Skip to main content

Thank you for visiting nature.com. You are using a browser version with limited support for CSS. To obtain the best experience, we recommend you use a more up to date browser (or turn off compatibility mode in Internet Explorer). In the meantime, to ensure continued support, we are displaying the site without styles and JavaScript.

Normal-state nodal electronic structure in underdoped high-Tc copper oxides

Abstract

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is characterized by small Fermi surface pockets that exist in the absence of symmetry breaking1,4,5,6,7,8 have been superseded by models based on the existence of a superlattice that breaks the translational symmetry of the underlying lattice7,8,9,10,11,12,13,14,15.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.

Access options

Rent or Buy article

Get time limited or full article access on ReadCube.

from$8.99

All prices are NET prices.

Figure 1: Fermi surface of YBa2Cu3O6.56 inferred from quantum oscillation measurements.
Figure 2: Quantum oscillations defining the Fermi surface of YBa2Cu3O6.56.
Figure 3: Fermi surface and geometry-dependent quantum oscillation damping for different crystal structures.
Figure 4: Quantum oscillation data compared with a staggered twofold Fermi surface model.

References

  1. 1

    Lee, P. A., Nagaosa, N. & Wen, X. G. Doping a Mott insulator: physics of high-temperature superconductivity. Rev. Mod. Phys. 78, 17–85 (2006)

    ADS  CAS  Article  Google Scholar 

  2. 2

    Grissonnanche, G. et al. Direct measurement of the upper critical field in cuprate superconductors. Nature Commun. 5, 4280 (2014)

    Article  CAS  Google Scholar 

  3. 3

    Wang, Y. et al. High field phase diagram of cuprates derived from the Nernst effect. Phys. Rev. Lett. 88, 257003 (2002)

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

  4. 4

    Yang, K.-Y., Rice, T. M. & Zhang, F.-C. Phenomenological theory of the pseudogap state. Phys. Rev. B 73, 174501 (2006)

    ADS  Article  CAS  Google Scholar 

  5. 5

    Anderson, P. W. et al. The physics behind high-temperature superconducting cuprates: the ‘plain vanilla’ version of RVB. J. Phys. Condens. Matter 16, R755–R769 (2004)

    CAS  Article  Google Scholar 

  6. 6

    Riggs, S. C. et al. Heat capacity through the magnetic-field-induced resistive transition in an underdoped high-temperature superconductor. Nature Phys. 7, 332–335 (2011)

    ADS  CAS  Article  Google Scholar 

  7. 7

    Doiron-Leyraud, N. et al. Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor. Nature 447, 565–568 (2007)

    ADS  CAS  PubMed  Article  Google Scholar 

  8. 8

    Sebastian, S. E., Harrison, N. & Lonzarich, G. G. Towards resolution of the Fermi surface in underdoped high-Tc superconductors. Rep. Prog. Phys. 75, 102501 (2012)

    ADS  PubMed  Article  CAS  Google Scholar 

  9. 9

    Harrison, N. & Sebastian, S. E. Protected nodal electron pocket from multiple-Q ordering in underdoped high temperature superconductors. Phys. Rev. Lett. 106, 226402 (2011)

    ADS  CAS  PubMed  Article  Google Scholar 

  10. 10

    LeBoeuf, D. et al. Electron pockets in the Fermi surface of hole-doped high-Tc superconductors. Nature 450, 533–536 (2007)

    ADS  CAS  PubMed  Article  Google Scholar 

  11. 11

    Chakravarty, S. & Kee, H.-Y. Fermi pockets and quantum oscillations of the Hall coefficient in high-temperature superconductors. Proc. Natl Acad. Sci. USA 105, 8835–8839 (2008)

    ADS  MathSciNet  PubMed  PubMed Central  MATH  Article  Google Scholar 

  12. 12

    Millis, A. J. & Norman, M. R. Antiphase stripe order as the origin of electron pockets observed in 1/8-hole-doped cuprates. Phys. Rev. B 76, 220503 (2007)

    ADS  Article  CAS  Google Scholar 

  13. 13

    Yao, H., Lee, D. H. & Kivelson, S. A. Fermi-surface reconstruction in a smectic phase of a high-temperature superconductor. Phys. Rev. B 84, 012507 (2011)

    ADS  Article  CAS  Google Scholar 

  14. 14

    Wu, T. et al. Magnetic-field-induced charge-stripe order in the high-temperature superconductor YBa2Cu3Oy . Nature 477, 191–194 (2011)

    ADS  CAS  PubMed  Article  Google Scholar 

  15. 15

    Chen, W.-Q., Yang, K.-Y., Rice, T. M. & Zhang, F. C. Quantum oscillations in magnetic-field-induced antiferromagnetic phase of underdoped cuprates: application to ortho-II YBa2Cu3O6. 5 . Europhys. Lett. 82, 17004 (2008)

    Article  CAS  Google Scholar 

  16. 16

    Sebastian, S. E. et al. Quantum oscillations from nodal bilayer magnetic breakdown in the underdoped high temperature superconductor YBa2Cu3O6+x . Phys. Rev. Lett. 108, 196403 (2012)

    ADS  PubMed  Article  CAS  Google Scholar 

  17. 17

    Sebastian, S. E. et al. Chemical potential oscillations from nodal Fermi surface pocket in the underdoped high-temperature superconductor YBa2Cu3O6+x . Nature Commun. 2, 471 (2011)

    ADS  Article  CAS  Google Scholar 

  18. 18

    Hossain, M. A. et al. In situ doping control of the surface of high-temperature superconductors. Nature Phys. 4, 527–531 (2008)

    CAS  Article  Google Scholar 

  19. 19

    Ghiringhelli, G. et al. Long-range incommensurate charge fluctuations in (Y,Nd)Ba2Cu3O6+x . Science 337, 821–825 (2012)

    ADS  CAS  PubMed  Article  Google Scholar 

  20. 20

    Chang, J. et al. Direct observation of competition between superconductivity and charge density wave order in YBa2Cu3O6. 67 . Nature Phys. 8, 871–876 10.1038/nphys2456 (2012)

    ADS  CAS  Article  Google Scholar 

  21. 21

    LeBoeuf, D. et al. Thermodynamic phase diagram of static charge order in underdoped YBa2Cu3Oy . Nature Phys. 9, 79–83 (2013)

    ADS  CAS  Article  Google Scholar 

  22. 22

    Hinton, J. P. et al. A new collective mode in YBCO observed by time-domain reflectometry. Phys. Rev. B 88, 060508 (2013)

    ADS  Article  CAS  Google Scholar 

  23. 23

    Bergemann, C., Mackenzie, A. P., Julian, S. R., Forsythe, D. & Ohmichi, E. Quasi-two-dimensional Fermi liquid properties of the unconventional superconductor Sr2RuO4 . Adv. Phys. 52, 639–725 (2003)

    ADS  CAS  Article  Google Scholar 

  24. 24

    Li, J.-X., Wu, C.-Q. & Lee, D.-H. Checkerboard charge density wave and pseudogap of high-Tc cuprate. Phys. Rev. B 74, 184515 (2006)

    ADS  Article  CAS  Google Scholar 

  25. 25

    Wang, Y. & Chubukov, A. V. Charge order and loop currents in hole-doped cuprates. Preprint at http://arXiv.org/abs/1401.0712 (2014)

  26. 26

    Castellani, C., Di Castro, C. & Grilli, M. Singular quasiparticle scattering in the proximity of charge instabilities. Phys. Rev. Lett. 75, 4650–4653 (1995)

    ADS  CAS  PubMed  Article  Google Scholar 

  27. 27

    Hayward, L. E., Hawthorn, D. G., Melko, R. G. & Sachdev, S. Angular fluctuations of a multi-component order describe the pseudogap regime of the cuprate superconductors. Science 343, 1336–1339 (2014)

    ADS  CAS  PubMed  Article  Google Scholar 

  28. 28

    Efetov, K. B., Meier, H. & Pépin, C. Pseudogap state near a quantum critical point. Nature Phys. 9, 442–446 (2013)

    ADS  CAS  Article  Google Scholar 

  29. 29

    Nie, L., Tarjus, G. & Kivelson, S. A. Quenched disorder and vestigial nematicity in the pseudo-gap regime of the cuprates. Proc. Natl Acad. Sci. 111, 7980–7985 (2014)

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  30. 30

    Sacuto, A. et al. New insights into the phase diagram of the copper oxide superconductors from electronic Raman scattering. Rep. Prog. Phys. 76, 022502 (2013)

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  31. 31

    Liang, R., Bonn, D. A. & Hardy, W. N. Evaluation of CuO2 plane hole doping in YBa2Cu3O6+x single crystals. Phys. Rev. B 73, 180505 (2006)

    ADS  Article  CAS  Google Scholar 

  32. 32

    Sebastian, S. E. et al. Metal-insulator quantum critical point beneath the high Tc superconducting dome. Proc. Natl Acad. Sci. USA 107, 6175–6179 (2010)

    ADS  CAS  PubMed  PubMed Central  Article  Google Scholar 

  33. 33

    Altarawneh, M. M., Mielke, C. H. & Brooks, J. S. Proximity detector circuits: an alternative to tunnel diode oscillators for contactless measurements in pulsed magnetic field environments. Rev. Sci. Instrum. 80, 066104 (2009)

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  34. 34

    Ramshaw, B. J. et al. Vortex lattice melting and Hc2 in underdoped YBa2Cu3Oy. Phys. Rev. B 86, 174501 (2012)

    Article  CAS  Google Scholar 

  35. 35

    Ruiller-Albenque, F., Alloul, H., Colson, D. & Forget, A. Determination of superconducting fluctuations in high-Tc cuprates. J. Phys. Conf. Ser. 449, 012010 (2013)

    Article  CAS  Google Scholar 

  36. 36

    Ramshaw, B. J. et al. Angle dependence of quantum oscillations in YBa2Cu3O6. 59 shows free-spin behaviour of quasiparticles. Nature Phys. 7, 234–238 (2011)

    ADS  CAS  Article  Google Scholar 

  37. 37

    Yamaji, K. On the angle dependence of the magnetoresistance in quasi-two-dimensional organic superconductors. J. Phys. Soc. Jpn 58, 1520–1523 (1989)

    ADS  CAS  Article  Google Scholar 

  38. 38

    Shoenberg, D. Magnetic Oscillations in Metals (Cambridge Univ. Press, 1984)

    Book  Google Scholar 

  39. 39

    Singleton, J. & Shoenberg, D. Band Theory and Electronic Properties of Solids (Oxford Univ. Press, 2001)

    Google Scholar 

  40. 40

    Wosnitza, J. Fermi surfaces of organic superconductors. Int. J. Mod. Phys. B 7, 2707–2741 (1993)

    ADS  CAS  Article  Google Scholar 

  41. 41

    Kartsovnik, M. V. High magnetic fields: a tool for studying electronic properties of layered organic metals. Chem. Rev. 104, 5737–5782 (2004)

    CAS  PubMed  Article  PubMed Central  Google Scholar 

  42. 42

    Yelland, E. A. et al. Quantum oscillations in the underdoped cuprate YBa2Cu4O8 . Phys. Rev. Lett. 100, 047003 (2008)

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  43. 43

    Bangura, A. F. et al. Quantum oscillations in the underdoped cuprate YBa2Cu4O8 . Phys. Rev. Lett. 100, 046004 (2008)

    Article  CAS  Google Scholar 

  44. 44

    Audouard, A. et al. Multiple quantum oscillations in the de Haas–van Alphen spectra of the underdoped high-temperature superconductor YBa2Cu3O6. 5 . Phys. Rev. Lett. 103, 157003 (2009)

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

  45. 45

    Sebastian, S. E. et al. A multi-component Fermi surface in the vortex state of an underdoped high-Tc superconductor. Nature 454, 200–203 (2008)

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  46. 46

    Singleton, J. et al. Magnetic quantum oscillations in YBa2Cu3O6. 61 and YBa2Cu3O6. 69 in fields of up to 85 T: patching the hole in the roof of the superconducting dome. Phys. Rev. Lett. 104, 086403 (2010)

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

  47. 47

    Sebastian, S. E. et al. Compensated electron and hole pockets in an underdoped high-Tc superconductor. Phys. Rev. B 81, 214524 (2010)

    ADS  Article  CAS  Google Scholar 

  48. 48

    LeBoeuf, D. et al. Lifshitz critical point in the cuprate superconductor YBa2Cu3Oy from high-field Hall effect measurements. Phys. Rev. B 83, 054506 (2011)

    ADS  Article  CAS  Google Scholar 

  49. 49

    Rourke, P. M. C. et al. A detailed de Haas–van Alphen effect study of the overdoped cuprate Tl2Ba2CuO6+δ . New J. Phys. 12, 105009/1–29 (2010)

    ADS  Article  CAS  Google Scholar 

  50. 50

    Helm, T. et al. Evolution of the Fermi surface of the electron-doped high-temperature superconductor Nd2–xCexCuO4 revealed by Shubnikov–de Haas oscillations. Phys. Rev. Lett. 103, 157002 (2009)

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  51. 51

    Carrington, A. Quantum oscillation studies of the Fermi surface of iron-pnictide superconductors. Rep. Prog. Phys. 74, 124507 (2011)

    ADS  Article  CAS  Google Scholar 

  52. 52

    Vignolle, B. et al. Quantum oscillations in an overdoped high-Tc superconductor. Nature 455, 952–955 (2008)

    ADS  CAS  Article  Google Scholar 

  53. 53

    Vignolle, B. et al. Quantum oscillations and the Fermi surface of high-temperature cuprate superconductors. C. R. Phys. 12, 446–460 (2011)

    ADS  CAS  Article  Google Scholar 

  54. 54

    Fazekas, P. Lecture Notes on Electron Correlation and Magnetism (World Scientific, 1999)

    Book  Google Scholar 

  55. 55

    Pines, D. & Noziéres, P. The Theory of Quantum Liquids: Normal Fermi Liquids (Addison-Wesley, 1989)

    Google Scholar 

  56. 56

    Bennett, A. J. & Falicov, L. M. g Factor in metallic zinc. Phys. Rev. 136, A998–A1002 (1964)

    ADS  Article  Google Scholar 

  57. 57

    Proshin, J. N. & Useinov, N. K. The spin flip in the theory of magnetic breakdown: magnetoresistance. Physica B 173, 386–388 (1991)

    ADS  Article  Google Scholar 

  58. 58

    Garcia-Aldea, D. & Charkravarty, S. Multiple quantum oscillation frequencies in YBa2Cu3O6+δ and bilayer splitting. New J. Phys. 12, 105005 (2010)

    ADS  Article  CAS  Google Scholar 

  59. 59

    Reynoso, A. A., Usaj, G. & Balseiro, C. A. Magnetic breakdown of cyclotron orbits in systems with Rashba and Dresselhaus spin-orbit coupling. Phys. Rev. B 78, 115312 (2008)

    ADS  Article  CAS  Google Scholar 

  60. 60

    Harrison, N. Near doping-independent pocket area from an antinodal Fermi surface instability in underdoped high temperature superconductors. Phys. Rev. Lett. 107, 186408 (2011)

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  61. 61

    Harrison, N. & Sebastian, S. E. Fermi surface reconstruction from bilayer charge ordering in the underdoped high temperature superconductor YBa2Cu3O6+x . New J. Phys. 14, 095023 (2012)

    ADS  Article  CAS  Google Scholar 

  62. 62

    Andersen, O. K., Liechtenstein, A. I., Jepsen, O. & Paulsen, F. LDA energy bands, low-energy Hamiltonians, t′, t″, t (k) and J . J. Phys. Chem. Solids 56, 1573–1591 (1995)

    ADS  CAS  Article  Google Scholar 

  63. 63

    Lee, P. A. From high temperature superconductivity to quantum spin liquid: progress in strong correlation physics. Rep. Prog. Phys. 71, 012501 (2008)

    ADS  Article  CAS  Google Scholar 

  64. 64

    Chakravarty, S., Laughlin, R. B., Morr, D. K. & Nayak, C. Hidden order in the cuprates. Phys. Rev. B 63, 094503 (2001)

    ADS  Article  CAS  Google Scholar 

  65. 65

    Wang, Z.-Q., Kotliar, G. & Wang, X.-F. Flux-density wave and superconducting instability of the staggered-flux phase. Phys. Rev. B 42, 8690–8693 (1990)

    ADS  CAS  Article  Google Scholar 

  66. 66

    Emery, V. J. & Kivelson, S. A. Importance of phase fluctuations in superconductors with small superfluid density. Nature 374, 434–437 (1994)

    ADS  Article  Google Scholar 

  67. 67

    Tranquada, J. M., Sternlieb, B. J., Axe, J. D., Nakamura, Y. & Uchida, S. Evidence for stripe correlations of spins and holes in copper oxide superconductors. Nature 375, 561–563 (1994)

    ADS  Article  Google Scholar 

  68. 68

    Chubukov, A. V. &. Morr, D. K. Electronic structure of underdoped cuprates. Phys. Rep. 288, 355–387 (1997)

    ADS  CAS  Article  Google Scholar 

  69. 69

    Varma, C. M. Non-Fermi-liquid states and pairing instability of a general model of copper oxide metals. Phys. Rev. B 55, 14554–14580 (1997)

    ADS  CAS  Article  Google Scholar 

  70. 70

    Schmalian, J., Pines, D. & Stojkovich, B. Weak pseudogap behavior in the underdoped cuprate superconductors. Phys. Rev. Lett. 80, 3839–3842 (1998)

    ADS  CAS  Article  Google Scholar 

  71. 71

    Kivelson, S. A. et al. How to detect fluctuating stripes in the high-temperature superconductors. Rev. Mod. Phys. 75, 1201–1241 (2003)

    ADS  CAS  Article  Google Scholar 

  72. 72

    Choy, T.-P. & Phillips Doped Mott insulators are insulators: hole localization in the cuprates. Phys. Rev. Lett. 95, 196405 (2005)

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

  73. 73

    Kyung, D.-B. et al. Pseudogap induced by short-range spin correlations in a doped Mott insulator. Phys. Rev. B 73, 165114 (2006)

    ADS  Article  CAS  Google Scholar 

  74. 74

    Li, C., Zhou, S. & Wang, Z. Inhomogeneous states with checkerboard order in the tJ model. Phys. Rev. B 73, 060501 (2006)

    ADS  Article  CAS  Google Scholar 

  75. 75

    Zaanen, J. et al. Towards a complete theory of high Tc . Nature Phys. 2, 138–143 (2006)

    Article  Google Scholar 

  76. 76

    Monthoux, P., Pines, D. & Lonzarich, G. G. Superconductivity without phonons. Nature 450, 1177–1183 (2007)

    ADS  CAS  PubMed  Article  PubMed Central  Google Scholar 

  77. 77

    Tewari, S., Zhang, C., Yakovenko, V. M. & Das Sarma, S. Time-reversal symmetry breaking by a (d+id) density-wave state in underdoped cuprate superconductors. Phys. Rev. Lett. 100, 217004 (2008)

    ADS  PubMed  Article  CAS  PubMed Central  Google Scholar 

  78. 78

    Anderson, P. W. & Casey, P. A. Hidden Fermi liquid: self-consistent theory for the normal state of high-Tc superconductors. Phys. Rev. Lett. 106, 097002 (2009)

    Google Scholar 

  79. 79

    Berg, E., Fradkin, E., Kivelson, S. A. & Tranquada, J. Striped superconductors: how the cuprates intertwine spin, charge and superconducting orders. New J. Phys. 11, 115004 (2009)

    ADS  Article  CAS  Google Scholar 

  80. 80

    Sedrakyan, T. A. & Chubukov, A. V. Pseudogap in underdoped cuprates and spin-density-wave fluctuations. Phys. Rev. B 81, 174536 (2010)

    ADS  Article  CAS  Google Scholar 

  81. 81

    Sushkov, O. P. Magnetic properties of lightly doped antiferromagnetic YBa2Cu3Oy . Phys. Rev. B 84, 094532 (2011)

    ADS  Article  CAS  Google Scholar 

  82. 82

    Rice, T. M., Yang, K.-Y. & Zhang, F. C. A phenomenological theory of the anomalous pseudogap phase in underdoped cuprates. Rep. Prog. Phys. 75, 016502 (2012)

    ADS  CAS  PubMed  Article  Google Scholar 

  83. 83

    Scalapino, D. J. A common thread: the pairing interaction for unconventional superconductors. Rev. Mod. Phys. 84, 1383 (2012)

    ADS  CAS  Article  Google Scholar 

  84. 84

    Sachdev, S. & La Placa, R. Bond order in two-dimensional metals with antiferromagnetic exchange interactions. Phys. Rev. Lett. 111, 027202 (2013)

    ADS  PubMed  Article  CAS  Google Scholar 

  85. 85

    Wang, Y. & Chubukov, A. Superconductivity at the onset of the spin-density-wave order in a metal. Phys. Rev. Lett. 110, 127001 (2013)

    ADS  PubMed  Article  CAS  Google Scholar 

  86. 86

    Tremblay, A.-M. S. Strongly correlated superconductivity. In Autumn School on Correlated Electrons: Emergent Phenomena in Correlated Matter (September 23–27, 2013, Forschungszentrum Julich, Germany) Preprint at http://arXiv.org/1310.1481 (2013)

  87. 87

    Hosur, P., Kapitulnik, A., Kivelson, S. A., Orenstein, J. & Raghu, S. Kerr effect as evidence of gyrotropic order in the cuprates. Phys. Rev. B 87, 115116 (2013)

    ADS  Article  CAS  Google Scholar 

  88. 88

    Gull, E., Parcollet, O. & Millis, A. J. Superconductivity and the pseudogap in the two-dimensional Hubbard model. Phys. Rev. Lett. 110, 216405 (2013)

    ADS  PubMed  Article  CAS  Google Scholar 

  89. 89

    Kivelson, S. A., Fradkin, E. & Emery, V. J. Electronic liquid-crystal phases of a doped Mott insulator. Nature 393, 550–553 (1998)

    ADS  CAS  Article  Google Scholar 

  90. 90

    Sau, J. D. & Sachdev, S. Mean field theory of competing orders in metals with antiferromagnetic exchange interactions. Phys. Rev. B 89, 075129 (2014)

    ADS  Article  CAS  Google Scholar 

  91. 91

    Laughlin, R. B. Hartree-Fock computation of the high-Tc cuprate phase diagram. Phys. Rev. B 89, 035134 (2014)

    ADS  Article  CAS  Google Scholar 

  92. 92

    Meier, H., Pepin, C., Einenkel, M. & Efetov, K. B. Cascade of phase transitions in the vicinity of a quantum critical point. Preprint at http://arXiv.org/1312.2010 (2013)

  93. 93

    Alexandrov, A. S. & Bratkovsky, A. M. de Haas-van alphen effect in canonical and grand canonical multiband fermi liquid. Phys. Rev. Lett. 76, 1308–1311 (1996)

    ADS  CAS  PubMed  Article  Google Scholar 

  94. 94

    Carrington, A. & Yelland, E. A. Band-structure calculations of Fermi-surface pockets in ortho-II YBa2Cu3O6. 5 . Phys. Rev. B 76, 140508 (2007)

    ADS  Article  CAS  Google Scholar 

  95. 95

    Elfimov, I. S., Sawatsky, G. A. & Damascelli, A. Theory of Fermi-surface pockets and correlation effects in underdoped YBa2Cu3O6. 5 . Phys. Rev. B 77, 060504 (2008)

    ADS  Article  CAS  Google Scholar 

  96. 96

    Melikyan Vafek, A. &. O. Quantum oscillations in the mixed state of d-wave superconductors. Phys. Rev. B 78, 020502 (2008)

    ADS  Article  CAS  Google Scholar 

  97. 97

    Varma, C. M. Magneto-oscillations in underdoped cuprates. Phys. Rev. B 79, 085110 (2009)

    ADS  Article  CAS  Google Scholar 

  98. 98

    Senthil, T. & Lee, P. A. Synthesis of the phenomenology of the underdoped cuprates. Phys. Rev. B 79, 245116 (2009)

    ADS  Article  CAS  Google Scholar 

  99. 99

    Pereg-Barnea, T., Weber, H., Refael, G. & Franz, M. Quantum oscillations from Fermi arcs. Nature Phys. 6, 44–49 (2010)

    ADS  CAS  Article  Google Scholar 

  100. 100

    Oh, H., Choi, H. J. & Louie, S. G. &. Cohen, M. L. Fermi surfaces and quantum oscillations in the underdoped high-Tc superconductors YBa2Cu3O6. 5 and YBa2Cu4O8 . Phys. Rev. B 84, 014518 (2011)

    ADS  Article  CAS  Google Scholar 

  101. 101

    Ong, N. P. Geometric interpretation of the weak-field Hall conductivity in two-dimensional metals with arbitrary Fermi surface. Phys. Rev. B 43, 193–201 (1991)

    ADS  CAS  Article  Google Scholar 

  102. 102

    Swanson, J. A. Saturation Hall constant of semiconductors. Phys. Rev. 99, 1799–1807 (1955)

    ADS  CAS  Article  Google Scholar 

  103. 103

    Ziman, J. M. Principles of the Theory of Solids (Cambridge Univ. Press, 1979)

    MATH  Google Scholar 

  104. 104

    Stark, R. W. & Reifenberger, R. Quantitative theory for the quantum interference effect in the transverse magnetoresistance of pure magnesium. J. Low Temp. Phys. 26, 763–817 (1977)

    ADS  CAS  Article  Google Scholar 

  105. 105

    Harrison, N. et al. Magnetic breakdown and quantum interference in the quasi-two-dimensional superconductor κ – (BEDT – TTF)2Cu(NCS)2 in high magnetic fields. J. Phys. Condens. Matter 8, 5415–5435 (1996)

    ADS  CAS  Article  Google Scholar 

  106. 106

    Basov, D. N. & Timusk, T. Electrodynamics of high-Tc superconductors. Rev. Mod. Phys. 77, 721–779 (2005)

    ADS  CAS  Article  Google Scholar 

  107. 107

    Allais, A., Chowdhury, D. & Sachdev, S. Connecting high-field quantum oscillations to the pseudogap in the underdoped cuprates. Preprint at http://arxiv.org/abs/1406.0503 (2014)

Download references

Acknowledgements

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.

Author information

Affiliations

Authors

Contributions

S.E.S., N.H., F.F.B., M.M.A. and P.A.G. performed high magnetic field measurements. R.L., D.A.B. and W.N.H. prepared single crystals. S.E.S., N.H. and G.G.L. analysed data and wrote the paper.

Corresponding authors

Correspondence to Suchitra E. Sebastian or N. Harrison.

Ethics declarations

Competing interests

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.

Extended Data Figure 3 Quantum oscillations up to 100 T.

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.

Extended Data Table 1 Model parameters for a staggered twofold Fermi surface model
Extended Data Table 2 Magnetic breakdown amplitude damping
Extended Data Table 3 Model parameters for a fundamental neck and belly Fermi surface model

PowerPoint slides

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

Further reading

Comments

By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.

Search

Quick links