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Observation of the Yamaji effect in a cuprate superconductor

Nature Physics volume 21, pages 1753–1758 (2025)Cite this article

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Abstract

The pseudogap state of high-temperature superconducting cuprates, known for its partial gapping of the Fermi surface above the superconducting transition temperature, is believed to hold the key to understanding the origin of Planckian relaxation and quantum criticality. However, the nature of the Fermi surface in the pseudogap state has remained a fundamental open question. Here we report the observation of the Yamaji effect, which appears as a peak in the c-axis resistivity at a specific angle of the applied magnetic field, in angle-dependent magnetoresistivity measurements above the critical temperature in the cuprate HgBa2CuO4+δ. The observation of the Yamaji peak is evidence for small Fermi-surface pockets in the normal state of the pseudogap phase. The small size of the pockets, each estimated to occupy only 1.3% of the Brillouin zone area, is not expected given the absence of long-range broken translational symmetry.

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Fig. 1: Observation of the Yamaji effect.
Fig. 2: Comparison of measured angle-dependent magnetoresistivity with a Boltzmann transport model.
Fig. 3: Magnetoresistivity for in-plane magnetic field.

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Data that support the findings of this study are provided in the Extended Data figures. Source data are provided with this paper.

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Acknowledgements

We thank M. R. Norman and S. Sachdev for comments on the manuscript. The high-magnetic-field measurements and sample preparation were supported by the US Department of Energy BES ‘Science of 100T’ grant. The National High Magnetic Field Laboratory Pulsed-Field Facility is funded under the National Science Foundation Cooperative Agreement no. DMR-2128556, by the State of Florida and the US Department of Energy. M.K.C. acknowledges support from LDRD 20210320ER for calculations of electrical transport in unconventional superconductors. M.K.C. acknowledges support from the National Science Foundation IR/D programme for research performed while serving at the National Science Foundation, and from the dedicated staff. Any opinion, findings and conclusions or recommendations expressed in this material are those of the author(s) and do not necessarily reflect the views of the National Science Foundation.

Author information

Authors and Affiliations

  1. Pulsed Field Facility, National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, NM, USA

    Mun K. Chan, Katherine A. Schreiber, Oscar E. Ayala-Valenzuela, Arkady Shekhter & Neil Harrison

  2. Los Alamos National Laboratory, Los Alamos, NM, USA

    Eric D. Bauer

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Contributions

M.K.C., K.A.S., O.E.A.-V. and N.H. developed the equipment and performed the pulsed-field measurements. M.K.C. and E.D.B synthesized the samples. M.K.C., K.A.S. and N.H. analysed and modelled the data. M.K.C., A.S. and N.H. interpreted the results and wrote the manuscript with critical input and review from all authors.

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Correspondence to Mun K. Chan.

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Nature Physics thanks Sylvia Lewin and the other, anonymous, reviewer(s) for their contribution to the peer review of this work.

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Extended data

Extended Data Fig. 1 θ dependence of magnetoresistivity.

Magnetoresistivity field sweeps underlying the θ dependence plots in Figs. 1 and 2. a–c show magnetoresistivity for the magnetic field tilted into ϕ = 0°, 23∘, and 45∘ respectively. T = 85 K.

Source data

Extended Data Fig. 2 ϕ dependence of magnetoresistivity.

a–d, Magnetoresistivity field sweeps underlying the ϕ dependence plots in Fig. 3b, covering approximately 180∘ angular range. θ = 90∘ for all curves, corresponding to magnetic field applied in the plane. Lower panels are the derivative with respect to field, showing a crossover to linear magnetoresistance in all curves. T = 85 K.

Source data

Extended Data Fig. 3 Fidelity of pulsed magnetic field measurements.

a, Measured Vx and Vy of the lock-in voltage corresponding to θ dependence of magnetoresistance at ϕ = 0∘. The out-of-phase component Vy is much smaller than the in-phase component Vx of the detected voltage. b, Representative field dependence of measured voltage comparing up (dots) and down (line) sweeps of the pulsed field. The two fall on top of each other except for a large spike starting around B ≈ 25 T on the upsweep due to the firing of the insert. Only down sweep data were analyzed and presented in the main text. c, Pulsed field profile of the 75 Tesla Duplex magnet.

Extended Data Fig. 4 Comparing Yamaji angle for Fermi surfaces with tetragonal planar symmetry.

a, Planar cross-sections of model Fermi surfaces. The grey curve is the large Fermi surface expected for 1 + p carriers in Hg120132,33. The Fermi surface cross sections were generated with the function \({k}_{\parallel }({\phi }_{k})=[1+\beta \cos (m{\phi }_{k})]{k}_{o}\) following ref. 30. ϕk is the azimuthal angle of the Fermi momentum. m = 4 for planar tetragonal symmetry. β controls the deviation from a circular shape. For the small Fermi pockets ko = 0.15 Å−1 and β = 0, 0.05 and 0.4 for the circle, diamond, and ‘+’ shaped Fermi surfaces respectively. b, Calculated Yamaji angles θYamaji as a function of ϕ30, with simple sinusoidal warping. Colors are matched to the corresponding Fermi surfaces shown in panel a. θYamaji was obtained for each ϕ by finding the zeros of the lowest order corrections to an un-warped Fermi surface cross-sectional area30:\(A={J}_{0}(\kappa )+{A}_{2}\cos (m\phi )\), where A2 = β(−1)m/2[(1 + m)Jm(κ) − κJm+1(κ)]. Jm(κ) are the Bessell functions of the mth kind, \(\kappa ={\rm{c}}{k}_{o}\tan (\theta )\), c is the c-axis lattice parameter. Finally, m and ko matches the values used to generate the specific Fermi surface cross-sections in a. The Yamaji condition for the leading term J0(κ) = 0 yields the relationship for the Yamaji peak used in the main text, \({\theta }_{{\rm{Yamaji}}}=\arctan [3\pi /(4c{k}_{o})]\) with ko replaced with kcal. The second term, which accounts for a non-circular cross-section, yields a relatively small \(\cos (m\phi )\) modulation about the first term. Therefore, our experimentally determined θYamaji constrains the size of the pocket with tetragonal (or higher, since the same argument holds for m = 6, 8, …) symmetry to the average Fermi momentum ko. The large Fermi surface has θYamaji ≈ 20∘, grey curve, that is clearly inconsistent with observations.

Extended Data Fig. 5 Contribution of two sets of orthogonal ellipses to the resistivity.

a, Simulated θ dependence of the magnetoresistivity δρc(θ), with ωcτ = 2.6, for each of the two orthogonally oriented ellipses (red and blue, see inset for schematic of the Fermi pockets color coded to match the curves) and the resultant combined contribution to δρc(θ) of both sets of pockets (grey). The plotted combined δρc(θ) curve includes an additional factor of two for easy comparison. For ϕ = 0∘, δρc(θ) is identical for the two sets of pockets. b, For ϕ = 23∘, δρc(θ) of each ellipse are no longer the same. While the Yamaji peak of the red curve is clearly discernible, the Yamaji effect manifests only as a broad kink in the blue curve. The θYamaji of the two ellipse orientations are displaced such that the Yamaji peaks interfere destructively and is unobservable in the combined δρc(θ) (grey). c, For ϕ = 4∘ the Yamaji peak of the red curve is sufficiently displaced from the kink in the blue curve such that the Yamaji effect is discernible as a small bump in the combined δρc(θ) (grey).

Extended Data Fig. 6 Comparing measurements of angle dependent magnetoresistivity to expectations for a bi-axial charge-density-wave reconstruction.

a, Magnetoresistivity δρc as a function θ for ϕ = 0∘, 23∘ and 45∘. b, Simulations for a reconstructed Fermi surface resulting from bi-axial charge-density-wave previously studied at low temperatures (T ≲ 4 K)12,39,40 This pocket has four-fold planar symmetry in agreement with the symmetry of the measured linear slope of magnetoresistivity a1(ϕ) for in-plane fields, but it cannot capture the observed evolution of the Yamaji effect shown in panel a. c & d, Simulations for slight variations of the curvature of the reconstructed CDW pocket, while keeping the enclosed area of the pocket the same. The insets shows a closeup of the reconstructed pocket (reproduced from panel b inset) in blue and the modified pocket in red. Experimentally, the Yamaji peak for ϕ = 0∘ is more pronounced than that for ϕ = 45∘. The opposite is always found for simulations of the CDW pocket.

Extended Data Fig. 7 Effect of changing ωcτ in modeling the Yamaji peak.

a, Close up of the Yamaji peak in δρc(θ) at ϕ = 0∘. It is compared to calculations from our model with varying ωcτ shown in panels b–d.

Source data

Source Data Fig. 1

Extracted from field sweeps in Extended Data Fig. 1.

Source Data Fig. 2

Extracted from field sweeps in Extended Data Fig. 2.

Source Data Fig. 3

Extracted from field sweeps in Extended Data Fig. 2.

Source Data Extended Data Fig. 1

Field sweep data underlying figures in main text.

Source Data Extended Data Fig. 2

Field sweep data underlying figures in main text.

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Chan, M.K., Schreiber, K.A., Ayala-Valenzuela, O.E. et al. Observation of the Yamaji effect in a cuprate superconductor. Nat. Phys. 21, 1753–1758 (2025). https://doi.org/10.1038/s41567-025-03032-2

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