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[Taichi Terashima](https://orcid.org/0000-0001-9239-0621), Yuki Tokumoto, Kotaro Hamano, [Takako Konoike](https://orcid.org/0000-0002-6037-5782), [Naoki Kikugawa](https://orcid.org/0000-0003-3975-4478), Keiichi Edagawa

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[Anomalous upper critical field in the quasicrystal superconductor Ta1.6Te](https://mdr.nims.go.jp/datasets/f04330b7-ac1c-4bcf-8ae1-b3d8617f78b4)

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Anomalous upper critical field in the quasicrystal superconductor Ta1.6Tenpj | quantummaterials ArticlePublished in partnership with Nanjing Universityhttps://doi.org/10.1038/s41535-024-00669-9Anomalous upper critical field in thequasicrystal superconductor Ta1.6TeCheck for updatesTaichi Terashima 1 , Yuki Tokumoto2 , Kotaro Hamano2, Takako Konoike 1, Naoki Kikugawa 3 &Keiichi Edagawa2Superconductivity in quasicrystals poses a new challenge in condensed matter physics. Wemeasured the resistance and acmagnetic susceptibility of a Ta1.6Te dodecagonal quasicrystal, whichis superconducting below Tc ~ 1 K. We show that the upper critical field increases linearly with a largeslope of − 4.4 T/K with decreasing temperature down to 0.04 K, with no tendency to level off. Theextrapolated zero-temperature critical field exceeds the Pauli limit by a factor of 2.3.We also observedflux-flow resistancewith thermally activated behavior and an irreversibility field that is distinct from theupper critical field.Wediscuss thesepeculiarities in termsof the nonuniformsuperconductinggap andspin-orbit interaction in quasicrystal structures.Quasicrystals (QCs), first reported by Shechtman et al. in 19841, lack aperiodic structure (translational symmetry), but their diffraction patternsexhibit sharp Bragg spots, which indicate rotational symmetries such asfive-, eight-, ten-, or twelve-fold symmetry, which are forbidden in peri-odic crystals. Since their discovery, many quasicrystals have beensynthesized2 and some natural quasicrystals have also been found3.Because many basic concepts in solid-state physics rely on latticeperiodicity and its associated Brillouin zone, the possibility of long-rangeelectronic order in quasicrystals is an intriguing prospect. Two studies havereported affirmative results: Kamiya et al. discovered superconductivity inAl–Zn–Mg icosahedral quasicrystals (i-QC)4, while Tamura et al. reportedlong-range magnetic order in Au–Ga–Gd and Au–Ga–Tb i-QCs5.The report ofKamiya et al. and earlierworks6–9 have inspired theoreticalinvestigations of quasicrystal superconductivity. Sakai et al., who studied anattractive Hubbard model on a Penrose lattice10, found a superconductingstate with a spatially inhomogeneous superconducting gap. They showedthat Copper pairs are spatially extended in the weak-coupling regime.Similar superconducting states with a spatially nonuniform gap are reportedelsewhere11–15. A nonuniform gap in quasicrystal superconductors can leadto peculiar superconducting properties. For instance, it suppresses theBogoliubov quasiparticle peak in the density of states, possibly causingunconventional current-voltage (I−V) characteristics16, and can imposeintrinsic vortex-pinning sites17. In addition, Sakai et al. argued that an exoticsuperconducting statewith a spatially sign-changingorder parameter,whichis reminiscent of the Fulde–Ferrell–Larkin–Ovchinnikov (FFLO) state18,may emerge at low temperatures and high fields in the temperature–fieldphase diagram of quasicrystal superconductors16.The low superconducting transition temperature Tc of the Al–Zn–Mgi-QC ( ~ 0.05 K) impedes detailed investigations of the superconductingproperties of this quasicrystal. Recently, Tokumoto et al. reported a Ta1.6Tedodecagonal quasicrystal (dd-QC) superconductor with a much higher Tc(0.98 K)19, enabling studies of quasicrystal superconductivity with variousprobes.The Ta1.6Te dd-QC is a layered material in which ~ 1 nm-thick Te-terminated layers are separated by van der Waals (vdW) gaps20–22. Toku-moto et al. demonstrated superconducting properties, namely, zero resis-tivity, diamagnetic shielding, and a specific-heat jump, in the Ta1.6Te dd-QC19. Combining the McMillan formula with specific-heat data, theyconcluded a weak-coupling superconductivity with an electron–phononcoupling constant λep of 0.52. They also measured the upper critical fielddown to T/Tc ~ 0.4. At the lowest temperature (T = 0.43 K), the Bc2 reached2.3 T, exceeding the Pauli limit (paramagnetic critical field) of magnetic-field strength Bpo = 1.8 T, where superconductivity is expected to bedestroyed by Zeeman-energy gain of the electron spins. Within the fra-mework of weak-coupling Bardeen–Cooper–Schrieffer (BCS) theory, Bpo(in Tesla) is given by 1.84Tc (in Kelvin)23. The temperature dependence ofBc2was almost linear in themeasured range, but it could also be describedbythe standardWerthamer–Helfand–Hohenberg (WHH) theory24 because ofthe limited temperature range. TheWHHcomputationofBc2(T) applies theweak-coupling BCS theory to a spherical Fermi surface.In this study, we measure the resistance R and ac magnetic suscept-ibility ac-χ (χ0 � iχ00) of theTa1.6Tedd-QCdown to0.04 K (T/Tc = 0.04)andilluminate peculiarities in the superconductivity of this quasicrystal. Espe-cially, we show that the upper critical field Bc2(T) increases linearly with1ResearchCenter forMaterialsNanoarchitectonics (MANA),National Institute forMaterials Science, Tsukuba, 305-0003, Japan. 2Instituteof Industrial Science, TheUniversity of Tokyo, Tokyo, 153-8505, Japan. 3Center for Basic Research on Materials, National Institute for Materials Science, Tsukuba, 305-0003, Japan.e-mail: TERASHIMA.Taichi@nims.go.jp; tokumoto@iis.u-tokyo.ac.jp; edagawa@iis.u-tokyo.ac.jpnpj Quantum Materials |            (2024) 9:56 11234567890():,;1234567890():,;http://crossmark.crossref.org/dialog/?doi=10.1038/s41535-024-00669-9&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41535-024-00669-9&domain=pdfhttp://crossmark.crossref.org/dialog/?doi=10.1038/s41535-024-00669-9&domain=pdfhttp://orcid.org/0000-0001-9239-0621http://orcid.org/0000-0001-9239-0621http://orcid.org/0000-0001-9239-0621http://orcid.org/0000-0001-9239-0621http://orcid.org/0000-0001-9239-0621http://orcid.org/0000-0002-6037-5782http://orcid.org/0000-0002-6037-5782http://orcid.org/0000-0002-6037-5782http://orcid.org/0000-0002-6037-5782http://orcid.org/0000-0002-6037-5782http://orcid.org/0000-0003-3975-4478http://orcid.org/0000-0003-3975-4478http://orcid.org/0000-0003-3975-4478http://orcid.org/0000-0003-3975-4478http://orcid.org/0000-0003-3975-4478mailto:TERASHIMA.Taichi@nims.go.jpmailto:tokumoto@iis.u-tokyo.ac.jpmailto:edagawa@iis.u-tokyo.ac.jpdecreasing temperature down to 0.04 K without leveling off, and that theestimated zero-temperature critical field Bc2(0) far exceeds the Pauli limit.ResultsSuperconducting transition and upper critical fieldThe study was performed on a polygrain sample [Fig. 1a] prepared viareaction sintering of TaTe2 and Ta (see Methods). The sample contains asmall amount of superconducting impurity, which causes a resistivity dropof ~ 2% at Tc ~ 3.2 K [see Supplementary Fig. 1a]. However, as the uppercritical field is small [ ~ 0.5 T at T = 1.88 K; see Supplementary Fig. 1b], thisimpurity negligibly disturbs the present measurements.We first examine the superconducting transition and upper criticalfield. Figure 1b shows the temperature dependences of the resistance R andac magnetic susceptibility below 1.5 K. The resistance at T = 1.1 K isRN = 88mΩ, approximately corresponding to a resistivity ρ of ~ 4mΩ cm.Tokumoto et al.19 reported a ρ of 1.7 mΩ cm at T = 300 K, which increasedby ~ 10% during cooling of the sample to 4.2 K. Considering the irregularsample shape [Fig. 1a], the two values fairly agree. The resistance sharplydrops below ~ 1 K, signaling a superconducting transition. The midpointtransition temperature is Tc = 0.97 K, which favorably agrees with the valueobtainedbyTokumotoet al (0.98 K).The10–90% transitionwidth is 0.03 K.The real part χ0 of the ac magnetic susceptibility begins deviating from thenormal-state value at ~ 0.95 K, roughly corresponding to the temperature atwhich R→ 0 as usual. The diamagnetic response confirms the super-conductivity of the sample.Figure 1c shows the R versus B curves measured at various tempera-tures. At all set temperatures except 0.22 K, the temperature was stabilizedwithin ~ 2% (at 0.22 K, the temperature stabilized within ~ 5%). Themagnetic field was swept up and down at each temperature. To avoid self-heating of the sample, the measurement current was reduced to 7 μA,corresponding to a current density of 1 × 10−3 Acm−2. For technical reasons,the field direction was offset by 20° from the parallel direction B∥I. At thehighest and second-highest set temperatures, 0.96 and 0.92 K, respectively,the resistance increased rapidly to ~ 80mΩ, followed by a gradual rise untilB reached ~ 0.8 T. The gradual increase was caused by the impurity phase,which has a higher Bc2 than the main quasicrystal phase at these tempera-tures. Lowering the temperature broadened the resistive transition: the10–90% transition width increased from ΔB = 0.12 T at 0.93 K to 0.72 Tat 0.04 K.We determined the upper critical field Bc2 from the R(B) data underthree criteria: 10%, 50%, and 90%of the normal resistanceRN. The resultantBc2’s are plotted in Fig. 2. The 50% data well agree with those of Tokumotoet al.19. Under each criterion, the Bc2 versus T plot was linear down toT = 0.04 K (t = T/Tc = 0.04). Straight-line fitting of the 50% result yielded aslope dBc2/dT of − 4.43(2) T/K and an extrapolated Bc2(0) of 4.21(1) T,corresponding to a coherence length of ξ = 88.4 Å. Note that this experi-mental Bc2(0) exceeds the Pauli limit Bpo = 1.8 T by a factor of 2.3. ApplyingWHHtheorywith the dirty limit and taking the aboveBc2 slope as the initialslopeofBc2, dBc2=dTjTc, we also calculated the theoreticalBc2 versusT curve(dotted line in Fig. 2). As the Pauli limitwas neglected in this calculation, thecalculated curve corresponds to the orbital critical field B�c2 and gives thelargest possible upper critical field in WHH theory. The experimental Bc2surpasses the theoretical curve at low temperatures.Flux-flow resistivity and irreversibility fieldWenowexamine theflux-flow resistivity and irreversibilityfield. Figure 3(a)plots the resistance as a function of the angle θ between the magnetic fieldand current in the resistive-transition region under two conditions: 3.5 Tand 0.13 K (I = 7 μA) and 1 T and 0.69 K (I = 20 μA). The observed resis-tances roughly follow the sin2ðθÞ curve [broken lines inFig. 3a], the expectedflux-flow resistance variation in the simple flux-motion model25. Figure 3bcompares R(B) curves for the two configurations B∥I and B⊥I at T = 0.11and 0.66 K (I = 7 μA). The clear differences between the two configurationsconfirm flux-flow resistance.We determined the irreversibility fields Birr byapplying the criterionof 1%ofRN to theB⊥I curves andplotted the results inFig. 2.806040200R (m)1.51.00.5T (K)0.5 RN"'806040200R (m)6543210B (T)0.5 RN0.9 RN0.1 RN(a)(b)(c)ac-Fig. 1 | Superconducting transition in the Ta1.6Te dd-QC. a Photograph of thesample. b Temperature dependences of resistance R and ac magnetic susceptibilityac-χ (χ0 � iχ00). cMagnetic-field dependence of resistance. At each set temperature(0.039, 0.054, 0.067, 0.092, 0.123, 0.219, 0.294, 0.433, 0.532, 0.642, 0.756, 0.802, 0.866,0.920, and 0.956 K from right to left), the field was swept up (red) and down (blue).43210Bc2 (T)1.00.80.60.40.20.0T (K)TcBc210%  50%  90% Tokumoto et al.Birr R vs T  R vs BFig. 2 | Superconducting phase diagram of the Ta1.6Te dd-QC. The upper criticalfield Bc2 data determined under the 10%, 50%, and 90% criteria (squares) arecompared with the Bc2 data of Tokumoto et al.19 (circles). The straight line is fitted tothe 50% Bc2 data and the dotted line describes the orbital critical field without thePauli limit based on WHH theory. The irreversibility fields Birr determined fromFigs. 3b and 4a are also shown (upward and sideways diamonds, respectively).https://doi.org/10.1038/s41535-024-00669-9 Articlenpj Quantum Materials |            (2024) 9:56 2Figure 4 aplots the temperature dependenceof the resistancemeasuredat B⊥I (I = 20 μA) with different field strengths, along with the zero-fieldcurve for comparison. The slight difference in the normal-state resistancebetween B = 0 and B⩾ 1 T is caused by the impurity, which is super-conducting at B = 0 but not at B⩾ 1 T. The transition broadens withincreasing applied field. As shown in Fig. 2, the Birr obtained under the 1%criterion is consistentwith those determined from theR(B) curves in Fig. 3b.Figure 4b plots the same data (omitting the zero-field curve) as Arrheniusplots. Before vanishing, the resistance exhibits thermally activated behaviorR∼ expð�Ea=TÞ [solid lines in Fig. 4b] resembling the thermally activatedflux-flow (TAFF) behavior observed (for example) in cuprates, iron-basedsuperconductors, and superconducting amorphous thin films26–28. Theactivation energy decreases from 24(2) K at 1 T to 2.7(1) K at 2.5 T.DiscussionWe now move on to discussion of our results. First, we compare thesuperconductivity of the Ta1.6Te dd-QC with that of the Al–Zn–Mg i-QC.The upper critical field in the Al–Zn–Mg i-QCwas determined down to T/Tc ~ 0.4 and was reportedly compatible withWHH theory4. The estimatedBc2(0)was 17mT,much smaller than thePauli limitBpo~ 90mT.The initialslope of Bc2, dBc2=dTjTc, was approximately− 0.5 T/K, one order-of-magnitude smaller than that of the Ta1.6Te dd-QC (− 4.4 T/K). Accordingto weak-coupling BCS theory, the initial slope in the dirty limit is propor-tional to the product of the normal-state resistivity and the electronicspecific-heat coefficient (per volume)23. The resistivity and specific-heatcoefficient are one order-of-magnitude and three times larger, respectively,in the Ta1.6Te dd-QC19 than in the Al–Zn–Mg i-QC4. This fact crudelyexplains the much larger Bc2 slope of the Ta1.6Te dd-QC than that of theAl–Zn–Mg i-QC. Notice that the large slope of− 4.4 T/K in the Ta1.6Te dd-QC can be compared with the initial slopes of Bc2 in a practical super-conductor Nb3Sn (− 2.6 T/K)29, a heavy-fermion superconductor UPt3(− 6.3 T/K)30, and an iron-based superconductor (Ba, K)Fe2As2 (− 5.4 T/K)31, for example.Wenowconsider the linearity of our experimentalBc2(T) curve.WithintheWHHtheory,which is aweak-couplingBCS theory for a spherical Fermisurface and isotropic gap, the Bc2(T) curve is always a concave function (i.e.,d2Bc2/dT2 < 0) irrespectiveof themagnitudesof the impurity scattering, spin-orbit coupling, and Pauli limit24. In the very strong-coupling region, it istheoretically anticipated that the Bc2(T) curve acquires a noticeable positivecurvature (d2Bc2/dT2 > 0) at intermediate temperatures32. A non-sphericalFermi surfacemay also brings a positive curvature near Tc33. However, thesepredictions are incompatible with the present experimentalBc2 curve, whichis linear in the entire temperature range below Tc. In addition, the observedlinearity is incompatible with the theoretically proposed FFLO-like state,which gives a Bc2(T) curve with an inflection point16.Experimentally, some superconductors belonging to either of the fol-lowing two classes are known to exhibit approximately linear Bc2(T)behavior: (i) highly disordered systems such as amorphous, metallic glass,and high-entropy alloy superconductors, and (ii) multi-band super-conductors such as borocarbide, MgB2, and iron-based superconductors.Examples of the former class are the superconducting amorphousalloys (Mo0.5Ru0.5)80P20, (Mo0.6Ru0.4)86B14, andMo30Re70,which give linearBc2(T) versusT curves down toT/Tc ~ 0.234. Si1−xAux amorphous alloys andTa–Nb-Hf–Zr–Ti high-entropy alloys35,36 exhibit similar linear Bc2(T)curves. However, it should be noted that linear Bc2(T) behavior is com-paratively rare in highly disordered superconductors37. Upward deviationsof Bc2(T) from the WHH predictions, including T-linear behavior, havebeen arguably ascribed to spatial electronic inhomogeneities on the order ofthe superconducting coherence length38.The Bc2(T) curves of multi-band superconductors exhibit various Tdependences ranging from a usual concave one (d2Bc2/dT2 < 0) to roughly806040200R (m)543210B (T)T = 0.11 KB  IB // IT = 0.66 KB  IB // I0.01RN(a)(b)706050403020R (m)900Angle  (deg)T = 0.69 KB = 1 TT = 0.13 KB = 3.5 TFig. 3 | Flux-flow resistance in the Ta1.6Te dd-QC. a Resistance as a function of theangle θ between the field and current. Measurements were performed at the statedtemperatures and fields in the resistive-transition region. The broken lines are fittedto sin2ðθÞ.bResistive-transition curves ofB∥I andB⊥I at the indicated temperatures.12410241002R (m)543211/T (1/K)806040200R (m)1.21.00.80.60.40.2T (K)B = 0B  IB = 2.5 T2 T1.5 T1 T0.01RN(a)(b)Fig. 4 | Temperature dependence of the resistance in magnetic fields.a Temperature dependence of resistance under B⊥I with different field strengths.The zero-field curve is also shown. b Arrhenius plot of the data plotted in (a)(omitting the zero-field data). The solid lines are linear fits.https://doi.org/10.1038/s41535-024-00669-9 Articlenpj Quantum Materials |            (2024) 9:56 3T-linear or convex one31,39–42. Theoretically, these variations can be ascribedto different gap sizes between different bands and the relative strengths ofintraband and interband scatterings43.These examples suggest that departure from the single-uniform-gappicture is a necessary condition of Bc2(T) deviation from WHH behavior.More specifically, the gap is spatially inhomogeneous in highly disorderedsuperconductors and nonuniform in k space in multi-band super-conductors. Consistent with this conjecture, theoretical studies have shownthat the gap in quasicrystal superconductors is intrinsicallyinhomogeneous10–12. However, the sufficient condition of linear Bc2(T) isunclear and requires theoretical elucidation.Next, we discuss why Bc2(0) exceeds the Pauli limit by a factor of 2.3.The Pauli limit is enhanced by electron–phonon coupling as (1+ λep)Bpo23.However, the reported electron–phonon coupling constant in the Ta1.6Tedd-QC is λep= 0.52, which is obviously insufficient to explain the largeBc2(0). We also note that in the presence of both the orbital effect and thePauli limit, the upper critical field is given byBc2 ¼ B�c2Bpo=ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ðB�c2Þ2 þ B2poq23. Evenwhen the orbital effect is absent (i.e.,B�c2 ! 1),Bc2 is limited toBpo=ffiffiffi2p. Because the experimentalBc2(T) curveshows no leveling-off tendency down to 0.04 K, the effective Pauli limit inthe Ta1.6Te dd-QC must far exceed the experimental Bc2(0).The spin-orbit interaction may be important. It induces spin-flipscattering and enforces finite spin susceptibility in superconductors even atzero temperature, thus reducing the effect of the Pauli limit. In the presenceof spin-orbit scattering, the effective Pauli limit Bp is enhanced toBp ¼ 1:33ffiffiffiffiffiffiλsopBpo23. The spin-orbit scattering parameter λso is given byλso = 2ℏ/(3πkBTcτso) (≃1.17ξo/lso), where τso and lso are the scattering timeand mean free path associated with spin-flip scattering and ξo is the BCScoherence length. Because these parameters are difficult to estimate in thepresent case (for crude estimation of required λso, see Supplementary Note2), we refer to highly disordered superconductors, in which the Bc2(0)sometimes (or likely) exceeds the Pauli limit: Bc2(0) in the superconductorsZr77Rh2344, Mo45Si5545, Hf80Fe2046, and (TaNb)0.16(ZrHfTi)0.8436 exceeds thePauli limit by ~ 7, 15, 50, and 9%, respectively. These excesses are howevermuch smaller than in the present case.When superconductors lack spatial inversion symmetry, the spin-orbitinteraction causes the antisymmetric spin-orbit interaction and intrinsicallysuppresses the effect of the Pauli limit47. For instance, in noncentrosymmetricsuperconductors such as CePt3Si48 and CeRhSi349 and interface super-conductors such as LAO/STO50 and EuO/KTO51, the Rashba-type anti-symmetric spin-orbit interaction operates because of the broken mirrorsymmetry and confines the electron spins to in-plane directions. Therefore,the Pauli limit is removed when the external field is applied perpendicular tothe plane, and the Bc2 becomes large. Another example is Ising super-conductivity in monolayer MoS252 and NbSe253, where the electron spins areconfined to the out-of-plane direction and hence the Pauli limit is removedfor in-plane fields.The spin-orbit interaction is stronger in heavier atoms than in lighteratoms, andTaandTe inour studiedquasicrystal areheavyatoms.Generally,atomic sites in quasicrystals possess no inversion symmetry. Therefore, wesuggest that spin-orbit interactionsmayplay an important role in enhancingthe upper critical field in the Ta1.6Te dd-QC.For the sake of completeness, we mention spin-triplet super-conductivity briefly. Large upper critical fields were reported for somecandidate spin-triplet superconductors suchasK2Cr3As354,CeRh2As255, andUTe256–58. However, there is no evidence for strong spin fluctuations med-iating triplet pairing in the Ta1.6Te dd-QC19 and hence the possibility of thetriplet pairing in the present case can likely be dismissed.We next consider the influence of possible anisotropy. The Ta1.6Te dd-QC has a layered structure with a vdW gap. However, the calculated elec-tronic band structure of a crystalline approximant Ta21Te13 is only mod-erately anisotropic with sizable dispersion along the out-of-planedirection22. This finding is likely related to the large thickness of the con-stituent Te-terminated layers (approximately 1 nm21,22), suggesting that theBc2 anisotropy in theTa1.6Te dd-QC,which is determinedby the square rootof the mass anisotropy, is limited. Experimentally, the resistive transitionunder I∥B broadens with decreasing temperature [Fig. 1c]. This behavior ispossibly explainedby anisotropy, but the transitionwidth remains a fractionof Bc2 even at 0.04 K, supporting a small anisotropy.Finally, we discuss the experimentally observed flux-flow resistanceand irreversibility field. In conventional low-Tc superconductors, the irre-versibility field is indistinguishable from the upper critical field. In contrast,the two critical fields in cuprates and some (relatively) high-Tc or low-dimensional superconductors are distinct and TAFF behavior is observed,highlighting the importance of thermal fluctuations in suchsuperconductors59,60. TAFF-like behavior [Fig. 4b], along with distinctirreversibility and upper critical fields (Fig. 2), were observed in the presentstudy. Thermal fluctuations in the Ta1.6Te dd-QC are likely limited by thelow Tc. However, the intrinsically inhomogeneous superconducting gapexpected in the Ta1.6Te dd-QC10–12 can affect the vortex phase diagram andtransport properties. The theoretically suggested peculiar I−Vcharacteristics61 and intrinsic pinning sites17 must also be considered.In summary, our resistance and ac susceptibility measurements of theTa1.6Te dd-QC down to 0.04 K revealed the following peculiarities of qua-sicrystal superconductivity: The upper critical field increases linearly as thetemperature reduces toT/Tc = 0.04,withno leveling-off tendency. The zero-temperature upper criticalfieldBc2(0) ismore than twice the Pauli limit. Theirreversibility field is distinct from the upper critical field despite the low Tc.The flux-flow resistance shows activated behavior. These observations arepossibly explained by the nonuniform gap distribution and spin-orbitinteraction in the quasicrystal structure. To confirm this conjecture, furtherexperimental and theoretical investigations are required. Lastly, we pointout that the large Bc2 slope (− 4.4 T/K) is of technological interest. If largeresistivity of quasicrystals tends to cause a large Bc2 slope, it is worthwhilesearching for practical superconductors in quasicrystals.MethodsSamples and measurementsThe Ta1.6Te samples were synthesized via reaction sintering19. Amixture ofTaTe2, Ta, and iodine (topromote the reaction)waspressed into apellet andsintered in a vacuum at 1273 K for six days. The synthesized samplesconsisted of disc-shaped grains with diameters ranging from a few to a fewhundred micrometers and thicknesses varying from a few tenths of amicrometer to a few micrometers. They were thoroughly characterizedusing electron diffractometry and X-ray diffractometry (XRD) analysesin ref. 19.For the present measurements, one synthesized pellet was chopped inair, and one piece was picked up and roughly shaped with sandpaper.Electrical contacts were formed with silver conducting paint.A dilution refrigerator equipped with a 20-T superconducting magnetwas used to produce low temperatures down to 0.04 K. For resistancemeasurements, the samplewasmountedona top-loadingprobe and rotatedin situ on a rotation platform. The resistance was measured using thestandard four-contact method with a low-frequency (f ~ 17Hz) ac current.For ac susceptibility measurements, the electrical wires were removed fromthe sample and the samplewasmounted on another top-loading probewitha pick-up coil.Data availabilityThe data that support the findings of this study are available from thecorresponding authors upon reasonable request.Received: 16 February 2024; Accepted: 14 July 2024;References1. Shechtman, D., Blech, I., Gratias, D. & Cahn, J.W.Metallic phasewithlong-range orientational order and no translational symmetry. Phys.Rev. Lett. 53, 1951 (1984).https://doi.org/10.1038/s41535-024-00669-9 Articlenpj Quantum Materials |            (2024) 9:56 42. Tsai A.-P. Discovery of stable icosahedral quasicrystals: progress inunderstanding structure and properties. Chem. Soc. Rev. 42,5352 (2013).3. Bindi, L., Steinhardt, P. J., Yao, N. & Lu, P. J. Natural quasicrystals.Science 324, 1306 (2009).4. Kamiya K. et al. Discovery of superconductivity in quasicrystal. Nat.Commun. 9, 154 (2018).5. Tamura R. et al. Experimental observation of long-range magneticorder in icosahedral quasicrystals. J. Am. Chem. Soc. 143,19938 (2021).6. Graebner, J. E. & Chen, H. S. Specific heat of an icosahedralsuperconductor, Mg3Zn3Al2. Phys. Rev. Lett. 58, 1945 (1987).7. Wong, K. M., Lopdrup, E., Wagner, J. L., Shen, Y. & Poon, S. J.Transport and superconducting properties of the Mg32(Al, Zn)49-type quasicrystalline and crystalline phases. Phys. Rev. B 35,2494 (1987).8. Wagner, J. L., Biggs, B. D., Wong, K. M. & Poon, S. J. Specific-heatand transport properties of alloys exhibiting quasicrystalline andcrystalline order. Phys. Rev. B 38, 7436 (1988).9. Azhazha V. et al. Superconductivity of Ti-Zr-Ni alloys containingquasicrystals. Phys. Lett. A 303, 87 (2002).10. Sakai, S., Takemori, N., Koga, A. & Arita, R. Superconductivity on aquasiperiodic lattice: Extended-to-localized crossover of Cooperpairs. Phys. Rev. B 95, 024509 (2017).11. Araújo, R. N. & Andrade, E. C. Conventional superconductivity inquasicrystals. Phys. Rev. B 100, 014510 (2019).12. LiuY.-B. et al. Cooper instability andsuperconductivity of thePenroselattice. Sci. China Phys. Mech. Astron. 65, 287411 (2022).13. Cao Y. et al. Kohn-Luttinger mechanism driven exotic topologicalsuperconductivity on the Penrose lattice. Phys. Rev. Lett. 125,017002 (2020).14. Liu, Y.-B., Shao, Z.-Y., Cao, Y. & Yang, F. Unconventionalsuperfluidity of superconductivity on Penrose lattice.Sci. China Phys.Mech. Astron 66, 290312 (2023).15. Liu, Y.-B., Zhou, J. & Yang, F. Nematic superconductivity and itscritical vestigial phases in the quasi-crystal. Preprint at arXiv https://arxiv.org/abs/2401.00750 (2024).16. Sakai, S. & Arita, R. Exotic pairing state in quasicrystallinesuperconductors under a magnetic field. Phys. Rev. Res. 1,022002(R) (2019).17. Nagai, Y. Intrinsic vortex pinning in superconducting quasicrystals.Phys. Rev. B 106, 064506 (2022).18. Matsuda, Y. & Shimahara, H. Fulde-Ferrell-Larkin-Ovchinnikov statein heavy fermion superconductors. J Phys. Soc. Jpn. 76,051005 (2007).19. Tokumoto Y. et al. Superconductivity in a van der waals layeredquasicrystal. Nat. Commun. 15, 1529 (2024).20. Conrad, M., Krumeich, F. & Harbrecht, B. A dodecagonalquasicrystalline chalcogenide. Angew. Chem. Int. Ed. 37,1383 (1998).21. Conrad, M. & Harbrecht, B. Ta97Te60: A crystalline approximant of atantalum telluride quasicrystal with twelvefold rotational symmetry.Chem. Eur. J. 8, 3093 (2002).22. Cain, J. D., Azizi, A., Conrad, M., Griffin, S. M. & Zettl, A. Layer-dependent topological phase in a two-dimensional quasicrystal andapproximant. Proc. Natl. Acad. Sci. U.S.A. 117, 26135 (2020).23. Decroux,M.&Fischer,Ø. inSuperconductivity in TernaryCompoundsII, edited by Maple, M. B. and Fischer, Ø. (Springer-Verlag,Berlin, 1982).24. Werthamer, N. R., Helfand, E. & Hohenberg, P. C. Temperature andpurity dependence of the superconducting critical field, Hc2. III.electron spin and spin-orbit effects. Phys. Rev. 147, 295 (1966).25. Kwok W. K. et al. Direct observation of dissipative flux motion andpinning by twin boundaries in YBa2Cu3O7−δ single crystals. Phys.Rev. Lett. 64, 966 (1990).26. Palstra, T. T. M., Batlogg, B., Schneemeyer, L. F. & Waszczak, J. V.Thermally activated dissipation in Bi2.2Sr2Ca0.8Cu2O8+δ. Phys. Rev.Lett. 61, 1662 (1988).27. Jaroszynski J. et al. Upper critical fields and thermally-activatedtransport of NdFeAsO0.7F0.3 single crystal. Phys. Rev. B 78,174523 (2008).28. Graybeal, J. M. & Beasley, M. R. Observation of a new universalresistive behavior of two-dimensional superconductors in amagneticfield. Phys. Rev. Lett. 56, 173 (1986).29. Orlando, T. P., McNiff, E. J., Foner, S. & Beasley, M. R. Critical fields,Pauli paramagnetic limiting, and material parameters of Nb3Sn andV3Si. Phys. Rev. B 19, 4545 (1979).30. Chen J. W. et al. Upper critical magnetic field of the heavy-fermionsuperconductor UPt3. Phys. Rev. B 30, 1583 (1984).31. Yuan H. Q. et al. Nearly isotropic superconductivity in (Ba,K)Fe2As2.Nature 457, 565 (2009).32. Carbotte J. P. Properties of boson-exchange superconductors. Rev.Mod. Phys. 62, 1027 (1990).33. Kita, T. & Arai, M. Ab initio calculations of Hc2 in type-IIsuperconductors: Basic formalism and model calculations. Phys.Rev. B 70, 224522 (2004).34. Tenhover, M., Johnson, W. & Tsuei, C. Upper critical fields ofamorphous transition metal based alloys. Solid State Commun. 38,53 (1981).35. Furubayashi, T., Nishida, N., Yamaguchi, M., Morigaki, K. & Ishimoto,H. Superconducting properties of amorphous Si1−xAux near metal-insulator transition. Solid State Commun. 55, 513 (1985).36. von Rohr, F., Winiarski, M. J., Tao, J., Klimczuk, T. & Cava, R. J. Effectof electron count andchemical complexity in theTa-Nb-Hf-Zr-Ti high-entropy alloy superconductor. Proc. Natl. Acad. Sci. U.S.A. 113,E7144 (2016).37. Karkut, M. G. & Hake, R. R. Upper critical fields and superconductingtransition temperatures of some zirconium-base amorphoustransition-metal alloys. Phys. Rev. B 28, 1396 (1983).38. Carter, W., Poon, S., Hull, G. & Geballe, T. Enhanced critical fieldcurves of metastable superconductors. Solid State Commun. 39,41 (1981).39. Shulga S. V. et al. Upper critical field peculiarities of superconductingYNi2B2C and LuNi2B2C. Phys. Rev. Lett. 80, 1730 (1998).40. Bud’ko S. L. et al. Magnetoresistivity andHc2(T) inMgB2.Phys. Rev. B63, 220503(R) (2001).41. Hunte F. et al. Two-band superconductivity in LaFeAsO0.89F0.11 atvery high magnetic fields. Nature 453, 903 (2008).42. Terashima, T. et al. Anomalous Fermi surface in FeSe seen byShubnikov–de Haas oscillation measurements. Phys. Rev. B 90,144517 (2014).43. Gurevich A. Limits of the upper critical field in dirty two-gapsuperconductors. Physica C Supercond. 456, 160 (2007).44. Togano, K. & Tachikawa, K. Upper critical fields of noncrystallineZr77Rh23 alloy and intermetallic Zr2Rh compound. Phys. Lett. A 54,205 (1975).45. Ikebe, M., Muto, Y., Ikeda, S., Fujimori, H. & Suzuki, K.Superconductivity of bulk amorphous Mo-Si and Mo-C alloys.Physica B+C 107, 387 (1981).46. Tafra, E., Basletić, M., Ristić, R., Babić, E. & Hamzić, A. Enhancedsuperconductivity in Hf-base metallic glasses. J. Phys. Condens.Matter 20, 425215 (2008).47. Smidman, M., Salamon, M. B., Yuan, H. Q. & Agterberg, D. F.Superconductivity and spin-orbit coupling in non-centrosymmetricmaterials: a review. Rep. Prog. Phys. 80, 036501 (2017).48. Bauer E. et al. Heavy fermion superconductivity andmagnetic order innoncentrosymmetric CePt3Si. Phys. Rev. Lett. 92, 027003 (2004).49. Kimura, N., Ito, K., Aoki, H., Uji, S. & Terashima, T. Extremely highupper criticalmagneticfieldof thenoncentrosymmetric heavy fermionsuperconductor CeRhSi3. Phys. Rev. Lett. 98, 197001 (2007).https://doi.org/10.1038/s41535-024-00669-9 Articlenpj Quantum Materials |            (2024) 9:56 5https://arxiv.org/abs/2401.00750https://arxiv.org/abs/2401.00750https://arxiv.org/abs/2401.0075050. Rout, P. K., Maniv, E. & Dagan, Y. Link between the superconductingdome and spin-orbit interaction in the (111) LaAlO3/SrTiO3 interface.Phys. Rev. Lett. 119, 237002 (2017).51. Hua X. et al. Tunable two-dimensional superconductivity and spin-orbit couplingat theEuO/KTaO3(110) interface.npjQuantumMater.7,97 (2022).52. Lu J.M. et al. Evidence for two-dimensional Ising superconductivity ingated MoS2. Science 350, 1353 (2015).53. de la Barrera, S. C. et al. Tuning Ising superconductivity with layer andspin–orbit coupling in two-dimensional transition-metaldichalcogenides. Nat. Commun. 9, 1427 (2018).54. Bao J.-K. et al. Superconductivity in quasi-one-dimensional K2Cr3As3with significant electron correlations. Phys. Rev. X 5, 011013 (2015).55. Khim S. et al. Field-induced transition within the superconductingstate of CeRh2As2. Science 373, 1012 (2021).56. Ran S. et al. Nearly ferromagnetic spin-triplet superconductivity.Science 365, 684 (2019).57. Aoki D. et al. Unconventional superconductivity in UTe2. J. Phys.Condens. Matter 34, 243002 (2022).58. Lin W.-C. et al. Tuning magnetic confinement of spin-tripletsuperconductivity. npj Quantum Mater. 5, 68 (2020).59. Blatter, G., Feigel’man, M. V., Geshkenbein, V. B., Larkin, A. I. &Vinokur, V. M. Vortices in high-temperature superconductors. Rev.Mod. Phys. 66, 1125 (1994).60. Gurevich, A. Iron-based superconductors at high magnetic fields.Rep. Prog. Phys. 74, 124501 (2011).61. Takemori, N., Arita, R. & Sakai, S. Physical properties of weak-coupling quasiperiodic superconductors. Phys. Rev. B 102,115108 (2020).AcknowledgementsThis study was supported by the JST-CREST program (grant no.JPMJCR22O3; Japan), JSPS KAKENHI (Grant Numbers JP19H05821,JP23K04355, JP22H04485, and JP22K03537), and Tokuyama ScienceFoundation. MANA is supported by World Premier International ResearchCenter Initiative (WPI), MEXT, Japan.Author contributionsT.T. designed research. Y.T., K.H. and K.E. prepared and characterizedsamples. T.T., Y.T., K.H., T.K. and N.K. performed measurements. T.T.analyzed the data and wrote the manuscript with inputs from the otherauthors.Competing interestsThe authors declare no competing interests.Additional informationSupplementary information The online version containssupplementary material available athttps://doi.org/10.1038/s41535-024-00669-9.Correspondence and requests for materials should be addressed toTaichi Terashima, Yuki Tokumoto or Keiichi Edagawa.Reprints and permissions information is available athttp://www.nature.com/reprintsPublisher’s note Springer Nature remains neutral with regard tojurisdictional claims in published maps and institutional affiliations.Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in anymedium or format, as longas you give appropriate credit to the original author(s) and the source,provide a link to the Creative Commons licence, and indicate if changeswere made. 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To view a copy of thislicence, visit http://creativecommons.org/licenses/by/4.0/.© The Author(s) 2024https://doi.org/10.1038/s41535-024-00669-9 Articlenpj Quantum Materials |            (2024) 9:56 6https://doi.org/10.1038/s41535-024-00669-9http://www.nature.com/reprintshttp://creativecommons.org/licenses/by/4.0/ Anomalous upper critical field in the quasicrystal superconductor Ta1.6Te Results Superconducting transition and upper critical field Flux-flow resistivity and irreversibility field Discussion Methods Samples and measurements Data availability References Acknowledgements Author contributions Competing interests Additional information