Unconventional quantum vortex matter state hosts quantum oscillations in the underdoped high-temperature cuprate superconductors
Yu-Te Hsu, Máté Hartstein, Alexander J. Davies, Alexander J. Hickey, Mun K. Chan, Juan Porras, Toshinao Loew, Sofia V. Taylor, Hsu Liu, Alexander G. Eaton, Matthieu Le Tacon, Huakun Zuo, Jinhua Wang, Zengwei Zhu, Gilbert G. Lonzarich, Bernhard Keimer, Neil Harrison, Suchitra E. Sebastian
UUnconventional quantum vortex matter state hostsquantum oscillations in the underdopedhigh-temperature cuprate superconductors
Yu-Te Hsu, * M´at´e Hartstein, Alexander J. Davies, Alexander J. Hickey, Mun K. Chan, Juan Porras, Toshinao Loew, Sofia V. Taylor, Hsu Liu, Alexander G. Eaton, Matthieu Le Tacon, , Huakun Zuo, Jinhua Wang, Zengwei Zhu, Gilbert G. Lonzarich, Bernhard Keimer, Neil Harrison, † Suchitra E. Sebastian † Cavendish Laboratory, Cambridge University, J. J. Thomson Avenue, Cambridge CB3 0HE, UK Pulsed Field Facility, Los Alamos National Laboratory, Los Alamos,Mail Stop E536, Los Alamos, NM 87545, USA Department of Solid State Spectroscopy, Max Planck Institute for Solid State Research,Heisenbergstr. 1, D-70569 Stuttgart, Germany Institute for Quantum Materials and Technologies, Karlsruhe Institute of Technology,Hermann-v.-Helmholtz-Platz 1,D-76344 Eggenstein-Leopoldshafen, Germany Wuhan National High Magnetic Field Center and School of Physics, Huazhong Universityof Science and Technology, Wuhan 430074, China * current address: High Field Magnet Laboratory (HFML-EMFL) and Institute for Molecules and Materials,Radboud University, Toernooiveld 7, 6525 ED Nijmegen, Netherlands † To whom correspondence should be addressed. E-mail: [email protected]; [email protected] central question in the underdoped cuprates pertains to the nature of the pseudo-gap ground state. A conventional metallic ground state of the pseudogap region has beenargued to host quantum oscillations upon destruction of the superconducting order pa-rameter by modest magnetic fields. Here we use low applied measurement currents andmillikelvin temperatures on ultra-pure single crystals of underdoped YBa Cu O + G to un-earth an unconventional quantum vortex matter ground state characterized by vanishingelectrical resistivity, magnetic hysteresis, and non-ohmic electrical transport character-istics beyond the highest laboratory accessible static fields. A new model of the pseudo-gap ground state is now required to explain quantum oscillations that are hosted by thebulk quantum vortex matter state without experiencing sizeable additional damping inthe presence of a large maximum superconducting gap; possibilities include a pair densitywave. Previous electrical transport measurements in underdoped YBa Cu O + G reported the oc-currence of quantum oscillations above modest magnetic fields ⇡
22 T in a conventional metallicpseudogap ground state characterised by finite electrical resistivity, in which superconductivityis destroyed. Here we access the ground state of a new generation of pristine single crystalsof underdoped YBa Cu O + G (hole doping concentration ? = . . ) up to the highestaccessible static magnetic field of 45 T and down to 40 mK in the limit of low applied measure-ment current density ( ! ). We find instead the persistence of vanishing electrical resistivitycharacteristic of magnetic field-resilient superconductivity beyond the highest accessible staticmagnetic field of 45 T.In this work, we measure pristine single crystals of underdoped YBa Cu O + G (YBCO + G )characterised by prominent quantum oscillations with a significantly lower Dingle (impuritydamping) factor than the previous generation of single crystals (Fig. S1). We access the longi-tudinal voltage in the ground state by applying a range of applied measurement current densities2 down to low values of ⇡ . Acm , three orders of magnitude lower than previously usedvalues of ⇡ Acm . Fig. 1 A-H shows the effective resistivity ( d xx = + xx /( ) where ; is the distance between the voltage leads both in the case of ohmic and non-ohmic transportcharacteristics) as a function of magnetic field and temperature. For sufficiently low appliedmeasurement currents and temperatures, we find vanishing electrical resistivity that persistsbeyond the highest accessible static magnetic field of 45 T (Fig. 1 A-H ). A steeply rising phaseboundary of the resistive magnetic field scale up to which vanishing electrical resistivity persistsin the limit of low applied measurement current (i.e. ` r ( ! ) ) as a function of tempera-ture is shown in Fig. 2. A similarly sharply rising resistive magnetic field that shows no signof saturation down to temperatures . ) c ( ) was reported by Mackenzie et al. in pristinesingle crystals of Tl Ba CuO + X (Fig. S2), which also exhibit quantum oscillations. The low values of critical current as a function of ` and ) that we find to characterisehigh magnetic field-resilient superconductivity in high purity single crystals of underdopedYBa Cu O + G is consistent with a low density of pinning centres. Evidence of bulk super-conductivity in the ground state is further revealed by magnetic hysteresis due to bulk vortexpinning in magnetic torque. We find good agreement between the resistive magnetic field inthe limit of low applied measurement current (i.e. ` r ( ! ) ), and the high magnetic fieldscale up to which hysteresis from bulk vortex pinning persists in the magnetic torque (i.e. theirreversibility field ` irr = ` irr ( \ ) cos \ , where \ is the angle between the applied magneticfield and the crystalline -axis), up to the highest accessible static magnetic fields of 45 T asexpected for bulk superconductivity (Figs. 2 A , 4A, Materials and Methods, and Figs. S3-S5)(refs. 5–8). Single crystals of the same doping from different batches and grown by differentgroups show similarly high irreversible magnetic fields in agreement with the high resistivemagnetic fields we access in this work (Fig. S3).We provide further evidence that argues against an origin of magnetic field-resilient super-3onductivity in the ground state of underdoped YBa Cu O + G from high critical temperaturedoping inclusions. Superconducting homogeneity is indicated by the narrow superconduct-ing transition in the electrical resistivity as a function of temperature in high magnetic fields(Fig. 1 I,J ). Further support for superconducting homogeneity is provided by the sharpness ofthe transition in the magnetic susceptibility ( j ) in very small magnetic fields (Figs. S6, S7),from which we infer a minimal volume fraction of any regions of the sample with a value of ) c greater than the mean ) c (defined by the step in j ). The observation of low critical temperaturesat high magnetic fields further argues against the inclusions of higher dopings as responsiblefor the persistence of superconductivity up to high magnetic fields. The systematic doping evo-lution of the high-field superconducting region, which reaches higher critical temperatures withincreasing doping also supports the intrinsic bulk character of the high magnetic field–resilientsuperconductivity (Fig. 2).In order to discern the nature of the high-field superconducting ground state, we performa study of the voltage-current characteristics, signatures of which are used to characterizeregimes of superconducting vortex physics. We find a striking and systematic non-ohmicvoltage-current dependence at high magnetic fields and low temperatures (Fig. 3
A-E ). Giventhat previous measurements in the vortex state were largely confined to low magnetic fieldsand high temperatures, we use a model of unconventional quantum vortex matter developedto treat the high magnetic field region to compare with our measurements in the high magneticfield–low temperature region of underdoped YBa Cu O + G . We find the measured non-ohmicvoltage-current dependence can be well captured by a model of quantum vortex matter basedon self-organisation of vortices in a magnetic field.
10, 11
We use the term ‘quantum vortex mat-ter’ to describe the vortex regime in high magnetic fields and low temperatures where quantumfluctuations are expected to be relevant, as opposed to the more conventional vortex regimeat low magnetic fields and high temperatures. We extract a temperature scale ( ) HFF ) as-4ociated with the melting of quantum vortex matter into a vortex liquid (Fig. 3 F ) for variousmagnetic fields, and find that the quantum vortex matter–vortex liquid phase boundary agreeswell with the extracted resistive magnetic field ` r below which the voltage drops to a van-ishingly small value for a range of temperatures (Fig. 2 A ). Our findings thus unearth a bulkquantum vortex matter ground state that persists up to at least 45 T and evolves to a vortex liq-uid with increasing temperature as shown in the phase diagrams (Figs. 2 A-C , 4 E , SI AppendixFig. S8). A similar superconducting phase diagram driven by quantum fluctuations has been re-ported in two-dimensional materials families such as the organic superconductors, and may besimilarly expected in the strongly interacting quasi-two dimensional cuprate superconductors.Similarly non-BCS (Bardeen-Cooper-Schrieffer)-like magnetic field-resilient superconductiv-ity with positive curvature of the resistive magnetic field was reported in high purity singlecrystals of Tl Ba CuO + X . An interplay of superconductivity and a density wave has beenfurther proposed to yield a steep magnetic field – temperature slope of the superconductingphase boundary, as observed in our experiments.The finite resistivity previously accessed above a modest magnetic field scale ⇡ T inunderdoped YBa Cu O + G can be attributed to the use in pulsed magnetic field experimentsof applied measurement current densities three orders of magnitude higher than present mea-surements, and even larger eddy current densities (Methods) at elevated temperatures (Fig. S9),yielding vortex dissipation. Previous heat capacity measurements were performed at elevatedtemperatures and do not access low enough temperatures to capture the unconventional quantumvortex matter regime (Fig. S9, ref. 14). Features in thermal conductivity previously interpretedas a signature of the upper critical magnetic field in YBa Cu O + G (ref. 1) meanwhile differfrom signatures of the upper critical magnetic field as observed in other type-II superconductors(Fig. S10), prompting its alternative interpretation as a density wave transition in YBa Cu O + G (ref. 5). 5he superconducting phase diagram in high-magnetic field–low-temperature space for theunderdoped cuprates revealed by our present measurements is shown in Fig. 4 E and Fig. S11.We find the quantum vortex matter region characterized by vanishing electrical resistivity in the ! limit and non-ohmic voltage-current characteristics (coloured shading) to steeply riseat low temperatures, persisting beyond the highest laboratory accessible static magnetic fieldsof 45 T. The newly uncovered high magnetic field superconducting phase diagram supercedesprevious proposals involving a finite electrical resistivity ground state (Fig. S8 A,B ). Previousproposals include a BCS-like type-II superconducting phase diagram in which a Meissner su-perconducting state rapidly enters a conventional metallic region
1, 14 at high magnetic fields viaa vortex solid (Shubnikov phase) region (Fig. S8 A , S12), and the alternative possibility of a vor-tex liquid ground state at high magnetic fields characterized by finite electrical resistivity
5, 15, 16 (Fig. S8 B ).We gain insight into the character of the quantum vortex matter ground state of the pseu-dogap by examining the quantum oscillations that are hosted in this region characterized byhysteretic magnetic torque (zero applied measurement current) evidencing vortex pinning,
5, 6 vanishing electrical resistivity in the ! limit, and non-ohmic electrical transport (Fig. 4 A,B ).Quantum oscillations in the electrical resistivity also appear in the quantum vortex matter re-gion, upon the application of sufficiently elevated currents for finite resistivity to be inducedfrom vortex dissipation (Fig. 4 D ). We compare quantum oscillations in the superconductingregion of underdoped YBa Cu O + G with those observed in other type-II superconductors in-cluding NbSe , V Si, Nb Sn, YNi B C, LuNi B C, UPdAl , URu Si , CeCoIn , CeRu , ^ -(BEDT-TTF) Cu(NCS) , MgB , and others, for which theories have been developed ofquantum oscillations in the mixed state (e.g. refs. 19, 20). To estimate the extent of supercon-ducting damping of quantum oscillations, we compare the ratio of the Landau level spacing \ l c to the maximum superconducting gap (here l c = / < ⇤ is the cyclotron frequency and < ⇤
6s the cyclotron effective mass) at magnetic fields where superconducting damping reduces thequantum oscillation amplitude (corrected for the Dingle damping factor) by a factor of two. In the case of underdoped YBa Cu O + G , we obtain an upper bound for this ratio at the lowestmagnetic field value at which quantum oscillations are observed. A low value of \ l c / / . is estimated for underdoped YBa Cu O + G , taking the maximum superconducting gap at zeromagnetic field ⇡ meV from complementary measurements (consistent with the highmagnetic field resilience of superconductivity), < ⇤ = < e ( < e is the free electron mass), and ` =
20 T. This ratio in underdoped YBa Cu O + G is an order of magnitude smaller thanthe ratio \ l c / ⇡ . for conventional type-II superconductors including NbSe , V Si, Nb Sn,YNi B C, LuNi B C, CeRu , MgB (see details in Table S1). Quantum oscillations thus persistin the presence of a large maximum superconducting gap, displaying minimal superconductingdamping in the case of underdoped YBa Cu O + G , unlike conventional type-II superconduc-tors. Similarly low ratios of \ l c / as underdoped YBa Cu O + G are found in unconventionaltype-II superconductors such as URu Si ( \ l c / ⇡ . for ab and ⇡ . for c ), ^ -(BEDT-TTF) Cu(NCS) ( \ l c / ⇡ . ), and UPd Al ( \ l c / ⇡ . for ab and ⇡ . for c ) (seedetails in Table S1).Models of quantum oscillations in the presence of a spatially uniform superconducting gapassociate a low ratio of \ l c / in unconventional type-II superconductors with an anisotropicd-wave superconducting gap, compared to a higher ratio of \ l c / in the case of conventionaltype-II superconductors characterized by an isotropic superconducting gap. Inspection ofquantum oscillations in the quantum vortex matter state of underdoped YBa Cu O + G , however,reveal distinguishing characteristics that are challenging to reconcile with models of spatiallyuniform superconductivity. Firstly, in models of spatially uniform superconductivity, whethercharacterized by isotropic superconducting gapping over the full Fermi surface or d-wave su-perconducting gapping over the full Fermi surface except at a gapless nodal point, the quantum7scillation amplitude is expected to exhibit a reduced temperature variation at low temperaturesdue to the vanishing of in-gap states.
9, 19–21, 24, 29
In contrast, the quantum oscillation ampli-tude increases at low temperatures in underdoped YBa Cu O + G , consistent with the Lifshitz-Kosevich form even within the quantum vortex matter state, signalling Fermi-Dirac statisticsof low energy excitations within the gap (Fig. 4 C ). Secondly, models of spatially uniform su-perconductivity are expected to yield increased damping both as the system transitions fromthe ‘normal’ metallic regime in which the superconducting order parameter is destroyed tothe vortex liquid regime in which vortices are mobile, and as the system further transitionsto the quantum vortex matter regime in which vortices are collectively pinned (Fig. 4 E ). Inour present experiments, we access quantum oscillations as the system transitions from a mo-bile vortex liquid state to the pinned quantum vortex matter state (Fig. 4 B,E ). In contrast tothe expectation from models of spatially uniform superconductivity, no discernible additionaldamping beyond that in the usual Lifshitz-Kosevich description is observed as the quantum os-cillations evolve from the vortex liquid regime to the quantum vortex matter regime (Fig. 4 B , C ,Fig. S14, Fig. S13). These properties of the quantum oscillations we observe in the quantumvortex matter regime reveal the coexistence of finite gapless excitations with a large maximumsuperconducting gap.The observed coexistence can potentially be explained by nonuniform models of supercon-ductivity that are spatially modulated at a finite wavevector, such as the pair density wave (PDW)recently reported in experiments such as scanning-tunneling microscopy.
31, 31–36, 36–38, 38–44
Un-like models of spatially uniform superconductivity that are characterised by nodal points,
9, 19–21, 24, 29 models of finite wavevector superconductivity result in ‘nesting’ over only a portion of theFermi surface and consequently yield a partially gapped Fermi surface and lines of gaplessexcitations. PDW models display a nodal–antinodal dichotomy in which the antinodes aregapped by a large maximum superconducting gap, while gapless ‘Fermi arcs’ occur near the8odes.
31, 32, 34, 37
The reconstruction of the gapless nodal ‘Fermi arcs’ in PDW models yielda sharply defined nodal Fermi pocket, providing a possible explanation for our observation ofquantum oscillations hosted in a quantum vortex matter ground state, which are largely un-damped by the large maximum superconducting gap. Alternatively, a nodal Fermi pocket hasbeen modelled to arise from Fermi surface reconstruction by biaxial charge density wave or-der. Our observations potentially point to quantum oscillations in the quantum vortex matterregime due to the interplay of superconductivity and biaxial charge density wave order.Any model of quantum oscillations in the unconventional quantum vortex matter groundstate of the pseudogap region must also explain features such as the isolated nodal Fermi pocketfound by complementary observations of forward–sawtooth form quantum oscillations, a lowmeasured value of linear specific heat capacity in high magnetic fields, the high magneticfield saturation of quantities such as the specific heat capacity and the spin susceptibility fromthe Knight shift. An open question pertains to the extent to which vortex physics persistsover the broader doping, temperature, and magnetic field range of the pseudogap region of theunderdoped cuprate phase diagram.
31, 34, 48 aterials and Methods Sample preparation for transport measurements
The electrical transport is measured on pristine detwinned oxygen-ordered single crystals ofYBa Cu O + G grown by the flux technique. Samples with typical dimensions of (0.8 - 1.5)mm ⇥ (0.5 - 1.0) mm ⇥ (0.03 - 0.08) mm were selected for the electrical transport measure-ments. Gold pads of standard six-contact geometry were deposited onto the top surface with160 nm thickness and to the sides with 80 nm thickness using thermal evaporation methods.Top and side views of a typical transport sample are shown in SI Appendix Fig. S1. Sampleswith gold pads were annealed at temperatures above 500 C with flow of high-purity oxygen( > G and meanwhile allow the gold pads to diffuse into thebulk of the crystal. All measurements in this work were performed with current flowing alongthe ˆ -axis, with crystals detwinned under uniaxial stress of 100 MPa at 250 C. Cu-O chain su-perstructures were formed in samples under vacuum conditions of below 3 ⇥ mbar. Sam-ples with current contact resistances of ⇡ ⌦ , made using gold wires attached with DuPont4929N, were used for high-field measurements. SI Appendix Fig. S2 shows the superconduct-ing transitions in the susceptibility for the measured samples, with transition widths similar toprevious reports. Hole doping ? is inferred from the critical temperature ) c , defined as themid-point of the superconducting transition. Critical current densities inferred from magnetic torque and resistivity
Assuming the current to be uniformly distributed throughout the sample and to be flowing pre-dominantly within the CuO planes, the critical current density is estimated using c ⇡ " / cA where " is the hysteresis of the magnetisation between the up and down field sweeps in unitsof Am , A ⇡ p ;F / is the effective radius of the sample and where ; , F and C are the length,width and thickness of the sample along the ˆ -axis, ˆ -axis and ˆ -axis, respectively. In the10lectrical transport measurements, we also assume the current to be uniformly distributed, fromwhich we obtain the current density of ⇡ / FC . We find an order of magnitude agreement be-tween the critical current density estimated from torque hysteresis and from electrical transportmeasurements (Fig. S5). Evidence for bulk superconductivity
An important question concerning the observation of superconductivity that persists up to highmagnetic fields is whether the superconductivity is of bulk character. Fig. 1
I,J presents measure-ments of the electrical resistivity versus temperature for ? = . , . . The narrow absolutewidth of the superconducting transition in electrical resistivity at high magnetic fields indicatesno significant increase in inhomogeneity of the superconducting state in strong magnetic fieldswhen ) c is suppressed. Furthermore, an inclusion of a small superconducting volume fractionof a higher doping, were it to exist, would manifest in magnetic susceptibility measurements,which is not observed, as shown in SI Appendix Fig. S3.The observation of significant vortex pinning in the magnetic torque (i.e. hysteresis inFig. 4 A , SI Appendix Fig. S4 C ) accompanying vanishing electrical resistivity in the ! limit below the superconducting transition indicates a bulk superconducting state. No suchsharp transition or hysteretic signature in the bulk magnetic torque is expected to occur forsurface or filamentary superconducting states, which furthermore typically exhibit small fi-nite electrical resistivity rather than the vanishing electrical resistivity that is observed. Theabove findings, and the systematic nature and sample independence of our results point to theintrinsic, bulk character of the magnetic field-resilient superconducting state in the pseudogapground state characterised by low critical temperature and low critical current that we find to bepersistent beyond the highest accessible static magnetic fields of 45 T.11 greement with complementary measurements In addition to the sensitivity to elevated temperatures, we find the superconducting state char-acterized by vanishing electrical resistivity in the ! limit to give way to a vortex liquidstate with finite but low electrical resistivity upon using larger measuring current densities of ⇡ Acm as used in earlier studies of electrical resistivity, and by performing measure-ments in rapidly changing pulsed magnetic fields as previously reported, that generate largeeddy currents in the sample of the order of ⇡ Acm . The phase boundary demarcated byresistive magnetic field ( ` r ) above which the vanishing resistivity vortex matter ground stateevolves into a finite resistivity vortex liquid as a function of applied measurement current andtemperature shows agreement with results of previous measurements (Fig. S9) at high appliedmeasurement currents and high temperatures. Our measurements are in agreement with the ob-served p ` field dependence of the linear specific heat coefficient extending up to the highestaccessible static magnetic fields, which is characteristic of a vortex state. Other specific heatmeasurements at high magnetic fields
14, 52 are limited to relatively high temperatures of 2 K, notlow enough to capture the steep upturn observed for ` r at low temperatures. Fig. S10 showsthe feature in thermal conductivity in underdoped YBa Cu O + G that has been interpreted as asignature of the upper critical magnetic field. A comparison with the thermal conductivity as afunction of magnetic field in other type-II superconductors shows differences in characteristicsignatures at the upper critical magnetic field that exhibits an upward slope starting from zeromagnetic field. These differences have led to the feature in thermal conductivity in under-doped YBa Cu O + G to be interpreted instead as a signature of a density wave onset at highmagnetic fields. ow temperature growth of quantum oscillation amplitude compared withLifshitz-Kosevich expansion The Fermi-Dirac distribution yields a temperature-dependent quantum oscillation amplitudein the Lifshitz-Kosevich (LK) form. This low temperature growth of quantum oscillationamplitude is given by: ' T = - sinh - , with - = c : B ) < ⇤ / \ ` , where : B is Boltzmann’s constant, ) is temperature, < ⇤ is thequasiparticle effective mass, is the electron charge, and \ is the reduced Planck constant. For small ) , a series expansion of the temperature dependence term yields: ' T ⇡ - + O ⇣ - ⌘ , For small ) , therefore, the quantum oscillation amplitude linearly increases with decreasing - approaching the ) ! limit. The low temperature growth in quantum oscillation amplitude iscaptured by the relative change of quantum oscillation amplitude at a finite temperature ( ) ) with respect to the amplitude at the lowest measured temperature , given by: ( ) ) = ( ) ) = - . A plot of ( ( ) ))/ against - would therefore yield a straight line with a gradient equal to / at low temperatures for low energy excitations within the gap. In contrast, in the absence oflow energy excitations, gapped quantum oscillation models would yield a much reduced changein amplitude as a function of - at low temperatures well below the gap temperature scale. The inset to Fig. 4 C shows the growth in quantum oscillation amplitude plotted against - witha quasiparticle effective mass < ⇤ / < e = . (ref. 55). The rapid low temperature growth ofthe quantum oscillation amplitude yields a linear slope of 0.20(2) at low temperatures, in notable13ontrast to the expectation of little to no growth in the case of gapped quantum oscillations inthe low temperature limit. A full temperature dependence of the quantum oscillation amplitudeup to a temperature of 18 K is shown in Fig. S14. Author contributions
Y.-T.H., J.P., T.L., M.L.T., B.K. prepared samples; Y.-T.H., M.H., A.J.D., A.J.H., M.K.C.,S.V.T., H.L., A.G.E., H.Z., J.W., Z.Z., N.H., S.E.S. performed measurements; Y.-T.H., M.H.,A.J.D., A.J.H., H.L., G. G. L., N.H., S.E.S. analyzed data; N.H., S.E.S. wrote the paper withinput from all the co-authors.
Acknowledgements
Y.-T.H., M.H., A.J.D., A.J.H., S.V.T., H.L. and S.E.S. acknowledge support from the RoyalSociety, the Winton Programme for the Physics of Sustainability, Engineering and Physical Sci-ences Research Council (EPSRC; studentship and grant numbers EP/R513180/1, EP/M506485/1and EP/P024947/1), and the European Research Council under the European Unions SeventhFramework Programme (Grant Agreement numbers 337425 and 772891). A portion of mag-netic measurements were carried out using the Advanced Materials Characterisation Suite in theUniversity of Cambridge, funded by EPSRC Strategic Equipment Grant EP/M000524/1. S.E.S.acknowledges support from the Leverhulme Trust by way of the award of a Philip LeverhulmePrize. H.Z., J.W. and Z.Z. acknowledge support from the National Key Research and Develop-ment Program of China (grant no. 2016YFA0401704). We are grateful for helpful discussionswith colleagues including P. W. Anderson, L. Benfatto, J. Blatter, J. C. S. Davis, N. Doiron-Leyraud, M. Eisterer, E. M. Forgan, R. H. Friend, D. Geshkenbein, P. Kim, S. A. Kivelson, M.H. Julien, D. H. Lee, P. A. Lee, T. Maniv, D. R. Nelson, M. R. Norman, N. P. Ong, C. Pepin,M. Randeria, S. Sachdev, J. Schmalian, T. Senthil, L. Taillefer, C. M. Varma, H. H. Wen. A14ortion of this work was performed at the National High Magnetic Field Laboratory (NHMFL),which is supported by NSF Cooperative Agreement DMR-1157490, the State of Florida, andthe Department of Energy (DOE). M. K. C. and N. H. acknowledge support from the DOEBasic Energy Sciences project: ‘Science of 100 tesla’. We thank S. A. Kivelson for suggestingthe application of the vortex model of Huse et al. to our data. We are grateful for experimentalsupport at NHMFL, Tallahassee from J. Billings, E. S. Choi, B. L. Dalton, D. Freeman, L. J.Gordon, D. E. Graf, M. Hicks, S. A. Maier, T. P. Murphy, J.-H. Park, K. N. Piotrowski, andothers. We thank S. Lacher and C.T. Lin for assistance with synthesis of high-quality singlecrystals.
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Physica C: Superconductivity and its applications , 450–457 (2001).[77] Dai, P., Mook, H. A., Hunt, R. D. & Do˘gan, F. Evolution of the resonance and incommen-surate spin fluctuations in superconducting YBa Cu O + G . Physical Review B , 054525(2001).[78] Yu, G., Li, Y., Motoyama, E. & Greven, M. A universal relationship between magneticresonance and superconducting gap in unconventional superconductors. Nature Physics , 873 (2009). 23 µ A 100 µ A 30 µ A 10 µ A H (T) T = 0.44 K BA C T = 4.0 K xx ( m Ω ⋅ c m ) T = 0.04 K p = 0.132 T = 0.44 K T = 1.3 K T = 4.0 K E H (T) F µ A 100 µ A 30 µ A 10 µ A G p = 0.108 T = 10 K D T = 10 K H xx ( m Ω ⋅ c m ) T (K)
44 T35 T42 T25 T µ A I
44 T35 T42 T T (K)
25 T J Figure 1: Onset of finite electrical resistivity as a function of magnetic field for different appliedmeasurement currents. ( A - H ) In-plane electrical resistivity d xx of YBa Cu O + G measured onsweeping the magnetic field ( ` H k ˆ ) at different temperatures ) , hole dopings ? , and appliedmeasurement currents as indicated. Here d xx = + xx /( ) is an effective resistivity in thenon-ohmic regime. Brown arrows indicate the resistive magnetic field ( ` r ) above which theelectrical resistivity becomes finite in the limit of low applied measurement current ( ` A ( ! ) ). = ⇡ on assuminga limit of uniform current distribution. Superconductivity characterized by vanishing electricalresistivity in the ! limit is revealed to persist up to the highest accessible magnetic fields.At the lowest temperature of 40 mK, currents higher than 300 ` A are expected to lead to self-heating effects, and hence only measurements with applied currents below 100 ` A are shown.( I - J ) In-plane electrical resistivity d xx transition into the superconducting state characterized byvanishing electrical resistivity in the ! limit shown at different values of magnetic field(symbols show pulsed field data and dashed lines connecting symbols are guides to the eye,solid lines show DC field data). Width of the superconducting transition is similarly narrow tothat observed at zero magnetic fields. Legend in panel I also applies to panel J .24
20 40 60010203040 A YBCO ( p = 0.108) I = 10 µ A I = 30 µ A I = 1 mA I = 10 mA T HFF H r ( T ) T (K) T c (0 T) T c (0 T) YBCO ( p = 0.116) B I = 10 µ A I = 30 µ A I = 1 mA I = 10 mA T (K) T c (0 T) YBCO ( p = 0.132) C I = 30 µ A I = 10 mA T (K) H irr (SI) H irr (ref. 11) H r , H i rr ( T ) T (K) H r ( T ) T (K) H r ( T ) T (K) (ref. 5) Figure 2: Magnetic field-resilient superconductivity revealed at low temperatures and low ap-plied measurement currents. Resistive magnetic field ` r measured for YBa Cu O + G at threehole dopings ( ? = 0.108, 0.116, and 0.132) using static magnetic field scans at different fixedtemperatures and currents indicated by circles (here, a current of = ⇡ ), obtained from Fig. 1. Magnetic field-resilient superconductivityis observed to persist up to the highest magnetic fields for low applied measurement currents.Similarly non-saturating values of resistive magnetic field down to the lowest temperatureswere reported in Tl Ba CuO + X (Fig. S2). The temperature scale associated with the meltingof quantum vortex matter into a vortex liquid ( ) HFF ) are indicated by diamonds in panel a. Thebulk character of superconductivity at high magnetic fields is indicated by the similar phaseboundary obtained from magnetic torque measurements (inset to panel A ).25 -2 -1 0-5-4-3-2-1 -2 -1 0-5-4-3-2-1-2 -1 0-5-4-3-2-1 -2 -1 0-5-4-3-2-1-2 -1 0-5-4-3-2-1 A l og ( V ) ( m V ) [ H = T ] log( I ) (mA) p = 0.108 B l og ( V ) ( m V ) [ H = T ] log( I ) (mA) D l og ( V ) ( m V ) [ H = T ] log( I ) (mA) FE l og ( V ) ( m V ) [ H = T ] log( I ) (mA) C l og ( V ) ( m V ) [ H = T ] log( I ) (mA) T (K)
45 T 42 T 38 T 34 T 30 T 26 T
Figure 3: Non-ohmic voltage-current dependence at the highest magnetic fields and lowesttemperatures. ( A - E ) In-plane longitudinal voltage measured for YBa Cu O + G ( ? = . )as a function of applied measurement current, at temperatures ranging from 20 K down to0.04 K at different applied magnetic fields. Voltage values measured at the same temperatureare connected by dashed lines. Solid lines in the insets correspond to fits based on a modelof quantum vortex matter by Huse, Fisher, and Fisher, characterized by non-ohmic current-voltage dependence described by + / exp [ ( ) / ) ` ] , where is the current density, and ) and ` are fitting parameters. ( F ) The exponent ` as a function of temperature, with ) HFF definedas the temperature when ` = . 26
25 30 35 40 450.000.050.100.1536 39 42 450.00.20.40.6 e quan t u m v o r t e x m a tt e r vortex solidvortex liquid m e t a lli c " no r m a l " pha s e Meissner phase M agne t i c f i e l d Temperature a C apa c i t an c e ( p F ) p = 0.108sample A H (T) T = 0.04 K T = 1.0 K(VL) b p = 0.108sample B T = 0.04 K(QVM) C apa c i t an c e ( p F ) H (T) c A m p li t ude ( a . u . ) T (K) p = 0.10830 - 45 T ( A - A ( T )) / A X p = 0.108 T = 0.44 K d
10 Acm -2
15 Acm -2
20 Acm -2
25 Acm -2 H (T) xx ( µ Ω c m ) A BC D E M a g n e t i c fi e l d Temperature
Figure 4: Quantum oscillations co-existing with the vortex matter phase and the magneticfield–temperature phase diagram for YBa Cu O + G . ( A ) Quantum oscillations co-exist withhysteresis in magnetic torque (zero applied measurement current, \ = ) from vortex pinningextending up to the irreversibility field ` irr beyond 45 T, coincident with the vanishing elec-trical resistivity region that also extends beyond 45 T at ) = B ) Quantumoscillations in the quantum vortex matter (QVM) regime at the lowest temperature ) = ) = C ) Lifshitz-Kosevich (LK)temperature dependence of the quantum oscillation amplitude at the lowest measured temper-atures. The inset shows the growth of quantum oscillation amplitude continues to the lowestmeasured temperatures, as brought out by a low temperature expansion (Methods). ( ) ) isthe quantum oscillation amplitude at temperature ) , is the amplitude at the lowest mea-sured temperature, and - = c : B ) < ⇤ / \ ` is the temperature damping coefficient in theLK formula. ( D ) Shubnikov–de Haas oscillations after background subtraction in the quan-tum vortex matter regime upon applying elevated current densities to induce vortex dissipation.Here d xx = + xx /( ) is an effective resistivity in the non-ohmic regime. ( E ) New superconduct-ing phase diagram in which high magnetic field-resilient quantum vortex matter ground state isrevealed in the present measurements, melting to a vortex liquid with elevated temperature.27 igure S1: Electrical contacts on measured single crystals of YBa Cu O + G . Views of a highquality single crystal of YBa Cu O + G on which electrical resistivity measurements are per-formed; quantum oscillations on the same batch of single crystals were reported in ref. Topand side view of electrical contacts attached to the sample mounted on a quartz platelet.28 H r ( T ) T (K) Tl-2201 ( Mackenzie et al. ) H r ( T ) T (mK) (Mackenzie et al. ) Figure S2: Magnetic field-resilient superconductivity for Tl Ba CuO + G (Tl2001, T c ⇡ K)adapted from ref. 3. It exhibits a sharply rising resistive magnetic field that shows no sign ofsaturation down to temperatures . ) c ( ) , similarly to what we report for YBa Cu O + G .The employed current densities were from . to Acm , with lower current densitiesemployed at the lowest temperatures. Reprinted with permission from ref. 3. Copyright (1993)by the American Physical Society. 29 A sample Dsample D xx ( m Ω c m ) H (T) YBCO ( p = 0.108) sample C B xx ( m Ω c m ) H (T) 1.2 K2.3 K3 K4 K5 K7 K10 K15 K39 K55 K C C apa c i t an c e ( p F ) H (T) ≈ D H r (30 µ A) sample C H r (1 mA) sample C H r (1 mA) sample D H r (10 mA) sample C H irr sample D H irr (Yu et al. ) H r , H i rr ( T ) T (K) (Yu et al. ) Figure S3: Correspondence of resistive magnetic field ( ` r ) and irreversibility field from mag-netic torque hysteresis closure ( ` irr ). ( A,B ) In-plane electrical resistivity d xx measured onsweeping the magnetic field, at different temperatures (indicated by colour) using a current of1 mA for two samples of YBa Cu O + G with G = ? = C ) Torque magnetisationmeasurements of sample D showing hysteretic behaviour between rising and falling fields, re-vealing the presence of bulk vortex pinning in the same region where the electrical resistivityis vanishingly small, indicative of bulk superconductivity. Here d xx = + xx /( ) is an effectiveresistivity in the non-ohmic regime. Closure of the hysteresis loop in magnetic torque marksthe irreversibility field ( ` irr ). ( D ) ` r obtained from ( A,B ) and ` irr obtained from ( C )are seen to coincide for two samples of YBa Cu O + G with G = ? = ` irr as a function of temperature up to 45 T was reported in refs. 5, 6. Reprintedfrom ref. 5. 30 T (K) H r (10 µ A) H r (30 µ A) H r (100 µ A) H r (300 µ A) H r (1 mA) H r (3 mA) H r (5 mA) H irr H irr (Yu et al. ) YBCO ( p = 0.108) T (K) H r ( T ) , H i rr ( T ) A B
YBCO ( p = 0.116) T (K) YBCO ( p = 0.132) C (Yu et al. ) Figure S4: Dependence of resistive magnetic field ( ` r ) on applied measurement current as afunction of temperature. ( A-C ) Circles connected by lines indicate ` r observed on measur-ing the in-plane electrical resistivity as a function of applied static magnetic field at differentfixed temperatures and applied currents for YBa Cu O + G samples with G = = ⇡ . A clear increase in ` r is seen as the applied current is decreased, an effect that grows larger at lower temperatures.Downward triangles show the irreversibility magnetic field measured from hysteretic magnetictorque, reflecting the persistence of bulk pinned vortices up to ` irr . Good agreement is seenwith values of ` r measured from electrical resistivity at the lowest applied current density,confirming the bulk superconducting behaviour reflected by ` r ( ! ) . Reprinted fromref. 5. 31 j c ( A c m - ) H (T) xx (electrical resistivity) m (magnetic torque) YBCO ( p = 0.108) T = 0.04 K Figure S5: Critical current density inferred from the hysteretic torque magnetisation and resis-tivity measurements. Fig. 4 A shows hysteresis between rising and falling field in the measuredtorque magnetisation for hole doping ? = . , which is attributed to flux trapped by vor-tex pinning. We estimate a critical current density from these torque measurements, which isproportional to the extent of hysteresis, using the expression c ⇡ " / cA where " is thehysteresis of the magnetisation between the up and down field sweeps in units of Am , and A ⇡ p ;F / is the effective radius of the sample (see Methods). We compare the criticalcurrent density obtained from torque magnetisation to the critical current density obtained fromtransport measurements performed on the same sample, defined as the current density abovewhich the measured resistivity reaches m ⌦ cm, indicating the onset of finite resistivity,and find values obtained are similar to within an order of magnitude, consistent with a bulksuperconducting origin. 32 ( a . u . ) T (K) p = 0.108 0.116 0.132 Figure S6: Magnetic susceptibility measurements of the superconducting transition in variousdopings of YBa Cu O + G . The magnetic susceptibility j was measured with a small magneticfield of magnitude 0.2 mT. Sharp superconducting transitions are observed, indicative of highcrystal quality and superconducting homogeneity.33
50 55 60 65 70 75 80 850.000.010.02 Δ T c = 0.66 K Δ T c = 0.93 K d / d T ( a . u . ) T (K) p = 0.108 p = 0.116 p = 0.132 Δ T c = 0.57 K Figure S7: Width of the superconducting transition in underdoped YBa Cu O + G . The deriva-tive of the magnetic susceptibility j is shown for hole dopings 0.108, 0.116, and 0.132 revealingnarrow superconducting transitions. The full width at half maximum is found to be less than1 K for all three dopings. 34 uan t u m v o r t e x m a tt e r vortex solidvortex liquid m e t a lli c no r m a l pha s e Meissner phase M agne t i c f i e l d Temperature a cb metallicnormal phaseMeissner phase M agne t i c f i e l d Temperature
Shubnikov phase vortex solidvortex liquid m e t a lli c no r m a l pha s e Meissner phase M agne t i c f i e l d Temperature
A B C vortex liquid m e t a lli c n o r m a l p h a s e quantum vortex matterMeissner phase Figure S8: Alternative magnetic field–temperature phase diagrams for YBa Cu O + G . ( A ) BCS-like type-II superconductor, in which the superconducting order parameter is destroyed abovemodest magnetic fields, and the normal metallic phase is readily accessed down to the lowesttemperatures. ( B ) Proposal for a strongly interacting superconductor from refs.
16, 51 in whicha vortex liquid phase is accessed above modest magnetic fields. ( C ) New region of high mag-netic field-resilient superconductivity of a strongly interacting superconductor is revealed in thepresent measurements up to the highest accessible magnetic fields, melting to a vortex liquidwith elevated temperature.
56, 57
Schematic phase diagrams adapted from ref. 9.35
20 40 60051015202530354045 I = 10 µ A I = 30 µ A I = 1 mA I = 10 mA T HFF H vs (Grissonnanche et al .) C p anomaly (Ka mar ík et al .) C p anomaly (Marcenat et al .) YBCO ( p = 0.108) H r ( T ) T (K) Figure S9: Resistive magnetic field measured in static fields and low applied measurementcurrents, compared to high current electrical transport measurements and specific heat measure-ments. The resistive magnetic field ` r measured using static magnetic fields and down to lowapplied measurement currents revealing magnetic field-resilient superconductivity. The bluecircles correspond to ` r obtained from pulsed magnetic field measurements using high ap-plied measurement currents reported in ref. 58. Green and black squares represent the positionof the specific heat anomaly inferred from magnetic field scans reported in refs. 14, 52. Specificheat measurements are limited to relatively high temperatures of 2 K, not low enough to cap-ture the steep upturn observed for ` r at low temperatures. Reprinted from ref.52. Copyright(2018) by the American Physical Society. Reprinted from ref.14, which is licensed under CCBY 4.0¡https://creativecommons.org/licenses/by/4.0/legalcode¿. Reprinted from ref.58, whichis licensed under CC BY 3.0¡https://creativecommons.org/licenses/by-nc-nd/3.0/legalcode¿.36 B
30 K22.5 K 17.5 K40 K12.5 K κ xx ( W K - m - ) H (T) YBa Cu O ( p = 0.108) A T → NbSe CeIrIn UPt NbLiNi B CTl-2201 κ ( H ) / κ ( H c ) H / H c2 Figure S10: Thermal conductivity as a function of applied magnetic field for YBa Cu O . andtype-II superconductors. ( A ) Field dependence of the thermal conductivity of YBa Cu O . at different temperatures, with maximal values at 0 T. Reprinted from ref. 5. It does notshow the features observed for type-II superconductors, characterized by a gradual increasein thermal conductivity with applied magnetic field, indicating a transition from the vor-tex state to the normal state at ` c2 . ( B ) The thermal conductivity of type-II super-conductors as a function of applied magnetic field divided by the critical field ` c2 . Itshows a gradual increase of the thermal conductivity as a function of applied magneticfield within the vortex region. Reprinted from ref. 53, which is licensed under CC BY4.0¡https://creativecommons.org/licenses/by/4.0/legalcode¿.37
50 100 150 20001020304050 xx / lin M agne t i c f i e l d ( T ) Temperature (K) H r b p = 0.140 a quan t u m v o r t e x m a tt e r vortex solid vortex liquid m e t a lli c no r m a l pha s e Meissner phase M agne t i c f i e l d Temperature vortex liquid m e t a lli c n o r m a l p h a s e quantum vortex matterMeissner phase A B
Figure S11: Comparison of the phase diagram of a strongly interacting superconductor withthe phase diagram of underdoped YBa Cu O + G inferred from resistivity measurements. ( A )Schematic magnetic field–temperature phase diagram of a strongly interacting superconductor,with a new region of high magnetic field-resilient superconductivity and a vortex liquid regioncharacterized by suppressed resistivity extending over a broad region of magnetic field and tem-perature. ( B ) Magnetic field–temperature phase diagram constructed from the colour plot of thereduced electrical resistivity of YBa Cu O + G hole doping ? = . shown in Fig. S12. Theresistive magnetic field ` r forms a steep superconducting phase boundary resembling thatof the schematic phase diagram in ( A ). We also find an analogous broad region of suppressedresistivity corresponding to the vortex liquid region.38
50 100 150 2000.00.51.00 50 100 1500.00.51.0 xx / li n T (K) xx / lin lin H n p = 0.140 B H ( T ) T (K) H r C p = 0.140 xx / xx ( K ) T (K) A p = 0.140 Figure S12: Evolution of the reduced electrical resistivity region up to high temperatures due tostrong magnetic fields. ( A ) In-plane resistivity d xx of YBa Cu O + G hole doping ? = . as afunction of temperature for different magnetic fields applied perpendicular to the crystal plane( ` k ˆ ). A linear fit ( d lin = d + ) ) is made to approximate the temperature dependenceof resistivity at high temperatures (pink dashed line). ( B ) Reduced in-plane resistivity fromthe normal-state resistivity d xx / d lin as a function of temperature for different magnetic fields.Pink line indicates d xx / d lin = 1. Green circles denote the magnetic field ` = previously inter-preted as the onset of the normal state from measurements on the related material YBa Cu O (identified to have hole doping ? = . ) , now seen to clearly lie within the reduced re-sistivity vortex liquid region. ( C ) Magnetic field–temperature phase diagram constructed fromthe colour plot of the reduced in-plane electrical resistivity shown in ( B ). Red corresponds to d xx / d lin = , while dark blue indicates superconductivity characterized by vanishing electricalresistivity in the ! limit. Open circles indicate the finite electrical resistivity onset mag-netic field ` r . The intervening region of reduced resistivity appears in light blue. The dashedgreen line represents the previously interpreted upper boundary of the vortex-liquid phase fromref., which is now seen to clearly lie within the reduced resistivity vortex liquid region.39 A m K / A K µ H (T) = 9 o l n A pea k - t o - pea k µ H (T -1 ) T = 1.5 K = 36 o Figure S13: Comparison of quantum oscillation amplitudes in YBa Cu O . in different vortexregimes. Peak-to-peak quantum oscillation amplitude is plotted on a log scale as a functionof inverse magnetic field in the vortex liquid regime. Magnetic torque measurements wereperformed at 1.5 K using the piezocantilever technique in pulsed field up to 65 T, at a tilt angle \ = off -axis to eliminate beating patterns, on a YBa Cu O + crystal prepared from thesame batch as shown in the main text. Inset shows the ratio of quantum oscillation amplitudemeasured at 0.04 K (quantum vortex matter regime) and 1 K (vortex liquid regime) in this work,as shown in Fig. 4 B . The quantum oscillation amplitude ratio is found to be close to unity overthe entire measured magnetic field range. Reprinted from ref. 2.40 A m p li t ude ( a . u . ) T (K)
40 - 54 T p = 0.108 Figure S14: Lifshitz-Kosevich temperature dependence of quantum oscillation amplitude inYBa Cu O + G (hole doping ? = . ). Quantum oscillation amplitude measured over abroad temperature range 1.1 K / ) /
18 K, complementing the dilution fridge data be-low ) ⇡ C ; the dashed line shows a fit using a quasiparticle effective mass < ⇤ / < e = . . The quantum oscillation amplitude follows the Lifshitz-Kosevich form downto the lowest temperatures, as expected for quantum oscillations arising from gapless fermionicexcitations. Reprinted with permission from ref. 30. Copyright (2010) by the American Physi-cal Society. 41 able S1: Parameters characterising superconductors that display quantum oscillations in thesuperconducting region, including values of the critical temperature ) c , magnetic field ` where the quantum oscillation amplitude (corrected for the Dingle damping factor) is reducedby a factor of two, the effective mass < ⇤ , and the superconducting gap at zero magnetic field.These values are used to determine the ratio of the Landau level spacing to the superconductinggap. Compound ) c (K) ` (T) < ⇤ ( < e ) (meV) \ l c / Refs.NbSe . . .
61 1 . .
21, 59V Si
17 17 . . . .
17, 21, 60Nb Sn . . . . .
17, 60YNi B C . . .
35 2 . .
61, 62LuNi B C . . . . .
63, 64MgB . . .
46 2 . . . . .
55 1 . .
66, 67UPd Al . . .
24 0 .
68, 69URu Si ( -plane) . . . . .
70, 71URu Si ( -axis) . . . .
70, 72 ^ -(BEDT-TTF) Cu(NCS) . . . . Cu O .
61 20 ⇤ .
30 0 . .