Abstract
The nature of the pseudogap phase remains a major puzzle in our understanding of cuprate high-temperature superconductivity. Whether or not this metallic phase is defined by any of the reported broken symmetries, the topology of its Fermi surface remains a fundamental open question. Here we use angle-dependent magnetoresistance (ADMR) to measure the Fermi surface of the La1.6âxNd0.4SrxCuO4 cuprate. Outside the pseudogap phase, we fit the ADMR data and extract a Fermi surface geometry that is in excellent agreement with angle-resolved photoemission data. Within the pseudogap phase, the ADMR is qualitatively different, revealing a transformation of the Fermi surface. We can rule out changes in the quasiparticle lifetime as the sole cause of this transformation. We find that our data are most consistent with a pseudogap Fermi surface that consists of small, nodal hole pockets, thereby accounting for the drop in carrier density across the pseudogap transition found in several cuprates.
This is a preview of subscription content, access via your institution
Access options
Access Nature and 54 other Nature Portfolio journals
Get Nature+, our best-value online-access subscription
$29.99 /Â 30Â days
cancel any time
Subscribe to this journal
Receive 12 print issues and online access
$259.00 per year
only $21.58 per issue
Buy this article
- Purchase on SpringerLink
- Instant access to full article PDF
Prices may be subject to local taxes which are calculated during checkout
Similar content being viewed by others
Data availability
Source data are provided with this paper. Other experimental data presented in this paper are available at http://wrap.warwick.ac.uk/161600/. The results of the conductivity simulations are available from the corresponding author upon reasonable request.
Code availability
The code used to compute the conductivity is available from the corresponding author upon reasonable request.
References
Keimer, B., Kivelson, S. A., Norman, M. R., Uchida, S. & Zaanen, J. From quantum matter to high-temperature superconductivity in copper oxides. Nature 518, 179â186 (2015).
Hussey, N. E., Abdel-Jawad, M., Carrington, A., Mackenzie, A. P. & Balicas, L. A coherent three-dimensional Fermi surface in a high-transition-temperature superconductor. Nature 425, 814â817 (2003).
Platé, M. et al. Fermi surface and quasiparticle excitations of overdoped Tl2Ba2CuO6+δ. Phys. Rev. Lett. 95, 077001 (2005).
Vignolle, B. et al. Quantum oscillations in an overdoped high-Tc superconductor. Nature 455, 952â955 (2008).
Damascelli, A., Hussain, Z. & Shen, Z.-X. Angle-resolved photoemission studies of the cuprate superconductors. Rev. Mod. Phys. 75, 473â541 (2003).
Cooper, R. A. et al. Anomalous criticality in the electrical resistivity of La2âxSrxCuO4. Science 323, 603â607 (2009).
Daou, R. et al. Linear temperature dependence of resistivity and change in the Fermi surface at the pseudogap critical point of a high-Tc superconductor. Nat. Phys. 5, 31â34 (2009).
Grissonnanche, G. et al. Linear-in temperature resistivity from an isotropic Planckian scattering rate. Nature 595, 667â672 (2021).
Doiron-Leyraud, N. et al. Quantum oscillations and the Fermi surface in an underdoped high-Tc superconductor. Nature 447, 565â568 (2007).
Ramshaw, B. J. et al. Quasiparticle mass enhancement approaching optimal doping in a high-Tc superconductor. Science 348, 317â320 (2015).
Chan, M. K. et al. Extent of Fermi-surface reconstruction in the high-temperature superconductor HgBa2CuO4+δ. Proc. Natl Acad. Sci. USA 117, 9782â9786 (2020).
Blanco-Canosa, S. et al. Resonant X-ray scattering study of charge-density wave correlations in YBa2Cu3O6+x. Phys. Rev. B 90, 054513 (2014).
Gupta, N. K. et al. Vanishing nematic order beyond the pseudogap phase in overdoped cuprate superconductors. Proc. Natl Acad. Sci. USA 118, e2106881118 (2021).
Tranquada, J. M. et al. Coexistence of, and competition between, superconductivity and charge-stripe order in La1.6âxNd0.4SrxCuO4. Phys. Rev. Lett. 78, 338â341 (1997).
Ma, Q. et al. Parallel spin stripes and their coexistence with superconducting ground states at optimal and high doping in La1.6âxNd0.4SrxCuO4. Phys. Rev. Research 3, 023151 (2021).
Frachet, M. et al. Hidden magnetism at the pseudogap critical point of a cuprate superconductor. Nat. Phys. 16, 1064â1068 (2020).
Kunisada, S. et al. Observation of small Fermi pockets protected by clean CuO2 sheets of a high-Tc superconductor. Science 369, 833â838 (2020).
Bourgeois-Hope, P. et al. Link between magnetism and resistivity upturn in cuprates: a thermal conductivity study of La2âxSrxCuO4. Preprint at https://arxiv.org/abs/1910.08126 (2019).
Scheurer, M. S. et al. Topological order in the pseudogap metal. Proc. Natl Acad. Sci. USA 115, E3665âE3672 (2018).
Collignon, C. et al. Fermi-surface transformation across the pseudogap critical point of the cuprate superconductor La1.6âxNd0.4SrxCuO4. Phys. Rev. B 95, 224517 (2017).
Michon, B. et al. Thermodynamic signatures of quantum criticality in cuprate superconductors. Nature 567, 218â222 (2019).
Matt, C. E. et al. Electron scattering, charge order, and pseudogap physics in La1.6âxNd0.4SrxCuO4: an angle-resolved photoemission spectroscopy study. Phys. Rev. B 92, 134524 (2015).
Adachi, T., Noji, T. & Koike, Y. Crystal growth, transport properties, and crystal structure of the single-crystal La2âxBaxCuO4 (xâ=â0.11). Phys. Rev. B 64, 144524 (2001).
LeBoeuf, D. et al. Electron pockets in the Fermi surface of hole-doped high-Tc superconductors. Nature 450, 533â536 (2007).
Noda, T., Eisaki, H. & Uchida, S.-I. Evidence for one-dimensional charge transport in La2âxâyNdySrxCuO4. Science 286, 265â268 (1999).
Michon, B. et al. Wiedemann-Franz law and abrupt change in conductivity across the pseudogap critical point of a cuprate superconductor. Phys. Rev. X 8, 041010 (2018).
Chambers, R. G. The kinetic formulation of conduction problems. Proc. Phys. Soc. A 65, 458â459 (1952).
Yamaji, K. On the angle dependence of the magnetoresistance in quasi-two-dimensional organic superconductors. J. Phys. Soc. Jpn 58, 1520â1523 (1989).
Abdel-Jawad, M. et al. Anisotropic scattering and anomalous normal-state transport in a high-temperature superconductor. Nat. Phys. 2, 821â825 (2006).
Singleton, J. Studies of quasi-two-dimensional organic conductors based on BEDT-TTF using high magnetic fields. Rep. Prog. Phys. 63, 1111â1207 (2000).
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, 639â725 (2003).
Ramshaw, B. J. et al. Broken rotational symmetry on the Fermi surface of a high-Tc superconductor. npj Quantum Mater. 2, 8 (2017).
Kartsovnik, M. V. et al. Fermi surface of the electron-doped cuprate superconductor Nd2âxCexCuO4 probed by high-field magnetotransport. New J. Phys. 13, 015001 (2011).
Horio, M. et al. Three-dimensional Fermi surface of overdoped La-based cuprates. Phys. Rev. Lett. 121, 077004 (2018).
Vershinin, M. et al. Local ordering in the pseudogap state of the high-Tc superconductor Bi2Sr2CaCu2O8+δ. Science 303, 1995â1998 (2004).
Wu, T. et al. Magnetic-field-induced charge-stripe order in the high-temperature superconductor YBa2Cu3Oy. Nature 477, 191â194 (2011).
Allais, A., Chowdhury, D. & Sachdev, S. Connecting high-field quantum oscillations to zero-field electron spectral functions in the underdoped cuprates. Nat. Commun. 5, 5771 (2014).
Collignon, C. et al. Thermopower across the phase diagram of the cuprate La1.6âxNd0.4SrxCuO4: signatures of the pseudogap and charge density wave phases. Phys. Rev. B 103, 155102 (2021).
Doiron-Leyraud, N. et al. Hall, Seebeck, and Nernst coefficients of underdoped HgBa2CuO4+δ: Fermi-surface reconstruction in an archetypal cuprate superconductor. Phys. Rev. X 3, 021019 (2013).
Wen, X.-G. & Lee, P. A. Theory of underdoped cuprates. Phys. Rev. Lett. 76, 503â506 (1996).
Chakravarty, S., Laughlin, R. B., Morr, D. K. & Nayak, C. Hidden order in the cuprates. Phys. Rev. B 63, 094503 (2001).
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 (2011).
Storey, J. G. Hall effect and Fermi surface reconstruction via electron pockets in the high-Tc cuprates. Europhys. Lett. 113, 27003 (2016).
Badoux, S. et al. Change of carrier density at the pseudogap critical point of a cuprate superconductor. Nature 531, 210â214 (2016).
Li, Z.-X. and Lee, D.-H. The thermal Hall conductance of two doped symmetry-breaking topological insulators. Preprint at https://arxiv.org/abs/1905.04248 (2019).
Lewin, S. K. & Analytis, J. G. Angle-dependent magnetoresistance oscillations of cuprate superconductors in a model with Fermi surface reconstruction and magnetic breakdown. Phys. Rev. B 92, 195130 (2015).
Sénéchal, D. & Tremblay, A.-M. S. Hot spots and pseudogaps for hole- and electron-doped high-temperature superconductors. Phys. Rev. Lett. 92, 126401 (2004).
Scalapino, D. J. A common thread: the pairing interaction for unconventional superconductors. Rev. Mod. Phys. 84, 1383â1417 (2012).
Wu, W., Ferrero, M., Georges, A. & Kozik, E. Controlling Feynman diagrammatic expansions: physical nature of the pseudogap in the two-dimensional Hubbard model. Phys. Rev. B 96, 041105 (2017).
Gannot, Y., Ramshaw, B. J. & Kivelson, S. A. Fermi surface reconstruction by a charge density wave with finite correlation length. Phys. Rev. B 100, 045128 (2019).
Badoux, S. et al. Critical doping for the onset of Fermi-surface reconstruction by charge-density-wave order in the cuprate superconductor La2âxSrxCuO4. Phys. Rev. X 6, 021004 (2016).
Putzke, C. et al. Reduced Hall carrier density in the overdoped strange metal regime of cuprate superconductors. Nat. Phys. 17, 826â831 (2021).
Lizaire, M. et al. Transport signatures of the pseudogap critical point in the cuprate superconductor Bi2Sr2âxLaxCuO6+δ. Phys. Rev. B 104, 014515 (2021).
Shishido, H., Settai, R., Harima, H. & Ånuki, Y. A drastic change of the Fermi surface at a critical pressure in CeRhIn5: dHvA study under pressure. J. Phys. Soc. Jpn 74, 1103â1106 (2005).
Walmsley, P. et al. Quasiparticle mass enhancement close to the quantum critical point in BaFe2(As1âxPx)2. Phys. Rev. Lett. 110, 257002 (2013).
Uji, S. et al. Rapid oscillation and Fermi-surface reconstruction due to spin-density-wave formation in the organic conductor (TMTSF)2PF6. Phys. Rev. B 55, 12446 (1997).
Analytis, J. G. et al. Quantum oscillations in the parent pnictide BaFe2As2: itinerant electrons in the reconstructed state. Phys. Rev. B 80, 064507 (2009).
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, 561â563 (1995).
Goddard, P. A. et al. Angle-dependent magnetoresistance of the layered organic superconductor κ-(ET)2Cu(NCS)2: simulation and experiment. Phys. Rev. B 69, 174509 (2004).
Chakravarty, S., Sudbø, A., Anderson, P. W. & Strong, S. Interlayer tunneling and gap anisotropy in high-temperature superconductors. Science 261, 337â340 (1993).
Abrahams, E. & Varma, C. M. What angle-resolved photoemission experiments tell about the microscopic theory for high-temperature superconductors. Proc. Natl Acad. Sci. USA 97, 5714â5716 (2000).
Analytis, J. G., Abdel-Jawad, M., Balicas, L., French, M. M. J. & Hussey, N. E. Angle-dependent magnetoresistance measurements in Tl2Ba2CuO6+δ and the need for anisotropic scattering. Phys. Rev. B 76, 104523 (2007).
Tremblay, A.-M.S. in Strongly Correlated Systems (eds Avella, A. & Mancini, F.) 409â453 (Springer, 2012).
Sebastian, S. E. et al. Normal-state nodal electronic structure in underdoped high-Tc copper oxides. Nature 511, 61â64 (2014).
Chan, M. K. et al. Single reconstructed Fermi surface pocket in an underdoped single-layer cuprate superconductor. Nat. Commun. 7, 12244 (2016).
Riggs, S. C. et al. Heat capacity through the magnetic-field-induced resistive transition in an underdoped high-temperature superconductor. Nat. Phys. 7, 332â335 (2011).
Sachdev, S. & La Placa, R. Bond order in two-dimensional metals with antiferromagnetic exchange interactions. Phys. Rev. Lett. 111, 027202 (2013).
Acknowledgements
We acknowledge helpful discussions with J. Analytis, D. Chowdhury, N. Doiron-Leyraud, N. Hussey, M. Kartsovnik, S. Kivelson, D.-H. Lee, S. Lewin, A.-M. Tremblay, K. Modic, S. Musser, C. Proust and S. Todadri. A part of this work was performed at the National High Magnetic Field Laboratory, which is supported by the National Science Foundation (NSF) cooperative agreement no. DMR-1644779 and the State of Florida. P.A.G. acknowledges that this project is supported by the European Research Council (ERC) under the European Unionâs Horizon 2020 research and innovation programme (grant agreement no. 681260). J.Z. was supported by an NSF grant (MRSEC DMR-1720595). L.T. acknowledges support from the Canadian Institute for Advanced Research (CIFAR) as a Fellow and funding from the Natural Sciences and Engineering Research Council of Canada (NSERC; PIN: 123817), the Fonds de recherche du QuébecâNature et Technologies (FRQNT), the Canada Foundation for Innovation (CFI), and a Canada Research Chair. This research was undertaken thanks in part to funding from the Canada First Research Excellence Fund. Part of this work was funded by the Gordon and Betty Moore Foundationâs EPiQS Initiative (grant GBMF5306 to L.T.). B.J.R. and Y.F. acknowledge funding from the NSF under grant no. DMR-1752784.
Author information
Authors and Affiliations
Contributions
A.L., P.A.G., L.T. and B.J.R. conceived the experiment. J.Z. grew the samples. A.L., F.L., A.A., C.C. and M.D. performed the sample preparation and characterization. Y.F., G.G., A.L., D.G., P.A.G. and B.J.R. performed the high-magnetic-field measurements at the National High Magnetic Field Laboratory. Y.F., G.G., S.V., M.J.L. and B.J.R. performed the data analysis and simulations. Y.F., G.G., S.V., P.A.G., L.T. and B.J.R. wrote the manuscript with input from all the other co-authors. L.T. and B.J.R. supervised the project.
Corresponding author
Ethics declarations
Competing interests
The authors declare no competing interests.
Peer review
Peer review information
Nature Physics thanks the anonymous reviewers for their contribution to the peer review of this work.
Additional information
Publisherâs note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Extended data
Extended Data Fig. 1 Resistivity of Nd-LSCO at p = 0.24 near TSDW.
In-plane resistivity data at B = 35 T as a function of temperature (reproduced from ref. 20). The resistivity Ïxx (red line) is perfectly linear over this temperature range without any sign of an upturn or even a change in slope at TSDWâ=â13â±â1 K (black arrow) reported by Ma et al.15 at Bâ=â0 T. This suggests that either the SDW is not present in our samples or that the SDW vanishes in a magnetic field and thus does not interfere with our measurements performed at B = 45 T.
Extended Data Fig. 2 Best fit of Nd-LSCO p = 0.21 data with the large, hole-like, unreconstructed Fermi surface.
(a) ADMR data on Nd-LSCO pâ=â0.21 at Tâ=â25 K and Bâ=â45 T; (b, c) The best fits for the ADMR data in (a) using the band structure ARPES values for Nd-LSCO pâ=â0.24 with the chemical potential shifted across the van Hove point (at pâââ0.23) to pâ=â0.21, where the Fermi surface is hole-like. Insets represent the scattering rate distribution values over the hole-like Fermi surface at pâ=â0.21. In (b), the scattering is isotropic over the Fermi surface; in (c) we use the cosine scattering rate model (this figure differs from Fig. 3b because there we only shift the chemical potential, while here we show the best-fit using this model).
Extended Data Fig. 3 Calculation of ADMR for a period three CDW Fermi surface reconstruction.
Calculations using two different gap sizes are shown in (a) and (b), and using a d-wave form factor is shown in (c).
Extended Data Fig. 4 The Hall effect in Nd-LSCO at pâ=â0.21.
The data is taken at 30 K and is reproduced from Collignon et al.20. âh pocketâ is from the fit to the data shown in Fig. 4 of the main text; âh+e pocketâ is from a fit that includes both the hole and electron pockets after (Ï,Ï) reconstruction; âFermi arcsâ is from the fit in Fig. 3c,d of the main text; âe pocketâ is from just the electron pocket produced by (Ï,Ï) reconstruction, scaled down by a factor of 20 for clarity.
Extended Data Fig. 5 Variation in the Fermi velocity around the Fermi surface above and below pâ.
The red curve plots the magnitude of the Fermi velocity around the Fermi surface at pâ=â0.24, as shown in Fig. 2. The blue curve plots the same quantity for a single nodal hole pocket, as shown in Fig. 4 (the reduction in symmetry is because each nodal hole pocket is 2-fold symmetric). The total anisotropy in vF around the Fermi surface is a factor of 25 at pâ=â0.24, but just larger than a factor of 2 at pâ=â0.21.
Extended Data Fig. 6 ADMR experimental set up.
(a) An illustration of the sample mounting. The two samples here are mounted on a G-10 wedge to provide a Ï angle of 30â. Additional wedges provided angles of Ïâ=â15â and 45â; (b) ADMR as a function of θ angle fromâââ15â to 110â and Ïâ=â0 at Tâ=â20 K for Nd-LSCO pâ=â0.24, showing the symmetry of the data about these two angles.
Extended Data Fig. 7 ADMR dependence on the gap amplitude with (Ï,Ï) reconstruction.
ADMR calculations with a (Ï,Ï) reconstructed Fermi surface for different gap amplitudes at fixed isotropic scattering rate value 1/Ïâ=â22.88 psâ1. Note that this withinâââ40% of the nodal scattering rate at pâ=â0.24, consistent with a nodal hole pockets reconstructed from the larger Fermi surface.
Extended Data Fig. 8 ADMR dependence on the scattering rate amplitude with (Ï,Ï) reconstruction.
ADMR calculations with a (Ï,Ï) reconstructed Fermi surface for different isotropic scattering rate amplitudes at fixed gap value at Îâ=â55 K.
Supplementary information
Supplementary Information
Supplementary Figs. 1â3 and Discussion.
Source data
Source Data Fig. 1
Experimental dÏ/Ï as a function of θ for several ɸ values.
Source Data Fig. 2
Experimental dÏ/Ï as a function of θ for several ɸ values (Fig. 2a). Simulated dÏ/Ï as a function of θ for several ɸ values (Fig. 2b). Contours at three values of kz (Fig. 2c).
Source Data Fig. 3
Simulated dÏ/Ï as a function of θ for several ɸ values.
Source Data Fig. 4
Experimental dÏ/Ï as a function of θ for several ɸ values (Fig. 4a). Simulated dÏ/Ï as a function of θ for several ɸ values (Fig. 4b).
Rights and permissions
About this article
Cite this article
Fang, Y., Grissonnanche, G., Legros, A. et al. Fermi surface transformation at the pseudogap critical point of a cuprate superconductor. Nat. Phys. 18, 558â564 (2022). https://doi.org/10.1038/s41567-022-01514-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1038/s41567-022-01514-1
This article is cited by
-
Flat bands, strange metals and the Kondo effect
Nature Reviews Materials (2024)
-
Evolution from a charge-ordered insulator to a high-temperature superconductor in Bi2Sr2(Ca,Dy)Cu2O8+δ
Nature Communications (2024)
-
Fate of charge order in overdoped La-based cuprates
npj Quantum Materials (2023)
-
Electrons with Planckian scattering obey standard orbital motion in a magnetic field
Nature Physics (2022)