f-electron hybridised metallic Fermi surface in magnetic field-induced metallic YbB_{12}
H. Liu, A. J. Hickey, M. Hartstein, A. J. Davies, A. G. Eaton, T. Elvin, E. Polyakov, T. H. Vu, V. Wichitwechkarn, T. Förster, J. Wosnitza, T. P. Murphy, N. Shitsevalova, M. D. Johannes, M. Ciomaga Hatnean, G. Balakrishnan, G. G. Lonzarich, Suchitra E. Sebastian
𝑓𝑓 -electron hybridised metallic Fermi surface inmagnetic field-induced metallic YbB H. Liu, † A. J. Hickey, † M. Hartstein, A. J. Davies, A. G. Eaton, T. Elvin, E. Polyakov, T. H. Vu, V. Wichitwechkarn, T. F ¨orster, J. Wosnitza, , T. P. Murphy, N. Shitsevalova, M. D. Johannes, M. Ciomaga Hatnean, G. Balakrishnan, G. G. Lonzarich, Suchitra E. Sebastian ∗ Cavendish Laboratory, University of Cambridge,JJ Thomson Avenue, Cambridge, CB3 0HE, UK. Dresden High Magnetic Field Laboratory (HLD-EMFL)and W¨urzburg-Dresden Cluster of Excellence ct.qmat,Helmholtz Zentrum Dresden Rossendorf,Bautzner Landstrasse 400, Dresden, 01328, Germany. Institut f¨ur Festk¨orper- und Materialphysik,Technische Universit¨at Dresden, Dresden, 01062, Germany. National High Magnetic Field Laboratory, Tallahassee, Florida, 32310, USA. The National Academy of Sciences of Ukraine, Kiev, 03680, Ukraine. Center for Computational Materials Science, Naval Research Laboratory,Washington, DC, 20375, USA. Department of Physics, University of Warwick, Coventry, CV4 7AL, United Kingdom. ∗ To whom correspondence should be addressed: [email protected]. † These authors contributed equally to this work.1 a r X i v : . [ c ond - m a t . s t r- e l ] F e b he nature of the Fermi surface observed in the recently discovered familyof unconventional insulators starting with SmB and subsequently YbB is asubject of intense inquiry. Here we shed light on this question by comparingquantum oscillations between the high magnetic field-induced metallic regimein YbB and the unconventional insulating regime. In the field-induced metal-lic regime beyond 47 T, we find prominent quantum oscillations in the contact-less resistivity characterised by multiple frequencies up to at least 3000 T andheavy effective masses up to at least 17 𝑚 e , characteristic of an 𝑓 -electron hy-bridised metallic Fermi surface. The growth of quantum oscillation amplitudeat low temperatures in electrical transport and magnetic torque in insulatingYbB is closely similar to the Lifshitz-Kosevich low temperature growth ofquantum oscillation amplitude in field-induced metallic YbB , pointing to anorigin of quantum oscillations in insulating YbB from in-gap neutral low en-ergy excitations. The field-induced metallic regime of YbB is characterisedby more Fermi surface sheets of heavy quasiparticle effective mass that emergein addition to the heavy Fermi surface sheets yielding multiple quantum os-cillation frequencies below 1000 T observed in both insulating and metallicregimes. We thus observe a heavy multi-component Fermi surface in which 𝑓 -electron hybridisation persists from the unconventional insulating to the field-induced metallic regime of YbB , which is in distinct contrast to the unhy-bridised conduction electron Fermi surface observed in the case of the uncon-ventional insulator SmB . Our findings require a different theoretical modelof neutral in-gap low energy excitations in which the 𝑓 -electron hybridisationis retained in the case of the unconventional insulator YbB . ntroduction The origin of bulk quantum oscillations in bulk insulating unconventional insulators, first dis-covered in SmB [1], has been the subject of much debate [1–8]. Another recently discoveredunconventional insulator is the Kondo insulator YbB [4, 6], in which high magnetic fieldsdramatically reduce the electrical resistivity, causing the metallic ground state to be realisedbeyond 𝜇 𝐻 ≈
47 T [9, 10]. Quantum oscillation measurements in metallic YbB accessed inhigh magnetic fields thus uniquely enable us to make a comparison between quantum oscilla-tions in the unconventional insulating state and the field-induced metallic state.In this paper, we experimentally compare quantum oscillations in the unconventional insu-lating regime and the field-induced metallic regime of YbB accessed through high appliedmagnetic fields up to 68 T. In the field-induced metallic phase of YbB , we observe promi-nent quantum oscillations with a multiplicity of frequencies characterised by moderately heavyquasiparticle effective masses, which reflect an 𝑓 -electron hybridised metallic Fermi surface.In order to reliably extract information from the complex quantum oscillation spectrum com-prising multiple frequencies, we focus on (i) a comparison of the multiple quantum oscillationfrequencies observed in both magnetic torque and electrical resistivity of the unconventionalinsulating regime [4] and contactless resistivity of the field-induced metallic regime, (ii) thetemperature dependent quantum oscillation amplitude that can be used to distinguish betweengapped and gapless Fermi surface models in the unconventional insulating regime, and (iii)consequently shed light on the nature of hybridisation in the unconventional insulating andfield-induced metallic regimes. 3 Resonant frequency (MHz) (cid:1) H ( T ) T = 0 . 6 0 K (cid:1) ~ [ 0 0 1 ] ba D f (kHz) (cid:1) H ( 1 / T ) (cid:1) H ( T ) Fig. 1. Quantum oscillations in the field-induced metallic phase of YbB . Elec-trical resistivity of a single crystal of YbB measured with the contactless proximitydetector oscillator (PDO) technique. (a) PDO resonant frequency as a function of ap-plied magnetic field up to 68 T at a temperature of 0.60 K with the field aligned close tothe [001] crystallographic direction. The sharp change in resonant frequency at 𝜇 𝐻 ≈
47 T indicates the onset of the insulator-metal transition [9, 10]. Prominent oscillationscan be seen against the unsubtracted background above 𝜇 𝐻 ≈
50 T. (b)
Solid linesshow quantum oscillations measured with PDO up to 68 T with a smooth monotonicbackground subtracted at different temperatures, where Δ 𝑓 is the change in resonantfrequency after background subtraction. Dashed lines show Lifshitz-Kosevich simula-tions of quantum oscillations using multiple frequency components as listed in Table 1. Results
Figure 1a shows quantum oscillations in the contactless electrical resistivity of a single crystalof YbB measured using the proximity detector oscillatory (PDO) technique, at high magneticfields above the insulator-metal transition at 𝜇 𝐻 ≈
47 T [10]. Prominent quantum oscillationsare visible in the measured contactless electrical resistivity before background subtraction. Fig-ure 1b shows quantum oscillations after smooth, monotonic backgrounds have been subtractedfrom the contactless electrical resistivity (measured by the resonant frequency) above 50 T4
FFT amplitude (a.u.)
F r e q u e n c y ( k T ) (cid:2) ~ [ 0 0 1 ]5 0 T < (cid:1) H < 6 8 T a b F = 2 4 0 0 ( 1 0 0 ) T m e f f = 1 7 ( 3 ) m e F = 5 0 0 ( 2 0 0 ) T m e f f = 8 . 5 ( 1 ) m e F = 8 0 0 ( 2 0 0 ) T m e f f = 9 . 2 ( 2 ) m e FFT amplitude (a.u.) T ( K ) F = 1 3 0 0 ( 3 0 0 ) T m e f f = 1 2 . 1 ( 3 ) m e Fig. 2. Rich spectrum of multiple quantum oscillation frequencies in the field-induced metallic phase of YbB . (a) Fourier transforms of the subtracted quantumoscillations shown in Fig. 1b, for a field window of T < 𝜇 𝐻 < T at temper-atures between 0.6 – 2.0 K, where the applied field was aligned close to the [001]crystallographic direction. The horizontal dashed line indicates the FFT noise floor.Multiple distinct quantum oscillation frequency peaks between 500(200) T and at least3000(200) T are visible. (b)
Quantum oscillation amplitude obtained from the Fouriertransform peak height shown in (a) as a function of temperature for multiple representa-tive frequencies. Lifshitz-Kosevich temperature dependence fits, as shown by the solidlines, yield quasiparticle effective masses 𝑚 ∗ / 𝑚 e between 8.5(1) – 17(3) for the variousfrequencies. A summary of the multiple observed frequencies and their correspondingeffective masses is shown in Table 1. at various temperatures, where the quantum oscillation periodicity in inverse magnetic fieldcan be seen. Multiple frequency peaks between 500(200) T – 3000(200) T are revealed byFast Fourier Transforms (FFT) of the background-subtracted quantum oscillations, as shownin Fig. 2a. Plotting the quantum oscillation amplitude as a function of temperature down to0.6 K yields a Lifshitz-Kosevich (LK) temperature dependence with cyclotron effective masses 𝑚 ∗ / 𝑚 e between 8.5(1) – 17(3), as shown in Fig. 2b.5 iscussion Figure 3 shows multiple quantum oscillation frequencies in the insulating phase of YbB mea-sured through capacitive magnetic torque and contacted electrical transport [4]. Fig. 2a showsthe quantum oscillation frequency spectrum in the field-induced metallic phase, comprisingmultiple frequencies extending up to at least 3000(200) T. We note that even higher frequenciesmay exist, especially in view of the high value of linear specific heat 𝛾 ≈ mJ mol − K − measured in the field-induced metallic regime of YbB [11]. Multiple comparable quantumoscillation frequencies between approximately 300 – 800 T are measured in both the metallicand insulating phases (Table 1); while multiple frequencies were previously reported in the in-sulating phase of YbB measured through capacitive magnetic torque [4], these were missedin other reports of a single quantum oscillation frequency in the insulating phase of YbB [6].Such a quantum oscillation spectrum comprising multiple frequencies is expected from numer-ical Fermi surface simulations of metallic YbB involving hybridised 𝑓 -electrons [4]. In thesetheoretical simulations of the Fermi surface, multiple Fermi surface pockets located away fromthe centre of the Brillouin zone would be expected to yield a series of frequency branches; multi-ple frequencies would further be expected from a multiplicity of electron and hole pockets. Themultiple quantum oscillation frequencies experimentally observed in both the magnetic torqueand electrical resistivity of the unconventional insulating phase and the contactless resistivityof the field-induced metallic phase yield a complex quantum oscillation spectrum that unfortu-nately cannot be treated by analysis methods such as Landau level indexing in inverse magneticfield, complete mapping of the Fermi surface geometry, and other traditional Fermi surfacetreatments. In this work, therefore, we focus instead on a robust treatment involving com-parison between the multiple quantum oscillation frequencies and the temperature-dependenceof the quantum oscillation amplitudes observed in both the unconventional insulating [4] and6 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 501230 . 0 0 . 5 1 . 0 1 . 5 2 . 0 2 . 501234 FFT amplitude (a.u.)
F r e q u e n c y ( k T ) T = 0 . 3 K (cid:2) = 3 . 5 (cid:176) — [ 0 0 1 ]3 5 T < (cid:1) H < 4 5 T b T r a n s p o r t
FFT amplitude (a.u.)
F r e q u e n c y ( k T ) T = 0 . 3 K (cid:2) ~ [ 0 0 1 ]3 5 T < (cid:1) H < 4 5 T a T o r q u e
Fig. 3. FFT of quantum oscillations in insulating YbB . FFT of quantum oscil-lations measured on insulating YbB at a temperature of 0.3 K and field range of35 T < 𝜇 𝐻 <
45 T in (a) magnetic torque, with the applied field aligned close tothe [001] crystallographic direction, and (b) electrical resistivity, with the applied fieldaligned 3.5 ◦ away from the [001] crystallographic direction in the [001]-[111]-[110] ro-tation plane. The horizontal dashed lines indicate the FFT noise floors. Similar fre-quencies can be discerned in the two measured physical quantities. A summary ofthe multiple observed frequencies and their respective quasiparticle effective massesis shown in Table 1. An FFT decomposition involving LK simulations identifies a fre-quency ≈
450 T in electrical resistivity that is visible as a shoulder of the main peak. field-induced metallic regimes.Broad classes of models that have been proposed to explain bulk quantum oscillations inunconventional insulators include categories of gapped models, and models characterised byin-gap low energy excitations [12–23]. An analysis of the temperature-dependence of the quan-tum oscillation amplitude provides us with vital information to distinguish between classes ofgapped and gapless models to describe quantum oscillations in the unconventional insulating7hase.At the simplest level of weakly interacting gapped systems, the system is characterised bya single particle gap. Models in this category have for example been proposed for BCS su-perconductors [12] and for weakly interacting insulators [13, 14]. Examples of such behaviourare experimentally observed in unconventional superconductors [24–27]. For this category ofgapped models of quantum oscillations in weakly interacting insulators, the quantum oscillationamplitude exhibits a non-LK flattening or decrease at low temperatures [12, 13] (Supplemen-tary Information, Figure 4a lower inset). Other models of weakly interacting gapped systemsinvoke quantum oscillations arising from modulation of the gap resulting from an inverted bandstructure [15–17].This picture is modified in the case of strongly correlated insulators. These insulators,driven by strong interactions, are expected to be characterised by an in-gap density of states,as predicted by various theoretical models. For instance, models of single-band Mott insu-lators [18, 19] involve low energy excitations of chiefly spin character. Models of Majoranafermions proposed for Kondo insulators include those in refs. [20–22]. In these models, lowenergy excitations involve Majorana fermion bands, that can be a linear equal combination ofa canonical particle and anti-particle operators, crossing the chemical potential. Another modelhas been proposed for quantum oscillations from composite fermionic excitons in Kondo insu-lators [23]. In this case, mixed-valence insulators are proposed to host a fractionalised neutralFermi sea, which develops an emergent magnetic field in the presence of a physical magneticfield. In these various models, the Fermi surface may be expected to correspond to the unhy-bridised conduction electron band. Further, the quantum oscillation amplitude in these gaplessmodels is expected to increase at low temperatures, for instance obeying an LK form in the caseof low energy excitations characterised by Fermi-Dirac statistics (Supplementary Information,Figure 4a lower inset). 8igure 2 shows the quantum oscillation amplitude as a function of temperature in field-induced metallic YbB , growing in accordance to the LK form down to the lowest measuredtemperatures, as expected for a metal characterised by Fermi-Dirac statistics. We obtain the cy-clotron effective mass for multiple quantum oscillation frequencies in the field-induced metallicphase of YbB from an LK fit to the quantum oscillation amplitude as a function of tempera-ture (Fig. 2b). Table 1 shows a range of moderately high effective masses 𝑚 ∗ / 𝑚 e up to at least17(3) observed for multiple quantum oscillation frequencies up to at least 3000(200) T. Theheavy effective masses observed in the field-induced metallic phase indicate its correspondenceto an 𝑓 -electron hybridised metallic Fermi surface [11].The presence of neutral low-energy excitations in the gap would be expected to yield anincrease in quantum oscillation amplitude at low temperatures in strongly correlated models,which distinguishes them from gapped models of quantum oscillations in weakly interactinginsulators in which the quantum oscillation amplitude is expected to exhibit non-LK flatteningor decrease at low temperatures [12, 13] (Supplementary Information, Figure 4a lower inset).Figure 4 shows the temperature dependence of quantum oscillation amplitude for multiple rep-resentative frequencies in magnetic torque and electrical transport measured in the insulatingphase of YbB [4]. Similar to our observation in the metallic phase, the quantum oscillationamplitude of both magnetic torque and electrical resistivity in the insulating phase grows inaccordance with the LK form down to the lowest measured temperatures, below the gap tem-perature beneath which gapped models of quantum oscillations predict a non-LK flattening ordecrease in amplitude [12, 13]. LK fits to the quantum oscillation amplitude as a function oftemperature of quantum oscillation frequencies between 300 T and 800 T observed in the in-sulating phase yield moderately heavy effective masses 𝑚 ∗ / 𝑚 e between approximately 4.5 –9, which are similar to the effective masses observed in the field-induced metallic phase fora similar range of quantum oscillation frequencies (Table 1). The growth in quantum oscilla-9 b FFT amplitude (a.u.) T ( K ) (cid:2) = 3 . 5 (cid:176) — [ 0 0 1 ]3 5 T < (cid:1) H < 4 5 T F = e f f = 7 . 9 ( 8 ) m e T r a n s p o r t (cid:2) ~ [ 0 0 1 ]2 5 T < (cid:1) H < 4 5 T F = 7 0 0 ( 9 0 ) T m e f f = 7 ( 2 ) m e FFT amplitude (a.u.) T ( K ) a T o r q u e ( A A ( T )) / A X ( A A ( T )) / A X ( A A ( T )) / A X d = 0 d = 2 d = 4 d = 1 0 Fig. 4. Gapless low energy excitations yield low temperature Lifshitz-Kosevichgrowth of quantum oscillation amplitude in the insulating phase of YbB . (a) Am-plitude of the 700 T frequency quantum oscillations measured using cantilever torquemagnetometry as a function of temperature, with the applied field aligned close tothe [001] crystallographic direction. Measured quantum oscillation amplitude followsthe Lifshitz-Kosevich (LK) form (dashed line) down to lowest measured temperatures.(Lower inset) Low temperature model expansion of quantum oscillation amplitude fromrefs. [12, 13] shows non-LK activated behaviour for various finite gap sizes ( 𝛿 ≈ for YbB at 40 T [9]), in contrast to LK exponential growth expected for gapless lowenergy excitations ( 𝛿 = ). Model calculations are shown in Supplementary Informa-tion. 𝐴 ( 𝑇 ) is the quantum oscillation amplitude at temperature 𝑇 , 𝐴 is the amplitude atthe lowest measured temperature, 𝑋 = 𝜋 𝑘 B 𝑇 / ℏ 𝜔 c is the temperature damping coef-ficient in the LK formula [28], 𝛿 = 𝜋 Δ / ℏ 𝜔 c where Δ is the isotropic gap size and 𝜔 c isthe cyclotron frequency. (Upper inset) Growth of magnetic torque quantum oscillationamplitude at the lowest measured temperatures; experimental data (solid triangles) ex-hibits good agreement with gapless model simulation (dashed lines). (b) Amplitude ofthe 800 T frequency quantum oscillations measured using four-point contacted electri-cal transport as a function of temperature, with the applied field aligned 3.5 ◦ away fromthe [001] crystallographic direction in the [001]-[111]-[110] rotation plane. Measuredquantum oscillation amplitude follows LK form (dashed line) down to lowest measuredtemperatures. (Inset) Growth of electrical transport quantum oscillation amplitude atlowest temperatures; experimental data (solid circles) exhibits good agreement withgapless model simulation (dashed line). 𝑚 e ) Frequency(T) Mass( 𝑚 e ) Frequency(T) Mass( 𝑚 e )dHvA ( 𝜃 ∼ [ ] ) SdH ( 𝜃 = . ◦ ∠ [001]) PDO ( 𝜃 ∼ [ ] )150(90) 3.2(2)300(70) 4.5(5) 450(80) 6.1(6) 500(200) 8.5(1)700(90) 7(2) 800(90) 7.9(8) 800(200) 9.2(2)1300(200) 12.1(3)1700(200) 16(5)2300(200) 17(3)3000(200) 14(3) Table 1: Observed multiple quantum oscillation frequencies and effectivemasses in the insulating and metallic phases of YbB . Multiple quantum oscil-lation frequencies and cyclotron effective masses measured with capacitive torquemagnetisation (de Haas-van Alphen (dHvA) oscillations) and four-point contacted re-sistivity (Shubnikov-de Haas (SdH) oscillations) in the insulating phase of YbB , andwith proximity detector oscillator (PDO) contactless electrical transport in the magneticfield-induced metallic phase of YbB . The applied magnetic field was aligned close tothe [001] crystallographic direction for dHvA and PDO measurements, and was aligned3.5 ◦ from the [001] crystallographic direction in the [001]-[111]-[110] rotational planefor the SdH measurements. The FFT field range was T < 𝜇 𝐻 < T for PDO, T < 𝜇 𝐻 < T for the 700 T frequency in dHvA and 800 T frequency in SdH, and T < 𝜇 𝐻 < T for other frequencies in dHvA and SdH. tion amplitude down to the lowest measured temperatures is clearly evidenced in the two upperinsets in Fig. 4, which highlight low temperature growth of the torque and transport quantumoscillation amplitude measured in the insulating phase. This striking observation of a steepincrease in quantum oscillation amplitude down to the lowest temperature is in clear contrastto the non-LK flattening or decrease expected for gapped Fermi surface models, a simulationof which is shown in the lower inset of Fig. 4 for various gap values, exhibiting non-LK finiteactivation behaviour for a finite gap. We are thus able to identify quantum oscillation signaturesin the unconventional insulator YbB that reveal an origin from in-gap neutral low-energyexcitations, as yielded by correlated insulator models.11ur comparison of measured quantum oscillations between the unconventional bulk insulat-ing regime [4] and field-induced metallic regime of YbB shows that an application of magneticfields yields a spectrum of multiple quantum oscillation frequencies that appear prominently inmagnetic field-induced metallic YbB , encompassing similar frequencies below T ob-served in insulating YbB , but extending to higher frequencies up to at least 3000(200) T (Ta-ble 1). The comparable quantum oscillation frequency range observed in both metallic and in-sulating regimes is characterised by similar moderately heavy effective masses in both regimes,while higher frequencies in the field-induced metallic phase are characterised by heavy effectivemasses 𝑚 ∗ / 𝑚 e up to at least 17(3). This appearance of multiple additional heavy Fermi surfacesheets in the magnetic field-induced metallic regime of YbB would explain the steep increasein the linear specific heat at the field-induced insulator metal transition reported in [11].Our observation of a heavy Fermi surface with multiple quantum oscillation frequenciesin the unconventional insulating and high field-induced metallic regimes of YbB points to amulti-component Fermi surface characterised by 𝑓 -electron hybridisation that persists from theunconventional insulating regime to the high field metallic regime. We note a crucial distinc-tion between the band structure of unconventional insulators SmB [1] and YbB [4]. Whilein the case of SmB , a single half-filled unhybridised conduction 𝑑 -electron band crosses theFermi energy and hybridises with the 𝑓 -electron band to yield the Kondo gap (Figure 5a), thesituation is different in YbB . In the case of YbB , two partially filled unhybridised 𝑠 - 𝑝 con-duction electron bands that are cumulatively half-filled cross the Fermi energy with electron-likecharacter, and are gapped by hybridisation with the 𝑓 -electron band (Figure 5b). We find thisdifference leads to a distinct contrast between the case of the unconventional insulator YbB ,where heavy Fermi surface sheets are characterised by 𝑓 -electron hybridisation, and the caseof SmB , in which the observed light Fermi surface sheets correspond to an unhybridised con-duction electron band [1]. Our findings in YbB are a challenge to correlated models of in-gap12 L (cid:75) X W K -1-0.8-0.6-0.4-0.200.20.40.60.81 E n e r gy ( e V ) E F -2 E ne r g y ( e V ) E ne r g y ( e V ) E F YbB SmB Sm 4 f Sm 5 d Sm 5 d without d - f hybridisation Γ X M Γ XW L KW-1.5-1-0.500.511.52 -1-0.8-0.6-0.4-0.200.20.40.60.81 a b
Boron s - p states Yb f -states Yb d -states Boron s - p states without hybridisation Δ Fig. 5. Contrasting band structure of SmB and YbB . (a) Band structure of SmB from GGA calculations in ref. [29], zoomed in view near the Fermi energy 𝐸 F (full energyrange shown in ref. [29]). Sizes of yellow and blue dots denote weights of Sm-4 f andSm-5 d in various bands. Red dots denote metallic Sm-5 d orbitals without hybridisationwith Sm-4 f orbitals. (b) Calculated band structure of YbB shown with an expandedview around the Fermi energy 𝐸 F (full energy range shown in ref. [4]). Size of the circlesare proportional to the weight at each k -point, green circles are Yb f -states, orangecircles are Yb d -states, and violet circles are boron s-p -states. Red dots denote twopartially filled unhybridised boron s - p conduction electron orbitals without hybridisationwith Yb- f orbitals. In both cases, the Fermi surface yielded by the unhybridised bandis not simply connected, leading to a large number of expected frequencies [1, 4]. states that are expected to yield a Fermi surface corresponding to an unhybridised conductionelectron band. An alternative possibility is suggested by the close proximity of the underlyingbandstructure to a semimetallic bandstructure comprising heavy 𝑓 -electron hybridised electronand hole pockets (Fig. 5). For weak correlations between electrons and holes, metallic electri-cal conduction would be expected. In contrast, for strong correlations, the electrons and holesmay be expected to combine, such that they cannot be readily decoupled, thus impeding lon-13itudinal electrical conduction. Despite the electrically insulating behaviour in such a stronglycorrelated case where electrons and holes are coupled, the Lorentz force can still drive orbitalcurrents, which can yield quantum oscillations corresponding to a heavy 𝑓 -electron hybridisedsemimetallic Fermi surface of the kind observed.14 ethods Sample preparation
Source polycrystalline YbB powder was synthesised using borothermal reduction of 99.998%mass purity Yb O powder and 99.9% mass purity amorphous B at 1700 ◦ C under vacuum [30].The synthesized powder was isostatically pressed into a cylindrical rod and sintered at 1600 ◦ Cin Ar gas flow for several hours. Single crystals of YbB were grown by the traveling solventfloating zone technique under conditions similar to those in ref. [31] using a four-mirror Xe arclamp (3 kW) optical image furnace from Crystal Systems Incorporated, Japan. The growthswere performed in a reducing atmosphere of Ar with 3% H at a rate of 18 mm hr − with thefeed and seed rods counter-rotating at 20–30 rpm. Samples for all measurement techniqueswere cut to size using a wire saw and electropolished to remove heat damage and surface strain. Proximity detector oscillator
Contactless electrical transport measurements using the proximity detector oscillator (PDO)technique [32] were performed using a long-pulse magnet capable of generating up to 68 T atthe Hochfeld Magnetlabor Dresden (HLD) in Dresden, Germany. The capacitor bank-drivenmagnet has a pulse duration of 150 ms, and is fitted with a custom made He system with abase temperature of ≈
600 mK. The PDO circuit was made in accordance to ref. [32], using ahand-wound sensing coil with 10 turns. The raw frequency output from the PDO circuit was ∼
20 MHz, which was passed through a processing circuit before being recorded at ∼ Capacitive torque magnetometry
Torque magnetometry measurements were performed in DC magnetic fields at the NationalHigh Magnetic Field Laboratory in Tallahassee, Florida, USA. The 45 T hybrid magnet was15perated with a He system capable of reaching temperatures as low as 300 mK.Cantilevers were cut from 20 μ m or 50 μ m thick pieces of BeCu into flexible T-shapedpieces. Samples of dimensions approximately 1 × × were secured on the wide endof the cantilever using epoxy, which was thermally matched to the sample to minimise strain.The narrow end of the cantilever was secured down such that the wide end of the cantileverhovers above a Cu baseplate, forming the two plates of a capacitor. The change in capacitancebetween the two plates was measured using a General Radio analogue capacitance bridge witha lock-in amplifier. Density Functional Theory Calculations
Density functional theory bandstructures were calculated with the Wien2k augmented planewave plus local orbital (APW+lo) code [33]. The modified Becke-Johnson (mBJ) potential wasused, which is a semi-local approximation to the exact exchange plus a screening term [34] andwhich improves the band gap in many semiconductor materials. Application of mBJ resulted ina non-magnetic ground state with an indirect band gap of 21 meV and a direct gap of 80 meV,whereas the standard Perdew Burke Ernzerhof (PBE) potential produced a semimetal with over-lapping valence and conduction bands. Spin-orbit coupling was included via the second vari-ational method and resulted in a strong reordering of the bands. Self-consistent calculationswere converged using a k-mesh of × × followed by a non-self-consistent calculationwith a × × mesh for calculation of Fermi surfaces. The bandstructure for boron 𝑠 - 𝑝 states without hybridisation was calculated by shifting the 𝑓 -bands out of the energy range ofhybridisation using DFT+U [35]. 16 eferences
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H.L, A.J.H., M.H., A.J.D., A.G.E., and S.E.S. acknowledge support from the Royal Society, theLeverhulme Trust through the award of a Philip Leverhulme Prize, the Winton Programme forthe Physics of Sustainability, the Taiwanese Ministry of Education, EPSRC UK (studentship andgrant numbers EP/M506485/1, EP/P024947/1, EP/1805236, EP/2124504), the Royal Societyof Chemistry (researcher mobility grant M19-1108), and the European Research Council underthe European Unions Seventh Framework Programme (Grant Agreement numbers 337425 and772891). M.D.J. acknowledges support for this project by the Office of Naval Research (ONR)through the Naval Research Laboratory’s Basic Research Program. M.C.H. and G.B. acknowl-edge financial support from EPSRC, UK through Grant EP/T005963/1. We thank the team at20he National Academy of Sciences of Ukraine, Kiev for assistance in the preparation of poly-crystalline YbB . A portion of magnetic measurements were carried out using the AdvancedMaterials Characterisation Suite in the University of Cambridge, funded by EPSRC StrategicEquipment Grant EP/M000524/1.We acknowledge support from the Deutsche Forschungsgemeinschaft (DFG) through theW¨urzburg-Dresden Cluster of Excellence on Complexity and Topology in Quantum Matter-ct.qmat (EXC 2147, Project No. 390858490) as well as the support of the HLD at HZDR,a member of the European Magnetic Field Laboratory (EMFL). A portion of this work wasperformed at the National High Magnetic Field Laboratory, which is supported by NationalScience Foundation Cooperative Agreement No. DMR-1157490, the State of Florida and theDOE. 21 upplementary Information Non LK quantum oscillation amplitude temperature-dependence for gappedmodels compared with LK temperature-dependence for gapless models
Model simulations
To distinguish between gapless and gapped models of quantum oscillations in the unconven-tional insulating phase, here we simulate the quantum oscillation amplitude for various gapsizes. We use the formulation of ref. [12, 13], the ratio of the first harmonic between the gappedstate and the normal state is: 𝑀 g 𝑀 n = sinh ( 𝑋 ) 𝑋 ∫ ∞ cos (cid:18) 𝑋 𝜇𝜋 (cid:19) 𝜕 𝜇 (cid:32) 𝜇 √︁ 𝜇 + ( Δ / 𝑇 ) tanh (cid:32) √︁ 𝜇 + ( Δ / 𝑇 ) (cid:33) (cid:33) d 𝜇, (1)where 𝜇 is the chemical potential, Δ is the isotropic gap size, and 𝑋 is the temperature dampingcoefficient given by 𝑋 = 𝜋 𝑘 B 𝑇 𝑚 ∗ / 𝑒 ℏ 𝐵 . Here, 𝑘 B is Boltzmann’s constant, 𝑇 is temperature, 𝑚 ∗ is the quasiparticle effective mass, 𝑒 is the electron charge, ℏ is the reduced Planck constant,and 𝐵 = 𝜇 𝐻 is the applied magnetic field [28].If we set 𝑇 = 𝑋𝜔 𝑐 /( 𝜋 ) and Δ / 𝑇 = 𝜋 Δ / 𝜔 c 𝑋 = 𝜋𝛿 / 𝑋 , we find: 𝛿 = 𝜋 Δℏ 𝜔 c , (2)where 𝜔 c is the cyclotron frequency. We therefore find the ratio of the first harmonic betweenthe gapped state and the normal state to be: 𝑀 g 𝑀 n = sinh ( 𝑋 ) 𝑋 ∫ ∞ cos (cid:18) 𝑋 𝜇𝜋 (cid:19) 𝜕 𝜇 (cid:32) 𝜇 √︁ 𝜇 + ( 𝜋𝛿 / 𝑋 ) tanh (cid:32) √︁ 𝜇 + ( 𝜋𝛿 / 𝑋 ) (cid:33) (cid:33) d 𝜇. (3)Gapped model simulations of the non-LK form of quantum oscillation amplitude at low tem-peratures are shown in the lower inset to Fig. 4a for various gap sizes (i.e. various sizes of 𝛿 ),compared with the LK growth in quantum oscillation amplitude at low temperatures for gaplessmodels (i.e. 𝛿 = ). 22 odel comparisons with experimental data Upper insets to Fig. 4a and Fig. 4b of the main text show the growth in quantum oscillationamplitude of the 700 T frequency in magnetic torque and 800 T frequency in electrical resistiv-ity plotted against 𝑋 , respectively, in the unconventional insulating phase of YbB . The LKexponential low temperature growth of the measured quantum oscillation amplitude observedfor both electrical transport and torque magnetisation is in striking contrast to the non-LK fi-nite temperature activation expected for gapped models of quantum oscillations (lower inset toFig. 4a).For the insulating regime of YbB in which temperature dependent quantum oscillations aremeasured, the isotropic gap size at 40 T is given by Δ ≈
15 K [9], which yields 𝛿 ≈ for 𝑚 ∗ / 𝑚 e = for the quantum oscillation frequencies shown in Fig. 4. Simulations withvarious values of 𝛿 are shown in the lower inset of Fig. 4a in the main text [12, 13, 16]. Forthe gapless case ( 𝛿 = ), quantum oscillation amplitude simulations show an exponential LKgrowth at low temperature, while for the gapped case (finite 𝛿 , shown for values up to 𝛿 = ,similar to YbB ), quantum oscillation amplitude simulations show non-LK finite activationbehaviour at low temperature. A comparison of measured quantum oscillation amplitude growthat low temperature with model simulations thus evidences neutral gapless excitations in theunconventional insulating phase of YbB . Low temperature model expansion
A further simplification may be yielded at low temperatures by using a low temperature expan-sion. We perform a series expansion of the term sinh ( 𝑋 )/ 𝑋 corresponding to the temperaturedamping term 𝑅 T in the Lifshitz-Kosevich (LK) formula that describes the temperature depen-dence of quantum oscillations for particles obeying the Fermi-Dirac distribution [28].23or small 𝑇 , a series expansion of the temperature dependence term yields: 𝑅 T ≈ − 𝑋 + O (cid:16) 𝑋 (cid:17) . (4)The quantum oscillation amplitude therefore linearly increases with decreasing 𝑋 approachingthe zero 𝑇 limit. The low temperature growth in quantum oscillation amplitude is captured bythe relative change of quantum oscillation amplitude at a finite temperature 𝐴 ( 𝑇 ) with respectto the amplitude at the lowest measured temperature 𝐴 , given by: − 𝐴 ( 𝑇 ) 𝐴 = 𝐴 − 𝐴 ( 𝑇 ) 𝐴 = 𝑋 . (5)A plot of ( 𝐴 − 𝐴 ( 𝑇 ))/ 𝐴 against 𝑋 would therefore yield a straight line with a gradient equalto / at low temperatures for low-energy excitations within the gap. In contrast, in the absenceof low-energy excitations, gapped quantum oscillation models would yield a much reducedchange in amplitude as a function of 𝑋2