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M. Parzer, F. Garmroudi, A. Riss, [T. Mori](https://orcid.org/0000-0003-2682-1846), A. Pustogow, E. Bauer

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[Mapping Delocalization of Impurity Bands across Archetypal Mott-Anderson Transition](https://mdr.nims.go.jp/datasets/e8e1d6d6-fd25-47a2-810f-7ea9f429da56)

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Mapping Delocalization of Impurity Bands across Archetypal Mott-Anderson TransitionMapping Delocalization of Impurity Bands across Archetypal Mott-Anderson TransitionM. Parzer ,1,* F. Garmroudi ,2,† A. Riss,1 T. Mori ,3,4 A. Pustogow ,1 and E. Bauer11Institute of Solid State Physics, Technische Universität Wien, 1040 Vienna, Austria2Materials Physics Applications - Quantum, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA3International Center for Materials Nanoarchitectonics (WPI-MANA), National Institute for Materials Science (NIMS), Tsukuba, Japan4Graduate School of Pure and Applied Science, University of Tsukuba, Tsukuba, Japan(Received 16 November 2024; accepted 9 July 2025; published 6 August 2025)Tailoring charge transport in solids on demand is the overarching goal of condensed-matter research as itis crucial for electronic applications. Yet, often the proper tuning knob is missing and extrinsic factors suchas impurities and disorder impede coherent conduction. Here, we control the very buildup of an electronicband from impurity states within the pseudogap of ternary Fe2-xV1þxAl Heusler compounds via reducing theFe content. Our density-functional theory calculations combined with specific heat and electrical resistivityexperiments reveal that, initially, these states are Anderson-localized at low V concentrations 0 < x < 0.1.As x increases, we monitor the formation of mobility edges upon the archetypal Mott-Anderson transitionand map the increasing bandwidth of conducting states by thermoelectric measurements. Ultimately,delocalization of charge carriers in fully disordered V3Al results in a resistivity exactly at the Mott-Ioffe-Regel limit that is perfectly temperature-independent up to 700 K—more constant than constantan.DOI: 10.1103/fz9j-bj87Introduction—The periodic arrangement of atoms incrystals forms energy bands, where electrons behave asextended Bloch waves. In single atoms, however, electronsoccupy localized energy levels. It was Sir Nevill Mott whofirst approached the fundamental question in condensed-matter physics of band formation, transitioning from local-ized insulating states to delocalized metallic states. Heargued that as two atoms move closer, at a certain point,a critical threshold is reached where the screening of ions byneighboring electrons becomes strong enough to delocalizethem [1].In doped semiconductors, a similar, but more realistic,transition occurs. At low doping levels, electrons remaintrapped at impurity sites, but increasing dopant concen-tration leads to a critical concentration, xc, where aninsulator-metal phase transition occurs [1,2]. Additionally,correlations among the electrons can localize and split thenarrow impurity bands (IBs) by an energy gap Δ ¼ U −W,with U being the Coulomb interaction and W the bandwidth. In parallel to Mott’s work, Anderson developed atheory of disorder-induced localization of electrons inmatter due to their wavelike nature [3]. In its essence,Anderson localization describes the absence of diffusion ofwavelike objects due to the multiple scattering events andquantum interference of self-intersecting scattering paths[4–8]. In low-doped semiconductors, both Mott andAnderson localization—the latter arising from the inher-ently random distribution of impurity atoms on the orderedcrystal lattice—are believed to contribute to the insulatingnature of the IB [1,9]. Figure 1 illustrates both theselocalization mechanisms and the way in which delocaliza-tion has been theoretically predicted to take place: at thecritical concentration xc, two mobility edges Ec emerge inthe center of the IB. These critical energies separatelocalized states in the band tails from delocalized ones inthe center [10].The delocalization of IBs and their effect on electronictransport remain an important question, given the wide-spread use of dopant atoms with in-gap impurity states tocontrol the physical properties of semiconductors in variouselectronic devices [11–14]. However, up until now, exten-sive investigation of the Mott-Anderson transition (MAT) inIBs has frequently been hindered by the low solubility ofdopant atoms and the fact that the IBs usually hybridize withthe bulk conduction bands. This has hindered the unam-biguous analysis of the MAT in IBs, resulting, for instance,in the so-called exponent puzzle for the critical exponentdescribing the universality of the transition [15,16].Here, we successfully map the delocalization of IBs insemiconducting Fe2−xV1þxAl Heusler compounds, high-lighting the impact of disorder on the broadening of theimpurity band. This system offers exceptional tunability ofthe chemical composition (−1 ≤ x ≤ 2) within a similar*Contact author: michael_parzer@yahoo.de†Contact author: f.garmroudi@gmx.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 LETTERS 135, 066302 (2025)0031-9007=25=135(6)=066302(7) 066302-1 Published by the American Physical Societyhttps://orcid.org/0000-0003-3509-7474https://orcid.org/0000-0002-0088-1755https://orcid.org/0000-0003-2682-1846https://orcid.org/0000-0001-9428-5083https://ror.org/04d836q62https://ror.org/01e41cf67https://ror.org/026v1ze26https://ror.org/02956yf07https://crossmark.crossref.org/dialog/?doi=10.1103/fz9j-bj87&domain=pdf&date_stamp=2025-08-06https://doi.org/10.1103/fz9j-bj87https://doi.org/10.1103/fz9j-bj87https://creativecommons.org/licenses/by/4.0/https://creativecommons.org/licenses/by/4.0/cubic structure, ranging from Fe3Al with the partiallydisordered D03 structure up to V3Al with the fully dis-ordered A2 structure [see Fig. 2(a)]. Crucially, the IB ariseswithin the center of the gap and its bandwidth remainsremarkably narrow up to large concentrations of x (≈0.2 eVat x ¼ 0.074). This confines the electronic transport at lowtemperatures to the IB states, allowing one to map the MATfrom the fully insulating to the disordered and correlatedmetal regime. Recently, we studied electronic transport instoichiometric Fe2VAl, where a MAT arises from thermallyquenched antisite disorder and boosts thermoelectric per-formance [13]. However, the concentration of defects,which can be induced via thermal quenching, is limitedby rapid ordering during the quenching process and thefrozen disordered structures comprise a combination ofvarious different defects, promoting hybridization with thebulk bands and turning the system immediately metallic.Therefore, composition tuning in Fe2−xV1þxAl across thewhole range, −1 ≤ x ≤ 2, presents a much more controlledway of probing charge transport across the MAT.The Heusler compound Fe2VAl, crystallizing in the fullyordered L21 structure, has long been known as a promisingthermoelectric material for near-ambient temperatureapplications [18–21], owing to its narrow pseudogap inthe electronic structure. Over the recent years, varioussubstitution studies have been performed to optimize theposition of the Fermi energy and to tailor the band structureitself, resulting in outstanding thermoelectric power factorsthat rival or exceed those of state-of-the-art Bi2Te3 thermo-electrics [22–25].A highly unconventional doping dependence of thethermoelectric properties has been reported for off-stoichio-metric, self-substituted Fe2−xV1þxAl [26,27]. Figure S1 inSupplemental Material (SM) [28] compares the Seebeckcoefficient S as a function of the valence electron concen-tration per atom (VEC) for Fe2−xV1þxAl and varioussubstituted Fe2VAl-based compounds with different sub-stitution elements. Typically, for VEC < 6 (correspondingto p-type doping) S is expected to be positive, and forVEC > 6 (n-type doping) S < 0 is expected. This trend isindeed observed in all known substituted compounds exceptfor Fe2−xV1þxAl, where an entirely opposite behavioremerges. Moreover, Naka et al. reported a composition-induced metal-insulator quantum phase transition in V-richFe2−xV1þxAl and a ferromagnetic quantum critical point inFe-rich Fe2−xV1þxAl [17,54], yet the exact mechanismsbehind these transitions remain unclear. Here, we demon-strate that the origin of these quantum phase transitions andthe highly unconventional doping behavior can be tracedback to the delocalization of narrow IBs right next to EF.Crucially, using temperature-dependent thermoelectrictransport measurements we quantitatively map the widthof the delocalized IB as a function of the impurityconcentration.To obtain insight into the states around EF relevantfor charge transport, we performed density-functionaltheory (DFT) calculations on supercells of substitutedFe2−xV1þxAl with varying concentration of defects.Figure 2(b) displays the density of states (DOS) aroundEF for x ¼ 0, 0.037, 0.074. It is evident that forx ¼ 0 → 0.037, narrow and highly localized impurity statesemerge within the pseudogap at the lower edge of thedispersive conduction band (CB). As the concentration of Vantisites on the two Fe sublattices continues to increase,these states gradually broaden, eventually filling the entirepseudogap and transforming the system into a metal.However, the periodic boundary conditions (Bloch’s theo-rem) imposed by the supercell approach and the lack ofaccounting for the decoherence of the wave functionsprevent an accurate description of the Anderson-localizednature of these states. Thus, simple DFT calculations canonly predict that there exist impurity states but not whetherand to which extent they are localized.On the other hand, examining composition- and temper-ature-dependent electronic transport of Fe2−xV1þxAl,displayed in Figs. 2(c)–2(e), reveals clear evidence forthe Anderson-localized nature of these IBs as the low-temperature conductivity drops by several orders ofFIG. 1. Sketch of the Mott-Anderson transition. (a) Mott-typeand Anderson-type charge localization mechanisms in solids.(b) Semiconducting matrix with isolated impurities and (c) acorresponding localized IB. (d) Atomic potentials, hosting ex-tended Bloch states. (e) Semiconducting matrix with percolationof impurities, and (f) a partly delocalized IB.PHYSICAL REVIEW LETTERS 135, 066302 (2025)066302-2magnitude and exhibits a sharp minimum at x ≈ 0.1 [seeFig. 2(c)]. The inset in Fig. 2(c) compares the composi-tion-dependent DOS at the Fermi level DðEFÞ obtainedfrom DFT and specific heat data (C=T) [17] with theconductivity at T ¼ 2 K. While, both DFT calculationsand C=T data reveal a peak at x ¼ 0.1 associated with theIBs, σ2 K vanishes and only starts to increase at x > 0.1when the impurities start to form a “real,” delocal-ized band.Temperature-dependent resistivity curves for severaldifferent compositions across a wide range of temperaturesand compositional phase space are presented in Fig. 2(d).Focusing first on the end compounds, the magnetic metalFe3Al exhibits a metalliclike resistivity, dρ=dT > 0, welldescribed by the Bloch-Wilson law for T < TC (see SM[28]), with TC ≈ 740 K. On the other hand, V3Al with thefully disordered A2 structure (see Ref. [55]) shows anegligible, negative temperature dependence of ρðTÞ witha residual resistivity ratio of ≈0.98, aligning with thephenomenological Mooij criterion [29–31] (see alsoFig. S2 in SM [28]). Note that ρðTÞ of Fe3Al saturatestoward the value of V3AI, which lies almost exactly at theMott-Ioffe-Regel (MIR) limit (red dotted line), even in anonlogarithmic plot [inset Fig. 2(d)]. The MIR limit statesthat the charge carrier mean free path is limited by theinteratomic distance kFlmin ∼ 2π, where kF is the Fermiwave vector and lmin the minimal mean free path. Forspherical Fermi surfaces, Calandra and GunnarssonFIG. 2. Structure and electronic transport of Fe2−xV1þxAl system. (a) Crystal structures of ternary Heusler compound Fe2VAIcrystallizing in the fully ordered L21 structure, binary Fe3Al in the partly disordered D03 structures, and V3Al in the fully disordered A2structure. (b) Densities of states of Fe2−xV1þxAl from DFT supercell calculations with x ¼ 0, 0.037, 0.074. (c) Composition-dependentelectrical conductivity at T ¼ 2 K of Fe2−xV1þxAl over the whole compositional phase space, from −1 ≤ x ≤ 2. A distinct minimum isobserved at x ¼ 0.1. Inset shows a magnified view of the conductivity minimum alongside the density of states at the Fermi level derivedfrom specific heat data [17] and our DFT calculations, both showing a maximum instead. (d) Temperature-dependent resistivity ρðTÞ ofFe2−xV1þxAl close to the significant minimum (x ¼ 0.1–0.5), together with the two end points of the composition transition. Notably,Fe3Al shows metallic behavior, while the resistivity of V3AI is completely flat, with a residual resistivity ratio of ≈0.98, which is right atthe Mott-Ioffe-Regel (MIR) limit (dotted red line), to which Fe3Al converges at high temperatures. The inset shows ρðTÞ of the binarycompounds on linear scale. (e) Signatures of Mott-Anderson charge localization in ρðTÞ of Fe1.9V1.1Al, measured down to 0.39 K. Anextremely sharp upturn of ρðTÞ takes place below T ≈ 20 K, consistent in with Mott variable range hopping (inset) in a broad range oftemperatures.PHYSICAL REVIEW LETTERS 135, 066302 (2025)066302-3derived an expression [56,57],ρMIR ¼ 3π2ℏe2k2Flmin: ð1ÞAssuming kF ≈ π=a and using the experimental latticeparameter of V3Al, this yields ρMIR ≈ 187 μΩ cm [reddotted line in Fig. 2(d)], which almost perfectly coincideswith the temperature-independent resistivity value of V3AlρðTÞ ≈ const ≈ 180 μΩ cm up to 700 K. Notably, ρðTÞ ofthis fully disordered metal is more constant than inconstantan, with ρ300 K=ρ4 K ≈ 0.98 and even approachingits low-temperature resistivity above room tempera-ture, ρ700 K=ρ4 K ≈ 0.997.In the ternary system Fe2VAl, a deep pseudogap developsaround the Fermi energy, which leads to a semiconductor-like resistivity, dρðTÞ=dT, that can be mainly attributed tothe thermal activation of charge carriers across the narrowpseudogap [58,59], although recently it has been shown thatcharge carrier scattering may also play an important roleowing to residual intrinsic antisite exchange defects [60].Notably, ρðTÞ increases further by more than an order ofmagnitude when additional V atoms are substituted on theFe sites in Fe1.9V1.1Al, with a pronounced upturn at thelowest temperatures. Figure 2(e) shows a linear plot of ρðTÞof Fe1.9V1.1Al with measurements down to T ¼ 0.39 K.The extremely sharp increase of ρðTÞ below ≈20 K isconsistent with the presence of Anderson-localized statesaround EF [see inset, Fig. 2(e)], although at the lowesttemperatures, T ≲ 1.2 K, ρðTÞ deviates significantly fromvariable range hopping behavior and can instead bedescribed phenomenologically by a simple power lawρðTÞ ∝ T−α, with α ¼ 1.3–1.4, consistent with what hasbeen reported by Naka et al. [17] and also, e.g., in dopedsilicon near the MAT [61–63].At low temperatures, when thermally activated hoppingof electrons from one localized impurity to the next, whichare far apart in real space but close in energy, dominates, atemperature dependence ρðTÞ ∝ exp½ðT0=TÞ1=4� is expectedin three dimensions—commonly known as “Mott variablerange hopping.” Here, T0 is the characteristic temperature,which depends inversely on the localization lengthξL ¼�118DðEFÞkBT0�−1=3: ð2ÞThe localization length indicates the exponential decay ofthe wave function jΨðrÞj2 ∼ exp ð−jr − r0j=ξLÞ at an impu-rity site r0. By fitting the slope of ln ρ versus T−1=4 ofFe1.9V1.1Al at low temperatures, T0 is obtained as ≈490 K.Furthermore, using DðEFÞ, derived from C=T data fromRef. [17], leads to a localization length ξL ≈ 12 nm, equal-ing about 21 unit cells. If one considers that for Fe1.9V1.1Alabout one impurity is present in every three unit cells, thissuggests that already a significant overlap of the Anderson-localized wave functions exists. As will be demonstratedbelow, using temperature-dependent thermoelectric mea-surements, we confirm the existence of a delocalized regimeof the IB and quantitatively assess its bandwidth W, whichis not possible by merely investigating ρðTÞ. We alsoperformed measurements of the Hall mobility, which arepresented in SM [28] and which qualitatively agree withboth the conclusions drawn from the analysis of ρðTÞ aboveand SðTÞ below.Figure 3 shows SðTÞ of Fe2−xV1þxAl with0.05 ≤ x ≤ 0.5. For x ¼ 0.05, the impurity states remainfully localized and do not actively contribute themselves tothe transport properties. Instead, SðTÞ is governed by thedispersive CB states up to 230 K, where holes from thedispersive valence band (VB) states are thermally activatedand bipolar conduction takes over (see also SM [28]). Asx ≥ 0.1, a pronounced positive peak of SðTÞ develops atlow temperatures, which we assign to delocalization of theIB. To model the contribution of a (partly) delocalized IB tothe Seebeck coefficient, a band with a finite width W wasintegrated into the two-parabolic band modeling frameworkcommonly employed to understand charge transport inpristine Fe2VAl and conventional doping scenarios[23,32,64]. For parabolic bands, the energy-dependenttransport function ΣðEÞ ¼ DðEÞvðEÞ2τðEÞ increases lin-early at the band edge ΣðEÞ ∝ E [65].For the delocalized IB with finite widthW, we developeda model in which ΣðEÞ increases at either side of themobility edge withΣðE; TÞ ¼ Σ0ðTÞ�E − EckBT�ν: ð3ÞHere, ν is the critical exponent of the Anderson transitionthat can vary between 0.5 and 2, depending on compensa-tion and band hybridization [16]. In our model, the entiretyof the IB can be engineered using expressions followingEq. (3) to yield a single continuously differentiable function(for details, see SM [28]). Finally, the contribution of theentire IB to the transport properties can be evaluated fromthree independent parameters: (i) its bandwidth W, itsmaximum Σ0, and (iii) the position of EF.ΣðEÞ of the parabolic bands and the delocalized IB arethen used to numerically solve the transport integrals andcalculate the Seebeck coefficient,SðTÞ ¼ kBeTR∞−∞ ΣðEÞðE − μÞð−∂f=∂EÞdER∞−∞ ΣðEÞð−∂f=∂EÞdE ; ð4Þwith fðE; μ; TÞ and μðTÞ being the Fermi-Dirac distributionand the chemical potential, respectively; the numerator inEq. (4) represents the electrical conductivity. Since multiplebands contribute to the transport properties, the single-bandcontributions Si have to be weighted with their respectiveconductivities σi,PHYSICAL REVIEW LETTERS 135, 066302 (2025)066302-4Stot ¼PiSiσiPiσi; ð5Þwhere i ¼ fVB;CB; IBg. Equation (5) was fitted to theexperimental SðTÞ in the following way. First, Fe1.9V1.1Al,with the most pronounced second peak in SðTÞ, was fitted[Fig. 3(b)]. Here, the fit parameters are the two energy gapsbetween the IB center and the VB and CB, ΔVB and ΔCB,the effective mass ratio between the CB and VB,mCB=mVB,as well as W, Σ0, and the position of EF with respect to theIB center. In a second step, all these parameters, except forthe bandwidthW, were fixed. Then, all the measured curvesfor the remaining samples were modeled and perfectlyreproduced in a single-parameter fit, highlighting the robust-ness of our modeling approach and the obtained parametervalues. Further details regarding the fitting procedure andthe underlying charge transport model are presented inSM [28].With respect to the IB center, the energy gaps weredetermined as 120 meV toward the valence band and100 meV toward the conduction band edge, highlightingthat the impurity band forms in the middle of the gap, inagreement with our DFT results. The effective mass ratio ofthe two bands mCB=mVB is obtained as ≈2, in goodagreement with previous parabolic band modeling analysesof Fe2VAl-based thermoelectrics [64]. Most importantly,for Fe1.9V1.1Al, our analysis reveals an extremely narrowregion (W ≈ 24 meV) of the IB and EF positioned rightnext to the center of the IB, in striking agreement with theDFT-derived DOS. As x increases, our modeling schemeaccurately predicts the delocalization of the IB as Wprogressively increases up to 240 meV in Fe1.5V1.5Al, asshown in Fig. 3(c).In conclusion, our study provides a thorough investiga-tion of the Fe2−xV1þxAl Heusler system, focusing on thecomposition-induced Mott-Anderson transition. By exam-ining a wide range of compositions from Fe3Al to V3Al, weidentified a profound conductivity minimum in the V self-substituted Heusler compound Fe1.9V1.1Al. Temperature-dependent resistivity measurements near this stoichiometryreveal Mott variable range hopping behavior with a locali-zation length ξL ≈ 12 nm. The temperature-dependentSeebeck coefficient was measured over a wide range oftemperatures and compositions and a charge transportmodel was developed to accurately describe thermoelectrictransport in delocalized impurity bands. Analyzing variousSðTÞ datasets within this modeling framework, we success-fully capture and quantitatively map the delocalization ofIBs in Fe2−xV1þxAl. Our findings emphasize the value ofthermoelectric transport measurements in probing the elec-tronic structure and transport mechanisms in complexmaterials, especially when paired with appropriate multi-band fitting models. This work not only advances theunderstanding of Fe2VAl-based thermoelectrics and solvesthe issue of the unconventional doping behavior inFe2−xV1þxAl, but also provides a robust framework forstudying similar transitions in other materials.Acknowledgments—The authors thank V.Dobrosavljevic and S. Fratini for fruitful discussions.Financial support for M. P., F. G., A. R., T. M., and E. B.came from the Japan Science and Technology Agency(JST), program MIRAI, JPMJMI19A1. The computationalT (K)10 100 1000S(µV/K)-100-500500 200 400 600 800-60-40-2002040T (K) x0.0 0.1 0.2 0.3 0.4 0.5W (meV)050100150200250EΣ(E)WΣ0CBVBIBCBVB IBx = 0.05x = 0.1x = 0.2x = 0.3x = 0.4x = 0.5Fe2-xV1+x AlFe2-xV1+x Al(a) (b) (c)FIG. 3. Mapping delocalization of impurity bands via thermoelectric transport. (a) Temperature-dependent Seebeckcoefficient SðTÞ of Fe2−xV1þxAl with x ¼ 0.05–0.5. For x ¼ 0.05, the impurities merely act as fully localized donor states, placingEF right below the dispersive conduction band [see Fig. 2(b)] and leading to a negative SðTÞ over the entire temperature range. As xincreases up to 0.1, a distinct positive maximum develops around 120 K, which gets progressively smeared out and shifted toward highertemperatures with increasing x. (b) Experimental SðTÞ and least-square fits (solid lines) employing a three-band model with parabolicvalence and conduction bands and a delocalized impurity band confined to a width W. (c) Quantitative mapping of the compositiondependence ofW with a progressive broadening at x > 0.1. Inset shows a sketch of the energy-dependent transport distribution functionΣðEÞ of our model.PHYSICAL REVIEW LETTERS 135, 066302 (2025)066302-5results presented have been achieved using the ViennaScientific Cluster (VSC). The authors acknowledgeTU Wien Bibliothek for financial support through itsOpen Access Funding Programme. F. 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