitutes solid evidence for the emergence of asemiconducting nonmagnetic ground state in these sys-tems. This not only opens a new avenue for tuning theelectronic structure of full-Heusler compounds but alsobrings Fe2VAl-based compounds closer to practical ther-moelectric applications due to the substantial increase inS(T). Compared with the existing literature on p-typefull-Heusler compounds, the Seebeck values reported hereconstitute record-high values, exceeding those of existingmaterials by over 20%, without targeted material opti-mization. Therefore, we conclude that the ultrahigh off-stoichiometry with respect to the Z element constitutesa general approach to achieve stable full-Heusler semi-conductors with promising thermoelectric properties. Thisapproach significantly extends the phase space of ternaryHeusler compounds as functional materials and could leadto the discovery of multiple so-far-unexplored systemswith intriguing physical properties, helping to satisfy theneed for optimized electronic materials in the field ofenergy sciences. Not only is it remarkable that semicon-ducting states are found beyond the S-P principle but it isalso unique that such types of electronic band structurescan be found in heavily disordered structures.ACKNOWLEDGMENTSThe computational results presented have been achievedusing the Vienna Scientific Cluster (VSC). We acknowl-edge the TU Wien X-Ray Center (XRC) for the measure-ment time for the structural analysis of our samples. Weacknowledge the University Service Facility for Transmis-sion Electron Microscopy (USTEM) facility at TU Wienfor the electron-microscopy measurements. Funding camefrom the Japan Science and Technology Agency (JST)program MIRAI, via Grant No. JPMJMI19A1.M.P. and F.G. conceptualized the work and planned theoutline of the draft. M.R., R.P., and M.P. conceptualizedthe theoretical study and interpreted and discussed the DFTresults. M.P. did the DFT calculations. M.P. synthesizedthe samples and did the experiments. M.P., F.G., A.R.,A.P., E.C., and E.B. interpreted and discussed the experi-mental results. M. S.-P. did the TEM measurements as wellas the interpretation of the data. E.B. and T.M. supervisedthe work and organized the funding. All authors discussedthe work and modified the manuscript.The authors declare that they have no competinginterests.033006-8SEMICONDUCTING HEUSLER COMPOUNDS... PRX ENERGY 3, 033006 (2024)APPENDIX A: DFT RESULTS1. Different stoichiometriesTo study the impact of varying Al concentrations inFe2−2xV1−xAl1+3x, different stoichiometries were analyzedusing DFT calculations. The electronic band structures ofall calculated compositions ranging from x = 0 to x =0.22 are plotted in Fig. 6. The supercells were created onthe basis of the most stable antisite configuration (AC) ofFe1.78V0.89Al1.33, randomly adding or removing the respec-tive amount of substitutional Al antisites. For the differentAl concentrations, again different antisite configurationswere tested and the stability of the DOSs against changesin the ACs was analyzed. As an example, the stabilityof the electronic DOS for different antisite configura-tions of Fe1.7V0.85Al1.45 is demonstrated for the respectiveband structures of three different ACs in Fig. S1C of theSupplemental Material [23].APPENDIX B: STRUCTURALCHARACTERIZATION1. X-ray diffractionThe lattice parameters depicted in Fig. 4(a) wereextracted from X-ray diffraction using Rietveld refine-ment as implemented in the crystallographic softwarePowderCell. The fit of the model on the raw data ofFe1.78V0.89Al1.33 is shown as an example in Fig. 7.Although the absolute peak heights and widths of full-Heusler materials can be affected by grinding during prepa-ration for powder diffraction [131], the refinement showsexcellent agreement.2. Transmission electron microscopyTo ensure full homogeneity also at medium scales, theFe1.78V0.89Al1.33 sample was also analyzed using TEMwith lower magnifications. At the lower microscale, pre-cipitates of V4Al4C can be detected (see Fig. 8). Theirvolume fraction is far less than 1% and thus their influenceon the material properties is negligible. Furthermore, theenergy loss by channeled electrons (ELCE) method wasemployed to probe the distribution of the Al antisites onthe Fe and V sites, respectively. Using electron energy-lossspectroscopy (EELS), the Bloch-wave symmetry can beexploited to obtain site-selective information on the con-stituent elements. The different EELS spectra reveal anapparent trend of Al substitution at the Fe and V sites (seeFig. 9).APPENDIX C: ADDITIONAL MEASUREMENTDATA1. Magnetization measurementsTo measure the magnetization, cubic pieces of about 50mg were cut out of the samples and mounted on a brasssample holder to be used for the vibrating-sample magne-tometer (VSM). The commercially available VSM insertfor the physical-properties measurement system (PPMS)from Quantum Design was used to conduct the mea-surement. We used a vibration frequency of 20 Hz toachieve a good signal-to-noise ratio. First, a field sweepat room temperature from −9 to 9 T was measured. Thetemperature-dependent magnetization was then measuredfrom 300 to 2 K, with a cooling rate of 10 K/min, applyinga field of 0.1 T. Furthermore, magnetic field sweeps weredone from −9 to 9 T at 20 and 2 K with a sweeping rate of0.01 T/s, interrupting the cool-down. Another temperature-dependent sweep was done while warming up from 1.9 to300 K, with a rate of 3 K/min and a magnetic field of 1 T.As shown in Fig. 1, the synthesized Fe2−2xV1−xAl1+3xsamples in this work do not follow the S-P prediction oftheir electronic structure regarding the band gap as wellas the magnetic properties. While the S-P rule predicts amagnetic moment ofM = NV − 24 µB/f.u. , (C1)the semiconducting band structure with fully filled Fevalence bands and unfilled V-dominated conduction bandsprecludes spin polarization. Indeed, spin-polarized cal-culations were performed and showed no discrepancybetween majority and minority spins in the DOS. The mag-netic properties of pristine Fe2VAl found in the literaturevary strongly because of different synthesis methods andannealing conditions. It is generally established as a non-magnetic semimetal, containing magnetic antisite defects.For an FeV antisite pair, it has been shown experimentallyas well as theoretically that it exhibits a magnetic momentof 3.7 μB. These impurities can form magnetic clusters andlead to superparamagnetic behavior in the experimentalcompound [132].In this work, Fe and V atoms are actively replaced byAl, reducing the valence electron concentration from 24 e−per f.u. down to values as low as 21.6 e− per f.u. Followingthe S-P rule [Eq. (C1)], this should lead to strong magneticmoments of 1.3 and 2.4 µB per f.u. for x = 0.11 and x =0.2, respectively.However, the temperature-dependent magnetizationshown in Fig. 10(a), measured at H = 1 T, reveals verylow absolute values and apparent paramagnetic behavior,with the magnetization M rising in a 1/T manner towardlow temperatures. As the absolute values of the magneti-zation are very low, with around 0.006 µB per f.u. at 1 Tat 1.9 K, it is certain that the bulk of the sample is not fer-romagnetic, demonstrating the violation of the S-P rule inthese compounds.Moreover, the absolute magnetization drops notablywith increasing Al substitution, compared to the pristinematerial, as illustrated in the inset of Fig. 10(a). Thisbehavior indicates a reduction of magnetic Fe antisite033006-9MICHAEL PARZER et al. PRX ENERGY 3, 033006 (2024)1.01.50.5–0.5–1.0–1.50.01.01.50.5–0.5–1.0–1.50.01.01.50.5–0.5–1.0–1.50.01.01.50.5–0.5–1.0–1.50.01.01.50.5–0.5–1.0–1.50.01.01.50.5–0.5–1.0–1.50.0(a) (d)(b) (e)(c) (f)FIG. 6. The band structures of Fe2−2xV1−xAl1+3x for different Al concentrations. (a)–(f) The spectral functions A(k, EF) extractedfrom the unfolded band-structure calculations of Fe2−2xV1−xAl1+3x for x = 0.04, 0.07, 0.11, 0.15, 0.19 and 0.22. For x ≤ 0.11, a cleartransition from a pseudogap system to a semiconductor can be observed, while for higher x, the impurity bands are shifted into thegap, effectively reducing it. In general, the edge of the V-dominated band at the X point of the Brillouin zone is shifted toward higherenergies and broadens with increasing Al substitution. For very high antisite concentrations x, the band structure becomes decoherent,as the periodicity of the lattice is heavily disturbed.defects by increasing amounts of excess Al atoms, whichcan be made plausible by the excess of Al antisites sup-pressing other defects.The absence of ferromagnetism in the off-stoichiometricFe2−2xV1−xAl1+3x compounds is highlighted through acomparison with Mn2−yCoyAl full-Heusler compounds,which adhere to the S-P rule. As shown in Fig. 10(b),this comparison underscores the substantial difference inmagnetization between the two material systems.2. Analysis of the Seebeck coefficientThe temperature-dependent Seebeck-coefficient datawere fitted using a three parabolic band model as describedin Sec. II. The raw experimental data are shown in Fig. 11as the colored points. The respective curves resulting fromthe fits are plotted as solid lines in the same color. Theextracted fit parameters, which lead to the plotted theoret-ical curves, are presented in Table S2 of the SupplementalMaterial [23]. For completeness, the simple analysis with033006-10SEMICONDUCTING HEUSLER COMPOUNDS... PRX ENERGY 3, 033006 (2024)FIG. 7. The Rietveld refinement of the XRD pattern. An exam-ple of Rietveld refinement shown for Fe1.78V0.89Al1.33, withnormalized counts over the diffraction angle 2θ . The experimen-tal XRD pattern is depicted as the black line, while the fittedcrystal structure is depicted by the solid red line. The differencebetween experiment and theory is plotted as the green solid lineat the bottom. From the refinement, the lattice parameter of thesample has been determined to a = 5.81 Å.the Goldsmid-Sharp (GS) formula for estimating the bandgap in semiconductors was conducted. For a measurementcurve with the maximum Seebeck Smax and the respectivetemperature at the maximum Tmax, the band gap Eg of thesample can be estimated as [128]Eg ≈ 2e|Smax|Tmax. (C2)Applying this formula yields a similar behavior for thecomposition-dependent evolution of the band gap for theFIG. 8. A TEM image of Fe1.78V0.89Al1.33. Small precipitatesof V4Al4C are observable at medium magnifications. They arelikely caused by small amounts of carbon contained in the rawelements. Their volume fraction has been estimated to be smallerthan 1%.(a)(b)FIG. 9. The EELS analysis of the site occupancies of the Fe,V,and Al atoms. (a) The different momenta of the channeled elec-tron beam used to analyze the site occupancy in the sample.We measured between the 000 symmetry reflection and the 220symmetry of the crystal. (b) An analysis of the energy lossby channeled electrons in Fe1.78V0.89Al1.33. A similar trend ofV-Fe and Al-Fe site occupation for the angular-dependent elec-tron energy-loss spectra is observable, indicating an equal distri-bution of Al antisites on the respective lattice sites.Fe2−2xV1−xAl1+3x sample series, with a maximum evolv-ing around x = 0.11 [see Fig. 12(d)]. However, the abso-lute values of Eg are significantly underestimated by afactor of around 2 for the system presented here. It is wellknown that the GS formula can lead to erroneous resultsfor small-gap systems [133]. This is mainly due to theassumption of equal effective masses of the relevant bandsm2/m1 = 1, which is not the case in this study. Especiallyfor majority carriers with higher effective mass, this canlead to an underestimation of the band gap Eg . Therefore,this formula gives an easy but only crude estimation for theband gap. For the Fe2−2xV1−xAl1+3x compounds, there is alarge discrepancy between the band masses of the valenceand conduction bands. Hence, the parabolic band model isa better tool for estimating the electronic structure of thissystem, as also discussed in the main text.3. Analysis of electrical resistivityTo corroborate the experimental results in the maintext, the electrical resistivities of selected samples ofFe2−2xV1−xAl1+3x with x ranging from 0 tp 0.2 were033006-11MICHAEL PARZER et al. PRX ENERGY 3, 033006 (2024)(a)(c)(b)(d)FIG. 12. An analysis of the electrical resistivity of the Fe2−2xV1−xAl1+3x compounds. (a) The temperature-dependent electricalresistivity ρ(T) for different x in the range 0 ≤ x ≤ 0.2. At low temperatures, the resistivity behavior is reminiscent of bad metals,while at high temperatures, the resistivity drops due to the contribution of the conduction band. (b) The Arrhenius plot, showcasingactivated behavior across an energy gap at high temperatures, by plotting lnρ(T) versus 1/T. For x = 0 and x = 0.11; as an example,the fit is drawn as a straight solid line. (c) A comparison of the extracted band gaps from the DFT calculations, the analysis of S(T),and the analysis of ρ(T). The band gap obtained from the Arrhenius relation has been adjusted by substracting the energy differenceof the Fermi level EF to the valence-band edge from the extracted transport gap, as these samples are slightly doped. (d) The extractedband gap from the temperature-dependent Seebeck-coefficient data using the simple Goldsmid-Sharp formula [128].analyzed regarding their temperature-dependent behav-ior at high temperatures, where thermal activation acrossthe band gap leads to a semiconductinglike behav-ior (dρ/dT < 0). The measurement data, obtained fromRef. [105], are depicted in Fig. 12(a). To keep the plot man-ageable, only specific compositions along the transitionfrom x = 0 − 0.2 have been plotted. Apart from Fe2VAl(x = 0), all samples show metallic resistivity behavior atlow temperatures, as the Fermi level is intrinsically dopedinside the valence band due to FeV exchange antisitedefects. As discussed in Ref. [104], the residual resis-tivity ratio (RRR) of the metallic behavior is very lowand decreases with increasing x. At high temperatures, theconduction band becomes relevant for electronic transport,leading to a significant decrease of the resistivity, which isalso reflected in the Seebeck data at similar temperatures(see Fig. 11). This activated behavior can be fitted with theArrhenius relation, to obtain an estimate for the effectivetransport gap from the electrical-resistivity data:ρ(T) ≈ AeEg/2kBT, (C3)where kB is the Boltzmann constant and Eg and A are thefitting parameters. To visualize this relation, in Fig. 12(b)we show ln(ρ(T)) vs 1/T. Two fits for x = 0 and x = 0.11,which exhibit significant transport gaps, are depicted as033006-12SEMICONDUCTING HEUSLER COMPOUNDS... PRX ENERGY 3, 033006 (2024)(b)(a)FIG. 10. The magnetic properties of the Fe2−2xV1−xAl1+3xsamples. (a) The temperature-dependent magnetization for differ-ent Al concentrations x at a field of B = 1 T. The inset highlightsthe decrease of magnetization with rising Al at low temperatures.(b) A comparison of the magnetization of the Fe2−2xV1−xAl1+3xcompounds with the full-Heusler system Mn2−yCoyVAl [38],obeying the S-P rule. A clear distinction can be made, as the mag-netization values of the off-stoichiometric Heuslers presentedhere are magnitudes lower, ruling out ferromagnetic ordering inthese samples.representative examples. To compare the transport gapsobtained from the Arrhenius relation with the band gap,we have subtracted it with the position of the Fermi levelwith respect to the valence-band edge, obtained from theSeebeck fit. While it is a rough approximation, as onlyfew measurement points have been taken at the relevanttemperatures, the results shown in Fig. 12(c) are quite con-vincing. Again, the absolute values for the band gap Egobtained from fitting ρ(T) are in agreement with the valuesobtained from fitting the temperature-dependent Seebeck,as well as the values obtained from our DFT calculations.While the composition dependence shows some devia-tions, a clear maximum of Eg evolves around the com-position of x = 0.11. This is further confirmation of theFIG. 11. The Seebeck coefficient of the measuredFe2−2xV1−xAl1+3x samples with the respective fits. Thetemperature-dependent thermopower data of all measuredsamples of Fe2−2xV1−xAl1+3x. The points depict the measuredvalues, while the solid lines represent the curves of the fittingprocedure to approximate the effective band structure. For everyS(T) curve, the fit is very close to the experiment. 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Wu, Elec-tronic structure, magnetism, and transport properties of theHeusler alloy Fe2CrAl, J. Magn. Magn. Mater. 283, 409(2004).033006-17https://doi.org/10.1063/1.4905922https://doi.org/10.1007/s10853-016-0300-2https://doi.org/10.1038/s41598-018-37740-yhttps://doi.org/10.1016/j.intermet.2017.09.012https://doi.org/10.1016/j.jmmm.2004.06.013 I.. INTRODUCTION II.. MATERIALS AND METHODS A.. Computational details B.. Synthesis and characterization C.. Thermoelectric measurements D.. Analysis of transport data III.. ELECTRONIC STRUCTURE IV.. EXPERIMENTAL RESULTS V.. CONCLUSIONS . ACKNOWLEDGMENTS . APPENDIX A: DFT RESULTS 1.. Different stoichiometries . APPENDIX B: STRUCTURAL CHARACTERIZATION 1.. X-ray diffraction 2.. Transmission electron microscopy . APPENDIX C: ADDITIONAL MEASUREMENT DATA 1.. Magnetization measurements 2.. Analysis of the Seebeck coefficient 3.. Analysis of electrical resistivity . 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