Unusual high-field metal in a Kondo insulator
Ziji Xiang, Lu Chen, Kuan-Wen Chen, Colin Tinsman, Yuki Sato, Tomoya Asaba, Helen Lu, Yuichi Kasahara, Marcelo Jaime, Fedor Balakirev, Fumitoshi Iga, Yuji Matsuda, John Singleton, Lu Li
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Unusual high-field metal in a Kondo insulator
Ziji Xiang , ∗ Lu Chen , Kuan-Wen Chen , Colin Tinsman , Yuki Sato , Tomoya Asaba , , Helen Lu , YuichiKasahara , Marcelo Jaime , Fedor Balakirev , Fumitoshi Iga , Yuji Matsuda , John Singleton , † and Lu Li Department of Physics, University of Michigan, Ann Arbor, MI 48109, USA Department of Physics, Kyoto University, Kyoto 606-8502, Japan MPA-Q, Los Alamos National Laboratory, Los Alamos, NM, USA National High Magnetic Field Laboratory (NHMFL), MS E536,Los Alamos National Laboratory, Los Alamos, NM 87545 Institute of Quantum Beam Science, Graduate School of Science and Engineering, Ibaraki University, Mito 310-8512, Japan (Dated: March 1, 2021)
Within condensed-matter systems, strong electronic in-teractions often lead to exotic quantum phases. A re-cent manifestation of this is the unexpected observationof magnetic quantum oscillations [1–4] and metallic ther-mal transport [5, 6], both properties of systems with Fermisurfaces of itinerant quasiparticles, in the Kondo insu-lators
SmB and YbB . To understand these phenom-ena, it is informative to study their evolution as the en-ergy gap of the Kondo-Insulator state is closed by a largemagnetic field. We show here that both the quantum-oscillation frequency and the cyclotron mass display astrong field dependence in the resulting high-field metal-lic state in YbB . By tracking the Fermi-surface area,we conclude that the same quasiparticle band gives riseto the quantum oscillations in both insulating and metallicstates. These data are understood most simply using a two-fluid picture where unusual quasiparticles, contributinglittle or nothing to charge transport, coexist with conven-tional fermions. In the metallic state this leads to a heavy-fermion bad metal with negligible magnetoresistance, rel-atively high resistivity and a very large Kadowaki-Woodsratio, underlining the exotic nature of the fermion ensem-ble inhabiting YbB . In Kondo insulators (KIs), an energy gap is opened up bystrong coupling between a lattice of localized moments andthe extended electronic states. The resulting Kondo gap E g is usually narrow (typically E g ≃ −
20 meV), yet the rˆole itplays in charge transport is more complicated than that of thebandgap in conventional semiconductors. A low-temperature ( T ) saturation of the resistivity ρ has long been known in twoprototypical KIs, SmB and YbB [7, 8]; both are mixed-valence compounds with strong f − d hybridization that de-fines the band structure close to the Fermi energy. While thesaturation might suggest additional metallic conduction chan-nels, the high resistivity value within the weakly T -dependent“plateau” implies an unconventional nature for such chan-nels [4, 7].Recently, magnetic quantum oscillations, suggestive of aFermi surface (FS), and thus totally unexpected in an insu-lator, have been detected in both SmB and YbB [1–4]. ∗ [email protected] † [email protected] ‡ [email protected] Whilst some have attributed the oscillations in SmB to resid-ual flux [9], the flux-free growth process of YbB (see Meth-ods) excludes such a contribution. The oscillations in YbB are observed in both ρ (the Shubnikov-de Haas, SdH, effect)and magnetization M (the de Haas-van Alphen, dHvA, effect)at applied magnetic fields H where the gap is still finite. The T -dependence of the oscillation amplitude follows the expec-tations of Fermi-liquid theory [4]. Moreover, a contributionfrom gapless quasiparticle excitations to the heat capacity hasbeen detected in both KIs [5, 6]. In particular, YbB shows T -linear zero-field thermal conductivity, a characteristic ofitinerant fermions [6]. The agreement of the FS parametersderived from the quantum oscillations, heat capacity, and ther-mal conductivity suggests that the same quasiparticle band isresponsible [6].Despite this apparent consistency, the mystery remains:how can itinerant fermions exist in a gapped insulator andtransport heat but not charge? In response, many theo-retical models entered the fray, including topological sur-face states [10], magnetoexcitons [11], scalar Majoranafermions [12], emergent fractionalized quasiparticles [13, 14]and non-Hermitian states [15]. As these scenarios frequentlyenvisage some form of exotic in-gap states, it is potentiallyinvaluable to observe how the properties of KIs evolve as theenergy gap E g closes.The cubic rare-earth compound YbB is an excellent plat-form on which to carry out such studies. In YbB , theKondo gap ( E g ≈
15 meV [16]) is closed by large H , lead-ing to an insulator-to-metal (I-M) transition at fields rang-ing from µ H I-M ≃ H k [ ] ) to 55-59 T ( H k [ ] ) [4, 17, 18]. Kondo correlation does not break down atthe I-M transition, remaining strong to 60 T and beyond in thehigh-field metallic state; hence, this can be termed a Kondometal (KM) [19, 20]. In this study, we apply both transportand thermodynamic measurements, including ρ , penetrationdepth, M , and dilatometry, to YbB . By resolving quantumoscillations and tracking their T and H dependence in the KMstate, we trace the fate of the possible neutral quasiparticles atfields above the gap closure and expose their interactions withmore conventional charged fermions.In our YbB samples, M and magnetostriction data showthat the I-M transition occurs at µ H I-M = 46.3 T ( H k [ ] );the tiny valence increase of Yb ions at the transition suggestedby magnetostriction reinforces the KM nature of the high-fieldmetallic state (see Methods and Extended Data Fig. 1). Toprobe the electronic structure of the KM state, a proximity-detector-oscillator (PDO) was used (see Methods) for con-tactless SdH effect studies. The PDO technique is sensitiveto the sample skin depth, providing a direct probe of changesin the conductivity of the metallic KM state. The setup illus-trated in Fig. 1a (inset) was rotated on a cryogenic goniome-ter to achieve H -orientation-dependent measurements. Fig. 1asummarizes the H dependence of the PDO frequency f as H rotates from [100] to [110].The low conductivity of the KI state suggests that the MHzoscillatory field from the PDO coil will completely penetratethe sample [21]. Therefore, the response of the PDO is dom-inated by the sample skin depth only when the sample entersthe KM state and the conductivity increases significantly (seeMethods). The transition between these regimes is marked bya dip in the f versus H curves close to H I − M , above whichmagnetic quantum oscillations emerge. Fig. 1b displays ∆ f ,the oscillatory component of f , at various angles θ as a func-tion of 1 / H . The distinct oscillation pattern observed for H || [ ] ( θ = ) is preserved up to θ ≈ ◦ (Fig. 1b), beingstrongly modified at higher angles (Extended Data Fig. 2a).The oscillations in ∆ f represent a single series that is intrin-sically aperiodic in 1 / H ; attempts to fit them using a super-position of conventional oscillation frequencies fail to repro-duce the raw data (Supplementary Information). To demon-strate the point further, Fig. 1c compares Landau-level index-ing plots for the low-field KI and high-field KM states, bothwith H k [100]. The oscillatory component of the resistivity ∆ρ in the KI state is shown in the inset of Fig. 1c and in-dexed conventionally using integers for minima and half in-tegers for maxima in ρ . For the KM state, peaks in f corre-spond to peaks in conductivity [21], and are therefore indexedusing integers [22]. A further subdivision of the oscillations,reminiscent of a second harmonic, is likely due to Zeemansplitting of the quasiparticle levels [22]; these features aremarked with“+” and “-” assuming signs expected for conven-tional Zeeman shifts. In the KI state, the plot of Landau levelindex N versus 1 / H is a straight line, as expected for a field-independent quantum oscillation frequency in a nonmagneticsystem. By contrast, in the KM state the 1 / H positions ofthe oscillations have a nonlinear relationship with N whichwe shall describe below. (Here we note that the magnetic in-duction B ≈ µ H , since µ H is large and YbB has a weakmagnetization; see Methods).Nevertheless, despite their unusual periodicity, the T -dependences of individual oscillation amplitudes in the KMstate (Fig. 1d) closely follow the Lifshitz-Kosevich (LK)equation [22], ∆ f ( T ) ∝ π k B T / E c ∗ sinh ( π k B T / E c ∗ ) , (1)suggesting that they are almost certainly due to fermions.(Here, k B is the Boltzmann constant and E c ∗ the cyclotron en-ergy). However, the derived value of E c ∗ varies nonlinearlywith H , indicating that the cyclotron mass m ∗ = eB / E c ∗ is afunction of H (Fig. 2b, inset).The relationship between oscillation indices and magnetic field is described empirically by N + λ = F µ ( H N − H ∗ ) , (2)where an offset field of µ H ∗ = . µ H N . Here, λ is a phase factor, N is again the index, H N is the field at which the corresponding feature ( e.g., peak)occurs and F is the slope. Eq. 2 is symptomatic of a FS pocketthat progressively depopulates as H increases, for reasons thatwe will discuss below. In such cases, the Onsager relationship F ( B ) = ¯ h π e A ( B ) between the FS extremal cross-sectional area A and the frequency F of the corresponding quantum oscilla-tions still applies even when A changes with H (see Methods).Since B ≈ µ H , we write B ∗ = µ H ∗ and represent the fielddependence of Eq. 2 using a B -dependent frequency F KM ( B ) = F B − B ∗ B ; (3)this is associated with a B -dependent extremal area A ( B ) = A B − B ∗ B , where A = π e ¯ h F [23].Analyzed in these terms, our data indicate that the quantumoscillations seen in the KM phase are due to a FS pocket thatis the same as, or very closely related to, the FS pocket in theKI state which contributes quantum oscillations and a T -linearterm in the thermal conductivity and heat capacity [6]. Thisis strongly suggested by the field dependence of F KM (Eq. 3)shown in Fig. 2b. Even a cursory inspection reveals that Eq. 3,describing oscillations in the KM state, gives a frequency verysimilar to that of the KI-state oscillations when extrapolatedback to H I − M . This is true for all angles θ at which oscilla-tions were measured; using appropriate F values (ExtendedData Table 1) and substituting fields B = µ H I − M ( θ ) on thephase boundary (minima in PDO data; Fig. 1a) into Eq. 3gives the frequencies shown as magenta diamonds in Fig. 2c.The θ − dependence of F KM ( µ H I − M ) thus deduced tracks thebehaviour of the dHvA frequencies measured in the KI state,albeit with an offset ≈
100 T. This offset may be due to a dis-continuous change in F at the phase boundary; however, itcould also result from the potential uncertainty in determiningthe H -position of any phase transition associated with a va-lence change [25]. Substituting values of µ H I − M decreasedby ≈ . F KM withthe quantum-oscillation frequencies observed in the KI state(Fig. 2c, red diamonds).This consistency indicates that the novel quasiparticles de-tected in the KI state of YbB [4], which are probablycharge neutral [6], also cause the SdH effect in the KM state(see Methods for further discussion). The unusual nature ofthe KM state is further revealed by magnetotransport exper-iments. To reduce Joule heating as the KI state is traversed,we used a pulsed-current technique (see Methods). Currentis only applied to the sample when H > H I-M , as sketched inFig. 3a, inset. Below 10 K, very weak longitudinal magnetore-sistance (MR) is observed above 50 T (Extended Data Fig. 3)and is preserved up to ≈
68 T (Extended Data Fig. 2), permit-ting an analysis of the T -dependence of the resistivity ρ in theKM state. Fig. 3a and Fig. 3b show ρ at 55 T as a function of T and T , respectively. The ρ − T curve shows a maximum at T ∗ = 14 K. With decreasing T , a linear T -dependence devel-ops below 9 K and extends down to T ≈ T -behavior is established below T FL =2.2 K. The overall behavior of ρ ( T ) mimics that of a typicalKondo lattice where the Kondo coherence develops at T ∗ anda heavy Fermi-liquid state forms below T FL [26]. The residualresistivity ρ ( T → ) ∼ Ω · cm indicates that the KM statemay be classified as a “bad metal”.The negligible MR indicates that even if the FS pocket re-sponsible for the quantum oscillations is capable of carry-ing charge in the KM state, it plays a negligible rˆole in ρ (see Methods). Moreover, this FS pocket would only ac-count for 4-5% of the Sommerfeld coefficient obtained frompulsed-field heat-capacity measurements ( γ ≃
63 mJ mol − K − at 55 T [19], see Methods). Therefore, a separate FSof more conventional heavy fermions is required to accountfor the charge-transport properties and the large Sommerfeldcoefficient of the KM state. However, the heavy effectivemass of these fermions (see Methods) and the relatively high T > . A / γ ( A is the T -coefficientof resistivity) is surprisingly large. Using the value of A given by the fit in Fig. 3b, the estimated KW ratio is 1 . × − µΩ cm (K mol/mJ) , three and four orders of magni-tude larger than typical values for heavy-fermion compoundsand transition metals, respectively (Fig. 3c). Such an abnor-mal KW ratio cannot be addressed by the degree of degener-acy of quasiparticles, which tends to suppress the KW ratio inmany Yb-based systems [29, 30].The data in Figs. 1-3 imply an intriguing two-fluid picturein YbB that includes (i) a FS pocket of quasiparticles obey-ing Fermi-Dirac statistics but contributing little to the trans-port of charge; and (ii) more conventional charged fermions.For brevity, we refer to (i) as neutral fermions (NFs). TheseNFs cause quantum oscillations in both KI and KM states,whereas (ii) dominate the electrical transport properties andthe low- T heat capacity in the KM state. Under this two-fluiddescription, the I-M transition produces a sudden increase ofthe density of (ii), which changes from a thermally excitedlow-density electron gas in the KI state to a dense liquid ofheavy, charged quasiparticles in the KM state. This changehas two consequences. First, it dramatically enhances thecharge screening; this will in turn affect the interactions thatcontribute to the renormalization of the effective mass of theNFs [31]. This is the likely cause of the significant fall inmass as H increases through the I-M transition that is sug-gested if one extrapolates the m ∗ versus H plot (Fig. 2b, in-set) back to the phase boundary. Second, on entering the KMstate, the much enhanced number of states available close tothe Fermi energy will act as a “reservoir” into which quasi-particles from the NF FS can scatter or transfer (for analo-gous situations in other materials, see [32, 33]). As H in-creases past the I-M transition, it appears that the NF FS be-comes less energetically favourable; the availability of the KM“reservoir” means that quasiparticles can transfer out, leading to the falling quantum-oscillation frequency parameterized byEq. 3. As the NF FS shrinks, the effective mass increases(Fig. 2b), possibly due to nonparabolicity of the correspond-ing band and/or field-induced modification of the bandwidthor interactions contributing to the effective mass renormaliza-tion [31]. Also, the fact that the quantum oscillations causedby the NF FS are strongly affected by the I-M transition ( i.e., the frequency becomes H dependent in the KM phase) showsthat they are an intrinsic property of YbB , laying to rest sug-gestions that they are caused by a minority phase ( c.f. the Alflux proposal for SmB [9]) or surface effect.The survival of the NFs above the gap closure and their co-existence with charged excitations in a Fermi-liquid-like state[13, 34, 35] leads to an interesting scenario; in such a state,Luttinger’s theorem may be violated and a continuous vari-ation of FS properties is allowed [35]. In addition, the in-teraction of a relatively conventional FS with an ensemble ofquasiparticles unable to transport charge is a feasible prereq-uisite for ρ varying linearly with T (Fig. 3a) [36]. Finally,the scaling behaviour between the magnetization and the B -dependence of orbit area (see Supplementary Information),as well as the large “offset” field H ∗ needed to linearize theLandau diagrams (Fig. 2a), may suggest underlying nontriv-ial magnetic properties [37] in the KM state that await furtherinvestigation. DATA AVAILABILITY
The data that support the plots within this paper and otherfindings of this study are available from the corresponding au-thors upon reasonable request.
CODE AVAILABILITY
The code that support the plots within this paper and otherfindings of this study are available from the corresponding au-thors upon reasonable request.
ACKNOWLEDGEMENTS
We thank Lin Jiao, Liang Fu, Senthil Todadri, Tom Lan-caster, Paul Goddard, Hiroshi Kontani, Hiroaki Shishido andRobert Peters for discussions. This work is supported bythe National Science Foundation under Award No. DMR-1707620 (electrical transport measurements), by the Depart-ment of Energy under Award No. DE-SC0020184 (mag-netization measurements), by the Office of Naval Researchthrough DURIP Award No. N00014-17-1-2357 (instrumenta-tion), and by Grants-in-Aid for Scientific Research (KAK-ENHI) (Nos. JP15H02106, JP18H01177, JP18H01178,JP18H01180, JP18H05227, JP19H00649, JP20H02600 andJP20H05159) and on Innovative Areas “Quantum LiquidCrystals” (No. JP19H05824) from Japan Society for the Pro-motion of Science (JSPS), and JST CREST (JPMJCR19T5).A major portion of this work was performed at the NationalHigh Magnetic Field Laboratory, which is supported by Na-tional Science Foundation Cooperative Agreement No. DMR-1644779 and the Department of Energy (DOE). J.S. and M.J.thank the DOE for support from the BES program “Sciencein 100 T”. The experiment in NHMFL is funded in part bya QuantEmX grant from ICAM and the Gordon and BettyMoore Foundation through Grant GBMF5305 to Dr. Ziji Xi-ang, Tomoya Asaba, Lu Chen, Colin Tinsman, and Dr. Lu Li.We are grateful for the assistance of You Lai, Doan Nguyen,Xiaxin Ding, Vivien Zapf, Laurel Winter, Ross McDonald andJonathan Betts of the NHMFL.
AUTHOR CONTRIBUTIONS
F.I. grew the high-quality single crystalline samples. Z.X.,L.C., K-W.C., C.T., Y.S., T.A., H.L., F.B., J.S. and L.L. per-formed the pulsed field PDO and resistivity measurements.Z.X., T.A. and J.S. performed the pulsed field magnetometrymeasurements. L.C., C.T. and M.J. performed the pulsed fieldmagnetostriction measurements. Z.X., K-W.C., Y.K., Y.M.,J.S. and L.L. analyzed the data. Z.X., Y.M., J.S. and L.L. pre-pared the manuscript.
AUTHOR INFORMATION
The current address of M.J. is Physikalisch-TechnischeBundesanstalt, Braunschweig 38116, Germany. The au-thors declare no competing financial interests. Cor-respondence and requests for materials should be ad-dressed to Z.X.([email protected]), J.S.([email protected])and L.L.([email protected]). [1] Li, G. et al.
Two-dimensional Fermi surfaces in Kondo insulatorSmB . Science , 1208-1212 (2014).[2] Tan, B. S. et al.
Unconventional Fermi surface in an insulatingstate.
Science , 287-290 (2015).[3] Xiang, Z. et al.
Bulk rotational symmetry breaking in Kondoinsulator SmB . Physical Review X , 031054 (2017).[4] Xiang, Z. et al. Quantum oscillations of electrical resistivity inan insulator.
Science , 65-69 (2018).[5] Hartstein, M. et al.
Fermi surface in the absence of a Fermiliquid in the Kondo insulator SmB . Nat. Phys. , 166-172(2018).[6] Sato, Y. et al. Unconventional thermal metallic state of chargeneutral fermions in an insulator.
Nat. Phys. , 954 (2019).[7] Cooley, J. C., Aronson, M. C., Fisk, Z. & Canfield, P. C. SmB :Kondo insulator or exotic metal? Phys. Rev. Lett. , 1629(1995).[8] Kasaya, M., Iga, F., Takigawa, M. & Kasuya, T. Mixed valenceproperties of YbB . J. Magn. Magn. Mater. , 429-435(1985).[9] Thomas, S. M., Ding, X., Ronning, F., Zapf, V., Thompson, J.D., Fisk, Z., Xia, J. & Rosa, P. F. S. Quantum oscillations influx-grown SmB with embedded aluminum. Phys. Rev. Lett. , 166401 (2019).[10] Dzero, M., Sun, K., Galitski, V. & Coleman, P. TopologicalKondo insulators.
Phys. Rev. Lett. , 106408 (2010).[11] Knolle, J. & Cooper, N. R. Excitons in topological Kondo in-sulators: Theory of thermodynamic and transport anomalies inSmB . Phys. Rev. Lett. , 096604 (2017).[12] Erten, O., Chang, P.-Y., Coleman, P. & Tsvelik, A. M. Skyrmeinsulators: Insulators at the brink of superconductivity.
Phys.Rev. Lett. , 057603 (2017).[13] Chowdhury, D., Sodemann, I. & Sentil, T. Mixed-valence in-sulators with neutral Fermi-surfaces.
Nat. Commn. , 1766(2018).[14] Sodemann, I., Chowdhury, D. & Sentil, T. Quantum oscilla-tions in insulators with neutral Fermi surfaces. Phys. Rev. B ,045152 (2018). [15] Shen, H. & Fu, L. Quantum oscillation from in-gap statesand non-Hermitian Landau level problem. Phys. Rev. Lett. ,026403 (2018).[16] Okawa, M., Ishida, Y., Takahashi, M., Shimada, T., Iga, F., Tak-abatake, T., Saitoh, T. & Shin, S. Hybridization gap formationin the Kondo insulator YbB observed using time-resolvedphotoemission spectroscopy. Phys. Rev. B , 161108(R)(2015).[17] Sugiyama, K., Iga, F., Kasaya, M., Kasuya, T. & Date, M. Field-induced metallic state in YbB under high magnetic field. J.Phys. Soc. Jpn. , 3946-3953 (1988).[18] Terashima, T. T., Ikeda, A., Matsuda, Y. H., Kondo, A., Kindo,K. & Iga, F. Magnetization process of the Kondo insulatorYbB in ultrahigh magnetic fields. J. Phys. Soc. Jpn. ,054710 (2017).[19] Terashima, T. T., Matsuda, Y. H., Kohama, Y., Ikeda, A.,Kondo, A., Kindo, K. & Iga, F. Magnetic-field-induced Kondometal realized in YbB . Phys. Rev. Lett. , 257206 (2018).[20] Ohashi, T., Koga, A., Suga, S. & Kawakami, N. Field-inducedphase transitions in a Kondo insulator.
Phys. Rev. B , , 245104(2004).[21] Ghannadzadeh, S., Coak, M., Franke, I., Goddard, P. A., Single-ton, J. & Manson, J. L. Measurement of magnetic susceptibilityin pulsed magnetic fields using a proximity detector oscillator. Rev. Sci. Instrum. , 113902 (2011).[22] Shoenberg, D. Magnetic Oscillations in Metals. (CambridgeUniversity Press, Cambridge, England, 1984).[23] These results are consistent with the “back projection” ap-proach [24] for the measured SdH frequency (SupplementaryInformation). Furthermore, the H -dependence of the slope ofthe nonlinear Landau diagram shown in Fig. 1c scales with theback projection of the M curves, suggesting that the evolutionof the FS may be reflected in M (see Supplementary Informa-tion).[24] van Ruitenbeek, J. M., Verhoef, W. A., Mattocks, P. G., Dixon,A. E., van Deursen, A. P. J. & de Vroomen, A. R. A de Haas-van Alphen study of the field dependence of the Fermi surface in ZrZn , J. Phys. F: Met. Phys. , 2919-2928 (1982).[25] G¨otze, K. et al. , Unusual phase boundary of the magnetic-field-tuned valence transition in CeOs Sb . Phys. Rev. B ,075102 (2020).[26] Yang, Y-F. & Pines, D. Emergent states in heavy-electron ma-terials.
Proc. Natl. Acad. Sci. USA , E3060-E3066 (2012).[27] Jacko, A. C., Fjærestad, J. O. & Powell, B. J. A unified explana-tion of the Kadowaki-Woods ratio in strongly correlated metals.
Nat. Phys. , 422 (2009).[28] Hussey, N. E. Non-generality of the Kadowaki-Woods ratio incorrelated oxides. J. Phys. Soc. Jpn , 1107-1110 (2005).[29] Tsujii, N., Kontani, H., & Yoshimura, K. Universality in heavyfermion systems with general degeneracy. Phys. Rev. Lett. ,057201 (2005).[30] Matsumoto, Y., Kuga, K., Tomita, T., Karaki, Y. & Nakat-suji, S. Anisotropic heavy-Fermi-liquid formation in valence-fluctuating α -YbAlB . Phys. Rev. B , 125126 (2011).[31] Quader, K. F., Bedell, K. S. & Brown, G. E. Strongly interactingfermions. Phys. Rev. B , 156-167 (1987)[32] Harrison, N. et al. , Unconventional quantum fluid at highmagnetic fields in the marginal charge-density-wave system α − (BEDT-TTF) M Hg(SCN) ( M = K and Rb). Phys. Rev. B , 165103 (2004).[33] Singleton, J., Nicholas, R.J., Nasir, F. & Sarkar, C. K. Quantumtransport in accumulation layers on Cd . Hg . Te.
J. Phys. C:Solid State Phys,
35 (1986).[34] Senthil, T., Sachdev, S. & Vojta, M. Fractionalized Fermi liq-uids.
Phys. Rev. Lett. , 216403 (2003).[35] Senthil, T., Vojta, M. & Sachdev, S. Weak magnetism and non-fermi liquids near heavy-fermion critical points. Phys. Rev. B , 035111 (2004).[36] Patel, A. A., McGreevy, J., Arovas, D. P. & Sachdev, S. Magne-totransport in a model of a disordered strange metal. Phys. Rev.X , 021049 (2018).[37] The linear Landau diagrams shown in Fig. 2a with an “offsetfield” H ∗ subtracted from the applied magnetic field bears aqualitative similarity to the composite-fermion interpretation ofthe fractional quantum Hall effect (FQHE) observed in two-dimensional systems [38]. If we consider that this similarity isnot coincidental, then H ∗ can be regarded as an analogue of thegauge field of the composite fermions in the FQHE. In three-dimensional materials, an emergent gauge field, which can befelt by the charge carriers and manifest itself in transverse trans-port measurements, might originate from chiral spin texturesin a skyrmion lattice phase [39]. However, such a topologi-cally nontrivial magnetic structure has not yet been proposedfor YbB .[38] Leadley, D. R., Nicholas, R. J., Foxon, C. T. & Harris, J. J. Mea-surements of the effective mass and scattering times of compos-ite fermions from magnetotransport analysis. Phys. Rev. Lett. , 1906 (1994).[39] Nagaosa, N. Emergent electromagnetism in condensed matter. Proc. Jpn. Acad., Ser. B , 278-289 (2019).[40] Iga, F., Shimizu, N. & Takabatake, T. Single crystal growth andphysical properties of Kondo insulator YbB . J. Magn. Magn.Mater. , 337-338 (1998).[41] Daou, R., Weickert, F., Nicklas, M., Steglich, F., Haase, A. &Doerr, M. High resolution magnetostriction measurements inpulsed magnetic fields using fibre Bragg gratings.
Rev. Sci. In-strum. , 033909 (2010).[42] Jaime, M. et al. Fiber Bragg Grating Dilatometry in ExtremeMagnetic Field and Cryogenic Conditions.
Sensors. , 2572(2017). [43] Mushnikov, N. V. & Goto, T. High-field magnetostriction ofthe valence-fluctuating compound YbInCu . Phys. Rev. B ,054411 (2004).[44] Matsuda, Y. H., Kakita, Y. & Iga, F. The temperature de-pendence of the magnetization process of the Kondo insulatorYbB . Crystals , 26 (2020).[45] Yoshimura, K., Nitta, T., Mekata, M., Shimizu, T., Sakakibara,T., Goto, T. & Kido, G. Anomalous high-field magnetizationand negative forced volume magnetostriction in Yb − x M x Cu ( M = In and Ag) — evidence for valence change in high mag-netic fields. Phys. Rev. Lett. , 851 (1988).[46] Zieglowski, J., H¨afner, H. U. & Wohlleben, D. Volume mag-netostriction of rare-earth metals with unstable 4 f shells. Phys.Rev. Lett. , 193 (1986).[47] Goddard, P. A., Singleton, J., Lima Sharma, A. L., Morosan,E., Blundell, S. J., Bud’ko, S. L. & Canfield, P. C. Separationof energy scales in the kagome antiferromagnet TmAgGe: Amagnetic-fieldorientation study up to 55 T. Phys. Rev. B ,094426 (2007).[48] Goddard, P. A. et al. Experimentally determining the exchangeparameters of quasi-two-dimensional Heisenberg magnets.
NewJ. Phys , 083025 (2008).[49] Altarawneh, M. M., Mielke, C. H. & Brooks, J. S. Proxim-ity detector circuits: An alternative to tunnel diode oscillatorsfor contactless measurements in pulsed magnetic field environ-ments. Rev. Sci. Instrum. , 066104 (2009).[50] We also mention that a hypothetical model of the SdH effect in asystem containing gapless neutral fermions and gapped chargedbosons has been established by Sodemann et al, which predictsLK behavior above a certain T [14]. MethodsSample preparation and pulsed field facilities.
YbB sin-gle crystals were grown by the traveling-solvent floating-zonemethod [40]. The two samples studied in this work were cutfrom the same ingot and shown to have almost identical phys-ical properties, including the SdH oscillations below the I-Mtransition, in our previous investigations [4, 6]. The YbB sample characterized in the magnetostriction and the mag-netization ( M ) measurements corresponds to sample N1 inRef. [4] and sample N3 in Ref. [4], which is alsocrystal M of YbB samples were measuredin a capacitor-driven 65 T pulsed magnet at NHMFL, LosAlamos. In the PDO and MR measurements, fields were pro-vided by 65 T pulsed magnets and a 75 T Duplex magnet.Temperatures down to 500 mK are obtained using a He im-mersion cryostat. In the MR experiment, the sample was im-mersed in liquid He to achieve a more precise T in the range1 . ≤ T ≤ . Magnetostriction measurements.
The linear magnetostric-tion ∆ L / L of YbB was measured using a fibre Bragg grat-ing dilatometry technique [41, 42]. In our setup (ExtendedData Fig. 1), the dilatometer is a 2 mm-long Bragg gratingcontained in a 125 µ m telecom-type optical fibre. The ori-ented YbB single crystal was attached to the section of fibrewith the Bragg grating using a cyanoacrylate adhesive. Thecrystallographic [100] direction was aligned with the fibre,which is also parallel to H . Thus, we measure the longitudinalmagnetostriction along the a -axis of cubic YbB . The mag-netostriction ∆ L / L was extracted from the shift of the Braggwavelength in the reflection spectrum [42]. The signal froman identical Bragg grating on the same fibre with no sampleattached was subtracted as the background.In a paramagnetic metal, the high-field longitudinal mag-netostriction contains both M and M terms [43]. In thissense, the power-law H dependence of ∆ L / L with an expo-nent ≈ . M in YbB at 40 K [17]. As T lowers, ∆ L / L decreases and a nonmonotonic field dependence develops at30 K (Extended Data Fig. 1b). We note that the fast suppres-sion of ∆ L / L coincides with the sharpening of the I-M tran-sition in the derived susceptibility below 30 K [44], suggest-ing an additional energy scale in YbB that is much lowerthan the Kondo temperature T K ≈
240 K and the gap open-ing temperature T g ≈
110 K [16]. Below 5 K, ∆ L / L becomesquite small and a step-like feature is observed with an onsetat µ H = 46.3 T, perfectly aligned with the sudden increasein M (Extended Data Fig. 1c,d). We identify this characteris-tic field as the I-M transition field H I-M at which a metamag-netic transition also happens [18, 19, 44]. A negative volumemagnetostriction is characteristic of mixed-valence Yb com-pounds in which the volume of nonmagnetic Yb + (4 f ) is4.6% smaller than that of magnetic Yb + (4 f ). Thereforea volume decrease with increasing H is expected [45, 46].The step-like decrease at H I-M (Extended Data Fig. 1c) may therefore be evidence that the sudden shrinkage results froma valence transition of the Yb ions. Using a simple isotropicassumption, the change of volume magnetostriction at H I-M is δ ( ∆ V / V ) = δ ( ∆ L / L ) ≃ × − , corresponding to a va-lence increase of 0.00013. Such a small average valence en-hancement implies a quite weak and incomplete breakdownof the Kondo screening. Consequently, the state immediatelyabove H I-M is confirmed to be a KM in which mixed-valencefeatures persist.
Magnetization measurements. M was measured using acompensated-coil susceptometer [47, 48]. The 1.5 mm bore,1.5 mm long, 1500-turn coil is made of 50 gauge high-puritycopper wire. The sample was inserted into a 1.3 mm diameternon-magnetic ampoule that can be moved in and out of thecoil. When pulsed fields are applied, the coil picks up a volt-age signal V ∝ ( d M / d t ) , where t is the time. Numerical inte-gration is used to obtain M and a signal from the empty coilmeasured under identical conditions is subtracted. Pulsed-field M data are calibrated using the M of a YbB sampleof known mass measured in a Quantum Design VSM magne-tometer [6].As shown in Extended Data Fig. 1d, a metamagnetic tran-sition occurs at 46.3 T, coinciding with the onset of the step-like feature in the magnetostriction. This observation furtherconfirms the location of H I-M in our YbB samples. At thehighest H used in this experiment, M ≈ µ B /Yb, so that M contributes only ∼ B . Therefore we can ignore the M term and equate B to the external magnetic field, i.e., B ≈ µ H . Radio frequency measurements of resistivity using thePDO technique.
The PDO circuit [21, 49] permits conve-nient contactless measurements of the resistivity of metallicsamples in pulsed magnetic fields. In our experiments, a 6-8 turn coil made from 46 gauge high-purity copper wire istightly wrapped around the YbB single crystals and securedusing GE varnish. The coil is connected to the PDO, form-ing a driven LC tank circuit with a resonant frequency of22-30 MHz at cryogenic T and H =
0. The output signalis fed to a two-stage mixer/filter heterodyne detection sys-tem [21], with mixer IFs provided by a dual-channel BK-Precision Function/Arbitrary Waveform Generator. The sec-ond mixer IF was 8 MHz, whereas the first mixer IF was ad-justed to bring the final frequency down to ≈ f due to H is written as [4, 21] ∆ f = − a ∆ L − b ∆ R , (4)where a and b are positive constants determined by the fre-quency plus the capacitances, resistances and inductances inthe circuit, L is the coil inductance and R is the resistance ofthe coil wire and cables. In the case of a metallic sample, thecoil inductance L depends on the skin depth λ of the sample.If we assume that the sample magnetic permeability µ andthe coil length stay unchanged during a field pulse, we have ∆ L ∝ ( r − λ ) ∆λ , where r is the sample radius. At angular fre-quency ω , the skin depth is proportional to square root of theresistivity ρ : λ = s ρωµ . (5)Therefore, for a metallic sample, the resonance shift ∆ f re-flects the sample MR and the detected quantum oscillationsare due to the SdH effect. In YbB , the PDO measurementonly detects the signal from the sample in the high-field KMstate, i.e., when H > H I-M [4]. In the low-field KI state thesample is so resistive that the skin depth λ is larger than thesample radius r . As a result, ∆ f mainly comes from the MRof the copper coil [21]. The “dip” in PDO f in Fig. 1a con-sequently indicates where the skin depth is comparable to thesample radius and provides an alternative means to find H I − M .We note that the H I-M assigned to onset of the “dip” feature(see Extended Data Table 1) is ≈ . Onsager relationship for a field-dependent Fermi surface.
The Onsager relation [22] relates the frequency F of quantumoscillations to the FS extremal orbit area A : F = ¯ h π e A . Text-book derivations [22] invoke the Correspondence Principle togive an orbit-area quantization condition ( N + λ ) π eB ¯ h = A ,where N is a quantum number and λ is a phase factor. Thederivation makes no assumptions about A being constant, sofor a field-dependent A = A ( B ) , we can write ( N + λ ) π eB N ¯ h = A ( B N ) , (6)where B N is the magnetic induction at which the N th oscilla-tion feature (peak, valley, etc. ) occurs. Evaluating Eq. 6 for N and N + A ( B N + ) B N + − A ( B N ) B N = π e ¯ h . (7)From an experimental standpoint, in materials where F is con-stant, a particular feature of a quantum oscillation is observedwhenever N + λ ′ = FB ; λ ′ is another phase factor. Allowing F to vary ( i.e., F = F ( B ) ), evaluating the expression for N and N + F ( B N + ) B N + − F ( B N ) B N = . (8)A comparison of Eqs. 7 and 8 shows that the Onsager relationstill holds, i.e., F ( B ) = ¯ h π e A ( B ) .In Fig. 2a, B ∗ = µ H ∗ = 41.6 T is subtracted from the ap-plied field to yield linear Landau-index diagrams for θ ≤ ◦ . The resulting fit is described by N + λ = F ( B N − B ∗ ) (Eq. 2), which can be written in terms of a B -dependent fre-quency N + λ = F ( B N ) B N , where F ( B ) = F B ( B − B ∗ ) (Eq. 3). Fol-lowing the reasoning given above, this is associated with a B -dependent extremal area A ( B ) = A B − B ∗ B , with A = π e ¯ h F . (9) In the Supplementary Information, we show that the anal-ysis presented here is functionally equivalent to the “back-projection” method [23]. Note that we have not consideredthe Zeeman splitting of the peaks, so that Eqs. 3 and 9 describethe average of the spin-up and spin-down components. Resistivity measurements in the KM state.
From µ H = decreases by five ordersof magnitude [4]. As a result, MR measurements in pulsedfields are challenging. If a constant current is used duringthe entire field pulse, either the signal-to-noise ratio is poorin the KM state, or the high resistance in the KI state causesserious Joule heating. To solve this issue, we developed apulsed-current technique. The experimental setup is shownin Extended Data Fig. 3a. A ZFSWHA-1-20 isolation switchis used to apply current pulses with widths <
10 ms. Theswitch is controlled by two square wave pulses generated bya BK-Precision Model 4065 dual-channel function/arbitrarywaveform generator triggered by the magnet pulse. Thus, arelatively large electric current (2-3 mA) can pass through thesample during a narrow time window within the field pulse(Fig. 3a, inset).Current is applied only when YbB enters the KM state. Inour low- T MR measurements, the switch turns on at 47 T dur-ing the upsweeps and turns off at 47 T during the downsweeps.To reduce heating due to eddy currents, we measured thelongitudinal MR of a needle-shaped sample (length 6.5 mmbetween voltage leads; cross-sectional area 0.33 mm ). Asshown in Extended Data Fig. 3b, the downsweeps still suf-fer some Joule heating. Hence, in the main text we focus onthe upsweeps, which show very weak longitudinal MR andshould reflect the intrinsic electrical transport properties ofthe KM state of YbB . To further dissipate heat, we mea-sured the pulsed field MR with the sample in liquid He for1 . ≤ T ≤ . He sometimes gives a poorthermal impedance on the sample surface. For this reason, thelinear fit in Fig. 3b only uses data with the sample in liquid He.According to Eq. 9, A ( B ) for the pocket detected in the PDOmeasurement shrinks by ≃
45% from 50 T to 60 T. Assum-ing a spherical FS, this corresponds to an ≃
60% reductionin quasiparticle density n . Meanwhile, the cyclotron mass in-creases by ≃
60% (Fig. 2b, inset). Consequently, a textbookDrude expression ρ = m ∗ / ne τ ( τ is the relaxation time) pre-dicts that the resistivity would increase by a factor ≈ T -linear resistivityof the KM state between 4 K and 9 K are in agreement withthe predicted behaviour of a marginal Fermi liquid (withoutmacroscopic disorder) or an incoherent metal, potential evi-dence for the coexistence of fermions inactive in the chargetransport and more conventional charged fermions [36] . The Kadowaki-Woods ratio.
For the Sommerfeld coeffi-cient γ in the KM state of YbB , a pulsed-field heat ca-pacity study reports values of 58 mJ mol − K − and 67 mJmol − K − at 49 T and 60 T, respectively [19]. A linear in-terpolation gives γ = 63 mJ mol − K − at 55 T. Since γ =( π k /3) N ( E F ) and the density of states at the Fermi energy N ( E F ) = ( m ∗ / π ¯ h )(3 π n ) / , the Sommerfeld coefficient canbe written as: γ = π k B m ∗ k F π ¯ h , (10)where k F is the Fermi vector. As for the FS pocket de-tected in the SdH measurement, Eq. 3 gives F (55 T) = 231.9 T.In a spherical FS model, F = ¯ hk F / e , n = k F /3 π = ( eF / ¯ h ) / /3 π ( k F is the Fermi wavevector), therefore F =231.9 T corresponds to n = 1.99 × cm − . Also, for 55 T, m ∗ ≃ m e , based on Fig. 2b (inset). Putting these parametersinto Eq. 10, we estimate that the FS pocket responsible forthe SdH effect could contribute only 4.4% of the measured γ .Consequently, an additional band with a much larger densityof states must exist in the KM state of YbB .The unusually large KW ratio (Fig. 3c) suggests the some-what unusual nature of the heavy quasiparticles that domi-nate the charge transport and thermal properties of the KMstate. Analysis of the KW ratio shows that for a single-band,strongly correlated system, the value of A / γ does not de-pend on the strength of correlations but is instead solely deter-mined by the underlying band structure [27]. We consider thesimplest case of a single-band isotropic Fermi-liquid model.Using the same calculations as in Ref. [27], we have: A = π k B e ¯ h ( m ∗ ) k F . (11)Taken together, Eqs. 10 and 11 yield k F = .
126 nm − (corre-sponding to n = . × cm − ) and a rather large effectivemass of m ∗ = . m e . Such a heavy mass is unusual for Yb-based mixed-valence compounds, but explains the anomalousKW ratio as well as the absence of SdH oscillations due tothese quasiparticles in the PDO response. SdH oscillations due to charge-neutral quasiparticles.
Magnetic quantum oscillations are observed in both M and ρ in the KI phase of YbB [4]. The size of the FS and the effec-tive mass of the quasiparticles inferred from the oscillationsare consistent with the fermion-like contribution to the ther-mal conductivity [6]. A natural explanation, given the highelectrical resistivity, is that the thermal conductivity and quan-tum oscillations are due to charge-neutral fermions.Of the two oscillatory effects observed in the KI phase, thedHvA effect is the more fundamental. As pointed out by Lif-shitz, Landau and others [22] , it involves oscillations in athermodynamic function of state - M - that may be related tothe electronic density of states with a minimum of assump-tions. The fact that M in the KI phase oscillates as a functionof H can only be due to the oscillation of the fermion densityof states, and consequently their free energy.Even in conventional metals, quantum oscillations in ρ areharder to model quantitatively. A starting point was suggestedby Pippard [22]; the rate at which quasiparticles scatter willdepend on the density of states available via Fermi’s Golden Rule . Hence, if the quasiparticle density of states oscillates asa function of H , ρ will also oscillate proportionally, leadingto the SdH effect.Before modifying this idea to tackle the SdH effect in the KIphase, we remark that the H -dependent frequency of the os-cillations in the KM phase suggests that exchange of fermionsbetween charge-neutral and conventional FS sections occursreadily, probably via low-energy scattering. This is also sup-ported by the T -linear resistivity [36]. The rate at which thisscattering occurs will obviously reflect the joint density offermion states.Returning to the KI phase of YbB , ρ is thought to bedue to charge carriers thermally excited across the energy gap,plus contributions from states in the gap that lead to the ρ sat-uration at low T . Following the precedent of the KM phase, itis likely that fermions in the KI phase scatter back and forthbetween the charge-neutral states and the more conventionalbands. The situation is more complicated than Pippard’s origi-nal concept because scattering between two bands is involved.Nevertheless, the amount of scattering will be determined bythe joint density of states, and, because the density of states ofthe neutral quasiparticles oscillates with H , so will ρ . Unlikeconventional metals, the density of the charge carriers will be T -dependent, so the amplitude of the SdH oscillations in theKI phase will follow a different T dependence from that of thedHvA oscillations, as observed in experiments [4, 50].As described by Eq. 4, the PDO f in the KM phase is de-termined by the skin depth, i.e., the conductivity. How then,can oscillations due to the neutral fermions be manifested inthis signal? As described above, fermions will scatter backand forth between the charge-neutral states and more conven-tional bands; in the KM phase, our conductivity results and theheat capacity data of others suggests that the latter is a FS ofheavy charged fermions. Once again, scattering between twobands will occur, and the SdH effect in conductivity caused bythe oscillatory density of states of the neutral quasiparticles inmagnetic fields is detected in the PDO experiments. By con-trast, any intrinsic quantum oscillations due to the metallic FSwill be suppressed by a combination of the very heavy effec-tive mass and the relatively high T ( & .
30 40 50 601.921.962.002.042.08 [010] -0.2 10.9 15.5 20.7 25.7 34.6 45.5 f ( M H z ) H (T) [100]
H T = 0.6 K ac Kondo metal N valley peak (+) peak (-) H (T -1 ) N I-M transition
Kondo insulator
F = 697.8 10.7 T H (T -1 ) ( m c m ) + + + + - - - - b f ( k H z ) H (T -1 ) = -0.210.915.520.7 - + d
50 T 52 T 54 T 57 T S d H a m p li t ude f ( k H z ) T (K) = 10.9
50 52 54 56 580.50.60.70.80.9 E c * ( m e V ) H (T)
Fig. 1 . The SdH effect in YbB . a, The PDO frequency f for aYbB single-crystal sample, measured at T = 0.6 K at various tiltangles θ up to 62.5 T. The “dip” feature in f corresponds to the I-M transition which shifts to higher field at larger tilt angles. Solidlines and short-dashed lines are upsweeps and downsweeps, respec-tively. Inset: a photograph of the device we used in pulsed-fieldPDO experiments. The whole device was attached to a rotation stagewith the rotation axis normal to the (001)-plane. The tilt angle θ is defined as the angle between the field vector H and the crystallo-graphic [100] direction. b, Oscillatory component of the PDO fre-quency, ∆ f , obtained after a 4th-order polynomial background sub-traction from the raw data shown in a for different tilt angles from θ = . ◦ to θ = . ◦ . The numbers beside the SdH peaks are theLandau level indices. The signs “+” and “-” mark the spin-split Lan-dau sublevels with inferred Zeeman shift + g µ B B /2 (spin-down) and- g µ B B /2 (spin-up), respectively. The SdH effect is weaker on thedownsweeps (short-dashed lines), probably due to sample heating. c, Landau-level plots for YbB in the low-field KI state (blue dia-monds) and the high-field KM state (red circles), both under a mag-netic field along the [100] direction. Data points in the orange dashedcircle were taken in the 75 T Duplex magnet (Extended Data Fig. 2).The gray vertical bar denotes the I-M transition. The inset showsthe SdH oscillations in the KI state of YbB . d, T dependence of ∆ f at different fields. Data was taken at θ = 10.9 ◦ (Extended DataFig. 2b). Solid lines are the Lifshitz-Kosevich (LK) fits. Inset: thefield dependence of the fitted cyclotron energy E ∗ c . F KM ( H I-M -0.8 T) F KM ( H I-M ) F ( T ) | | ( ) dHvA in the KI state b
30 40 50 60 70100200300400500600700 F KI a KM F ( T ) H (T) H || [100] KI m* /m F KM
50 52 54 56 587891011 = 10.9 m * / m H (T) c H * = 41.6 T = 0.2 = 5.2 = 10.9 = 15.5 = 20.7 N
1/ ( H - H *) (T -1 ) Fig. 2 . The field-dependent Fermi surface in the metallic state.a,
The nonlinear Landau level plots shown in Fig. 1c become linearafter adding an offset of µ H = 41.6 T to the applied magnetic field.The linear fits yield slopes which are denoted by the parameter F in Eq. 3. The angular dependence of F is summarized in ExtendedData Table 1. The Landau diagrams are shifted vertically by index N = 1 for clarity. b, With field applied along the [100] direction,the quantum oscillations in the KI state exhibits a field-independentfrequency F KI , whereas in the KM state the SdH frequency F KM isdescribed by Eq. 3. Solid lines and short-dashed lines represent thefield range in which SdH oscillations are detected and absent, re-spectively. The light gray vertical bar marks the I-M transition. In-set: The field dependence of the cyclotron masses m ∗ inferred fromthe cyclotron energy E ∗ c shown in Fig. 1d. c, Angle dependence ofquantum oscillation frequencies in YbB . Blue circles are dHvAfrequencies in the KI state measured by torque magnetometry [4].Magenta and red diamonds are SdH frequencies F KM calculated us-ing Eq. 3 using the transition field µ H I-M (Extended Data Table 1)and B = µ H I-M -0.8 T, respectively. - - - - - - c YbNi B CYbNi Ge YbCu Si YbRh Si YbCu YbAg x Cu YbAgCu YbInAu YbInCu YbAlCu YbAl YbAl CeSn CePd CeNi Si UIn USn UPtUPt UAl UPt CeRu Si CeB CeCu Si CeCu CeAl V O La Sr CuO LiV O Na CoO Sr Ru O Sr RuO A / = 1 × -5 Transition MetalsCe- and U-basedHeavy FermionsYb-Compounds d -electron Oxides CeNi Pd NiPtFeRe A ( µ Ω c m K - ) (mJ mol -2 K -4 ) Os A / = 0.4 × -6 YbB ( µ H = 55 T) UBe α -YbAlB || a ( m Ω c m ) T (K) µ H = 55 T H || I || [100] ( T ) = + A TA = 0.16 0.01m Ω cm K -1 T * = 14 K I ( m A ) µ H ( T ) t (ms) MagneticFieldPulseCurrentPulse47T µ H = 55 T H || I || [100] He cryostat He cryostat b ( µ Ω c m ) ( T ) = + A T = 375 10 µΩ cm A = 61 4 µΩ cm K -2 T (K ) T FL = 2.2 K Fig. 3 . The temperature dependence of the resistivity in themetallic state. a,
The resistivity of YbB sample at µ H = 55 Tplotted as a function of T . Both the current and the magnetic fieldwere applied along the [100] direction. Blue and green hollow sym-bols are data measured using the pulsed current technique (see Meth-ods) in He liquid or gas and He liquid, respectively. The solidsymbols are data taken with a constant excitation in a He cryostat.A maximum in ρ appears at T ∗ =
14 K. The dashed line is a lin-ear fit of ρ ( T ) from 4 K to 9 K. The inset illustrates the magneticfield pulse and current pulse in our measurement in the time domain. b, The same data plotted against T . The dotted line is a linear fitto the He-liquid data, showing the behaviour of the T -dependencebelow T FL = c, The deviation of YbB from the Kadowaki-Woods relation. We use the value of the Sommerfeld coefficient γ reported for YbB in ref. [19]. The data points for transition met-als, dd