An outstanding problem in the field of high-transition-temperature (high-Tc) superconductivity is the identification of the normal state out of which superconductivity emerges in the mysterious underdoped regime1. The normal state uncomplicated by thermal fluctuations can be studied using applied magnetic fields that are sufficiently strong to suppress long-range superconductivity at low temperatures2,3. Proposals in which the normal ground state is characterized by small Fermi surface pockets that exist in the absence of symmetry breaking1,4,5,6,7,8 have been superseded by models based on the existence of a superlattice that breaks the translational symmetry of the underlying lattice7,8,9,10,11,12,13,14,15. Recently, a charge superlattice model that positions a small electron-like Fermi pocket in the vicinity of the nodes (where the superconducting gap is minimum)8,9,16,17 has been proposed as a replacement for the prevalent superlattice models10,11,12,13,14 that position the Fermi pocket in the vicinity of the pseudogap at the antinodes (where the superconducting gap is maximum)18. Although some ingredients of symmetry breaking have been recently revealed by crystallographic studies, their relevance to the electronic structure remains unresolved19,20,21. Here we report angle-resolved quantum oscillation measurements in the underdoped copper oxide YBa2Cu3O6 + x. These measurements reveal a normal ground state comprising electron-like Fermi surface pockets located in the vicinity of the nodes, and also point to an underlying superlattice structure of low frequency and long wavelength with features in common with the charge order identified recently by complementary spectroscopic techniques14,19,20,21,22.
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S.E.S. acknowledges support from the Royal Society, King’s College Cambridge, the Winton Programme for the Physics of Sustainability, and the European Research Council under the European Union’s Seventh Framework Programme (grant number FP/2007-2013)/ERC Grant Agreement number 337425-SUPERCONDUCTINGMOTT. N.H. and F.F.B. acknowledge support for high-magnetic-field experiments from the US Department of Energy, Office of Science, BES-MSE ‘Science of 100 Tesla’ programme. G.G.L. acknowledges support from Engineering and Physical Sciences Research Council (EPSRC) grant EP/K012894/1. P.A.G. is supported by the EPSRC and thanks the University of Oxford for the provision of a Visiting Lectureship. R.L., D.A.B. and W.N.H. acknowledge support from the Canadian Institute for Advanced Research, and the Natural Science and Engineering Research Council. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by NSF co-operative agreement number DMR-0654118, the state of Florida, and the DOE. We acknowledge discussions with many colleagues, including H. Alloul, C. Bergemann, A. Carrington, S. Chakravarty, A. Chubukov, E. M. Forgan, S. R. Julian, B. Keimer, S. A. Kivelson, R. B. Laughlin, M. Le Tacon, L. Taillefer, D.-H. Lee, P. A. Lee, P. B. Littlewood, A. P. Mackenzie, M. R. Norman, C. Pépin, C. Proust, M. Randeria, S. Sachdev, A. Sacuto, T. Senthil, J. P. Sethna, J. Tranquada and C. M. Varma. We are grateful for the experimental support provided by the ‘100 T’ team, including J. B. Betts, Y. Coulter, M. Gordon, C. H. Mielke, A. Parish, D. Rickel and D. Roybal.
The authors declare no competing financial interests.
Extended data figures and tables
Extended Data Figure 1 Measured projection of the magnetic field along the crystalline c axis of the sample.
Circles indicate the maximum Bcosθ measured at 65 T (cyan) and 85 T (red), obtained by means of a projection coil, while the dashed lines represent fits to a cosine function. The angular error is less than 0.2° for θ ≤ 66° and approximately 0.2° for 68° ≤ θ ≤ 71°. θ = 0°, 1.3°, 11.3°, 12°, 16.3°, 18°, 21.3°, 26.3°, 31.3°, 36.3°, 38°, 41.3°, 45.2°, 46.3°, 48°, 49°, 49.4°, 50.1°, 50.6°, 51.4°, 51.5°, 52°, 52.3°, 52.5°, 52.9°, 53.1°, 54.4°, 54.9°, 55.5°, 56°, 56.2°, −56.95°, 57.2°, −57.4°, −58.15°, 58.2°, −59.4°, 59.6°, 60.6°, 61.2°, −61.4°, 61.7°, 62.5°, 62.6°, −62.7°, −63.2°, 63.4°, 63.7°, −64.1°, 64.5°, 65.5°, 66°, 66.3°, 68.1°, 69.4° and 70.6°. Negative θ angles refer to a measured equivalent (180 − |θ|) angle as shown.
Extended Data Figure 2 Experimental quantum oscillations for different angles compared with simulations for a neck and belly model.
a, Measured oscillations in the contactless resistivity. b, Simulated oscillations at the same angles and fields as a for two Fermi surface cylinders exhibiting a fundamental neck and belly warping, for parameters used in ref 36 (listed in Extended Data Table 3) to simulate the restricted experimental range within the dashed line. Data in a and simulations in b have been scaled by for visual clarity. c, Symbols represent the absolute value of the cross-correlation between the quantum oscillation data in a with a simple sinusoid . F is matched to the periodicity of the oscillations at θ = 38°, where a single frequency dominates the measured quantum oscillations. Coloured lines indicate a simulation proportional to RwRs for a staggered twofold model using parameters in Extended Data Table 1 (magenta), and a neck and belly model using parameters from ref. 36 in Extended Data Table 3 (red). While the anti-resonance in the vicinity of θ = 60° yielded by the staggered twofold model is in good agreement with the experimental data, the striking Yamaji resonance in the vicinity of θ = 60° yielded by the neck and belly model is in marked contrast to experiment.
a, Schematic illustration of the small and large Fermi surface pocket sizes (and quantum oscillation Fourier frequencies) in underdoped YBa2Cu3O6 + x and overdoped Tl2Ba2CuO6 + δ, respectively16,52. b, Contactless electrical resistivity of YBa2Cu3O6.56 measured to 100 T, showing the resistive transition (at approximately 20 T) and quantum oscillations. The dominant quantum oscillations with a frequency of 530 T can be seen to be superimposed on slowly varying oscillations (red line), which we extract in the lower inset by subtracting the dominant oscillations and a linear background. The slowly varying oscillations are consistent with a low frequency of 90 ± 10 T. The upper inset shows the bilayer-split pockets expected for charge order in which the difference in area between two magnetic breakdown junctions corresponds to a frequency of approximately 90 T.
Extended Data Figure 4 Contour representation of a staggered twofold Fermi surface model compared to the experimental data.
a, Contour plot of the simulated quantum oscillation amplitude for a staggered twofold Fermi surface geometry represented by equations (2) and (4), using parameters in Extended Data Table 1 and shown in Fig. 4c. b, Contour plot of experimentally measured quantum oscillation amplitude; good agreement is seen with the model in a. The quantum oscillation amplitude is indicated by the colour scale (in arbitrary units) in the reciprocal field-angle plane; for clarity the ordinate is given as tanθ.
Extended Data Figure 5 Cross-correlation measured over different field ranges, compared to simulations for a staggered twofold Fermi surface geometry.
Real component of the cross-correlation between the quantum oscillation data over fixed ranges of Bcosθ for a range of measured θ angles, with a simple sinusoid cos(2πF/(Bcosθ) + ϕ). F and ϕ are matched to the periodicity and phase of the oscillations at θ = 38°, where a single frequency dominates the measured quantum oscillations. Black lines indicate the simulation (proportional to RwRs), where ΔFtwofold ≈ 15 T is the depth of the modulation, while square symbols indicate the experimental cross-correlation.
Extended Data Figure 6 Schematic of a nodal Fermi surface from charge order that is staggered perpendicularly to the bilayers.
a, Reconstruction of the Brillouin zone, with one instance of the pocket location indicated in the vicinity of the ‘T’ point in relation to the original Fermi surface (purple) and nodes in the superconducting wavefunction (from refs 8, 9, 60 and 61). Here the concentric arrangement of Fermi surfaces arises from bilayer splitting. We note that the in-plane shape of the Fermi pocket shown here is an illustration based on a non-interacting model calculation in refs 8, 9, 60 and 61. b, A three-dimensional view of a. While this schematic assumes achirality, a chiral model is not ruled out, as for instance proposed in ref. 87, where the form of order breaks mirror symmetry within each plane.
Extended Data Figure 7 Schematic of the Brillouin zone cross-section, showing magnetic breakdown orbits in a charge ordering scheme.
A cut through the kz = 0 plane of the Brillouin zone shows the six possible orbits resulting from magnetic breakdown tunnelling in a bilayer charge ordering scheme16,61, an illustrative Fermi pocket shape similar to Extended Data Fig. 6 is shown. a, The two Fermi surface cross-sections of frequency F1 = F0 − 2ΔFsplit and F6 = F0 + 2ΔFsplit that can result from bilayer splitting with in-plane ordering wavevectors and . The Γ and T symmetry points of the body-centred orthorhombic Brillouin zone of the charge order superstructure are depicted in blue. The gap separating bonding and antibonding surfaces is expected to be smallest at the nodes62. Panels b, c and d show the range of possible magnetic breakdown orbits, F2 = F0 − ΔFsplit, F3 = F0, F4 = F0 and F5 = F0 + ΔFsplit, as listed in Table 2.
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Sebastian, S., Harrison, N., Balakirev, F. et al. Normal-state nodal electronic structure in underdoped high-Tc copper oxides. Nature 511, 61–64 (2014). https://doi.org/10.1038/nature13326
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