Introduction

The close proximity of antiferromagnetic (AFM) order to an unconventional superconducting phase is a generic feature of strongly correlated superconductors. The coexistence and interplay of AFM order and superconductivity have led to theories based on spin-fluctuation mediated pairing interaction1. In cuprate high-temperature superconductors, magnetic interactions are at the heart of the debate for the pairing interaction. Although the parent compounds are antiferromagnetic Mott insulator, the presence of the pseudogap phase in hole-doped cuprates complicates the situation. Indeed, there is a variety of competing orders with superconductivity, such as charge order, stripe order or nematic phase, that nucleate inside the pseudogap2,3. The multi-layered cuprates provide a proving ground for studying such multiple phases. They have been thoroughly studied by NMR4, ARPES5,6,7 and Raman spectroscopy8. The highest superconducting transition temperature (Tc) at ambient pressure is observed for three CuO2 planes9,10, but the microscopic mechanism at the origin of this experimental observation is still under debate11,12,13,14. One way to understand this problem is to consider the substantial interplane coupling that could stabilize the AFM phase in the underdoped regime, thus boosting AFM fluctuations away from the ordered phase and close to optimal doping. Moreover, the interplane coupling could suppress phase fluctuations and hence increase Tc. Another important ingredient of multi-layered cuprates is the symmetry-inequivalent CuO2 planes. Indeed, the fact that the inner planes (IPs) are not adjacent to the charge reservoir has two consequences: (i) the inner CuO2 planes are protected from out-of-plane disorder and extremely clean4, and (ii) the fact that IPs are farther from the charge reservoir layer than outer planes (OPs) induces a charge imbalance between the different planes. This has been demonstrated by NMR measurements in several multi-layered cuprates (for a review, see ref. 4) and by ARPES measurements7 in optimally doped tri-layer Bi2Sr2Ca2Cu3O10+δ (Bi2223). Consequently, different competing orders can appear in the IPs and the OPs. And each of these orders could influence the Fermi surface (FS) from which high-Tc superconductivity emerges at optimal doping. Namely, AFM order is known to reconstruct the FS at low doping5, and charge order (CO) is also now recognized as a generic property of underdoped cuprates3. For instance, in underdoped YBa2Cu3Oy (YBCO) and HgBa2CuO6+δ (Hg1201), the observation of quantum oscillations (QOs) with small frequencies15,16 and negative Hall effect17,18 are a strong indication of the presence of a small closed electron pocket indicating a FS reconstruction. NMR19 and X-ray scattering20,21,22 measurements then found evidence of CO in YBCO and Hg1201. While the exact scenario for the FS reconstruction is still debated23, a biaxial CO can indeed lead to an electron pocket in the nodal region of the first Brillouin zone24.

Among cuprates, HgBa2Ca2Cu3O8+δ (Hg1223) holds the record of the highest superconducting transition temperature at ambient pressure (Tc = 133 K). It is a tri-layer cuprate and the narrow 63Cu-NMR linewidth4,25 clearly shows that the IP is extremely clean as it is homogeneously doped and screened from out-of-plane disorder by the OPs. In addition, Raman spectroscopy shows the typical signature of CO in optimally and underdoped Hg12238, and NMR measurements performed in an equivalent tri-layer cuprate4 suggest the critical doping at which AFM order in the IP disappears corresponds to an average carrier density p = 9% (Tc ≈ 80 K). In the following, the doping level of multi-layered cuprates has been estimated from Tc (see Methods) and represents an averaged doping level between the inner and the outer planes. Here, we investigate the transport properties of underdoped Hg1223 in the doping range p = 8–8.8% by means of contactless resistance and Hall effect measurements in pulsed fields up to 88 T. We discover quantum oscillations with small frequencies and a Hall coefficient that remains positive down to the lowest temperature, evidencing the presence of small reconstructed pockets of both holes and electrons, which strongly supports the coexistence of AFM and CO but on different CuO2 planes, with AFM on the IPs and CO on the OPs. An additional frequency corresponding to magnetic breakdown tunnelling between the inner and outer planes is also observed.

Results

Observation and analysis of quantum oscillations

Figure 1a shows the variation of the tunnel diode oscillator (TDO) circuit frequency (see Methods) as a function of magnetic field for Hg1201 at p = 9% and for two samples of Hg1223 at slightly different doping levels. In the latter, QOs are clearly observed above H = 40 T, confirming the high quality of the samples. A smooth background subtraction leads to the oscillatory part of the signal shown in Fig. 1b. While there is obviously only one QO frequency for Hg1201, the QO spectrum of Hg1223 is much more complex and contains several frequencies. This is confirmed by the discrete Fourier transform analysis depicted in Fig. 1c. In Hg1201, the discrete Fourier transform reveals a single frequency F = 850 T, in agreement with previous studies16,26. For Hg1223, neglecting the low frequencies that can be attributed to imperfect background subtraction (see discussion in the Supplementary Note 2), at least three frequencies can be isolated at F1 ≈ 350 T, F2 ≈ 500 T and F3 ≈ 850 T, where F3 − F2 ≈ F1. Some harmonics and frequency combinations are also present at higher frequencies. The temperature dependences of the QOs are shown in Supplementary Fig. 2. As expected from the Lifshitz-Kosevich theory27, the amplitude of QOs decreases as the temperature increases and vanishes above T ≈ 10 K.

Fig. 1: QO data.
figure 1

a Field dependence of the TDO frequency after the heterodyne circuit at low temperature in the monolayer Hg1201 (blue line) and in the tri-layer Hg1223 at different doping levels (black and red line). b Oscillatory part of the TDO signal after removing a smooth background (spline) from the data shown in panel a. c Discrete Fourier analysis of the oscillatory part of the TDO signal shown in panel b.

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A challenge in analysing these data is that the oscillation frequencies are low and there is a limited field range available. Therefore, the accurate determination of the value of the frequencies is ambiguous, in particular for the nearby frequencies F1 and F2. To assert the spectra of QOs, we performed fits to the data at different temperatures using the Lifshitz-Kosevich theory (see Supplementary Note 3 for the detailed procedure of the fit). In order to constrain the fits, we performed simultaneous fits to the dataset at different temperatures (from T = 1.4 K to T = 4.2 K at p = 8%), where all parameters are temperature independent except for the background. Figure 2 shows the raw data for the sample at p = 8% (symbols) and solid lines are the simultaneous fits in the temperature range T = 1.4–4.2 K (see Supplementary Fig. 3 for the p = 8.7% sample from T = 0.6 K to T = 2.9 K). The value of the frequencies deduced from the fitting procedure at p = 8% are F1 = 331 T, F2 = 500 T and F3 = 866 T, in good agreement with the values obtained by discrete Fourier transform at different temperatures (see Table 1). Both analyses confirmed that the oscillatory spectrum is composed of at least three frequencies linked by the relation F3 − F2 ≈ F1.

Fig. 2: Lifshitz-Kosevich fits.
figure 2

Field dependence of the TDO frequency in Hg1223 (p = 8%) at different temperatures (symbols). Solid lines correspond to the fits to the data using the Lifshitz-Kosevich theory plus a polynomial background in the field range 40 ≤ H ≤ 83 T and in the temperature range T = 1.4–4.2 K (see Supplementary Note 3 for details). The inset shows the Fourier analysis of the oscillatory part of the data at T = 1.4 K along with the contribution of F1 (blue), F2 (grey) and F3 (red), respectively. The green component corresponds to a frequency combination, which has been taken into account to improve the fits.

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Table 1 Sample family, Tc, doping p and QO frequency deduced from the discrete Fourier transform analysis.
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Hall effect measurements

In order to gain more insight into the Fermi surface of underdoped Hg1223, we performed Hall effect measurements up to 88 T at a doping level p = 8.8%. Figure 3 shows the temperature dependence of the normal-state Hall coefficient down to T = 1.5 K (the isotherms are shown in Supplementary Fig. 4). Remarkably, there is almost no temperature dependence of the Hall coefficient and it remains positive down to the lowest temperatures. This result contrasts with the Hall coefficient in underdoped YBCO17,28 and Hg120118, which changes signs and becomes negative at low temperatures. This has been interpreted as the signature of an electron pocket resulting from the FS reconstruction caused by the CO. In the case of underdoped YBCO, the CO is present in both CuO2 planes of the bilayer and QO measurements reveal the main frequency Fe = 540 T flanked by two nearby and equally spaced satellites Fe ± 90 T at a doping level p ≈ 11%29,30. This has been interpreted as magnetic breakdown tunnelling between bilayer-split pockets30,31,32 provided that the mirror symmetry between the planes of the bilayer is broken. In the single-layer Hg1201, only one QO frequency F ≈ 850 T has been detected so far16,26. The difference in the value of the main frequency between YBCO and Hg1201 is a direct consequence of a different CO wavevector, leading to different sizes of the reconstructed electron pocket.

Fig. 3: Hall data.
figure 3

Temperature dependence of the normal-state Hall coefficient RH, measured at high fields, in Hg1223 (p = 8.8%, red squares) and in Hg1201 (p = 8%, blue circles adapted from ref. 18). The Hall coefficient changes sign in Hg1201 while it remains positive (i.e. hole-like) down to the lowest temperature in Hg1223. Note that in Hg1223, as T → 0, RH ≈ 6.5 mm3/C, corresponding to an effective carrier density pH ≈ 8%. Error bars represent the noise generated by the 90 T coil (see Supplementary Note 4).

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Discussion

We now discuss different scenarios to explain our results in underdoped Hg1223. For the sake of simplicity, we treat the inner and the outer planes as independent even if they are coupled by an interlayer tunneling t⊥. For the Fermi surface reconstruction, we consider only one band within each CuO2 plane.

Scenario (1): Band structure calculations

Local-density-approximation calculations33 of the electronic structure of the stoichiometric compound HgBa2Ca2Cu3O8 reveal that the Fermi surface consists of three large hole-like tubular CuO2 sheets centered on the corner of the Brillouin zone plus a small electron-like Fermi surface located at the anti-node (see Supplementary Fig. 6a). The latter disappears with doping33. For a doping p ≈ 8%, the Fermi surface of the CuO2 sheets corresponds to 1 + p holes that translates into a QO frequency FLDA ≈ 15 kT much larger than the observed frequencies in our study.

Scenario (2): Charge order in the three CuO2 planes

In analogy with underdoped YBCO where a CO is present in both CuO2 planes, let us assume that a CO is present in the three CuO2 planes of Hg1223 (see a sketch of the scenario in Supplementary Fig. S6b). As the frequencies are not equally spaced, the model of magnetic breakdown tunnelling is inadequate to explain the spectrum of oscillation frequencies in Hg1223. Moreover, the Hall coefficient remains positive down to the lowest temperature, in contrast with underdoped Hg1201 and YBCO.

Scenario (3): AFM in the inner plane

Another scenario assumes an AFM metallic phase in the IP and a corresponding FS that contains both electron and hole pockets (see a sketch of this scenario in Supplementary Fig. 6c). While there is no direct evidence yet of an AFM order in underdoped Hg1223, such order has been detected by extensive NMR measurements in the IP of the tri-layer cuprate Ba2Ca2Cu3O6(F,O)2 (O223F) with Tc up to 81 K but not beyond4,34. Given the disorder-protected nature of IPs in multi-layered cuprates, let us assume that QOs originate from quasi-particles in the IP with Fhole = F3 ≈ 850 T and Felectron = F2 ≈ 500 T. In this scenario, the third frequency F1 = F3 − F2 would correspond to a magnetic breakdown between the hole and the electron pockets in the IP. However, in order to reproduce the size of the orbits corresponding to the observed frequencies F2 and F3, the AFM potential used in the calculation is 0.25 eV (see discussion in Supplementary Note 6 and Supplementary Fig. 6c). This value translates to a magnetic breakdown field unattainable, ruling out the possibility to observe magnetic breakdown between the hole and electron pocket. Finally, the presence of an electron pocket at the anti-node is difficult to reconcile with the presence of a pseudogap.

Scenario (4): AFM in the inner plane and charge order in the outer plane

Given the charge imbalance between IP and OP, the carrier density is always lower in the IP. Let us assume that an AFM metallic phase is present in the IP in analogy with the O223F compound4,34. But compared to scenario (3), the FS in the IP consists solely of hole pockets at the nodes corresponding to Fhole = F2 ≈ 500 T. This is in agreement with recent ARPES and QOs studies showing the metallic character of the AFM phase at low doping in the IPs of a 5-layer cuprate5. Kunisada et al. found two QO frequencies F(IP0) = 147 T and F(IP1) = 318 T corresponding to an effective carrier density p = 2.1% and p = 4.5%, respectively. In our study, the hole frequency F2 ≈ 500 T translates to a carrier density p = 7.2%, in good agreement with the estimation given by NMR measurements in a 3-layer cuprate with Tc = 76 K, where p(IP) ≈ 7.4% and p(OP) ≈ 8.7%4. In addition, as shown by recent Raman spectroscopy measurements in underdoped Hg12238, we assume that the CO sets in the OP as the effective doping is higher. It induces a Fermi surface reconstruction leading to an electron pocket at the node corresponding to Felectron = F3 ≈ 850 T, in analogy with the monolayer Hg1201. In Fig. 4a, we sketch the real space structure corresponding to this scenario: an AFM order in the IP coexisting with a charge order in the OPs. Figure 4b shows the resulting Fermi surface consisting of both electron (orange) and hole (purple) pockets. The third frequency FMB = Fe − Fh would correspond to magnetic breakdown tunnelling between OP and IP. Note that the amplitude of the different frequencies depends on the value of t⊥ and the broadening of the Landau level due to disorder31. But how can we reconcile this scenario with a positive Hall effect? Let us focus on the low-temperature value of the Hall coefficient and assume the low-field limit for the two-band model of the Hall effect (see Supplementary Note 5). The Hall coefficient is given by: \({R}_{H}=\frac{{\sigma }_{h}{\mu }_{h}-{\sigma }_{e}{\mu }_{e}}{{({\sigma }_{h}+{\sigma }_{e})}^{2}}\), where σ and μ are the conductivities and mobilities, respectively. Given the carrier densities deduced from quantum oscillation frequencies, a Hall coefficient RH ≈ 6.5 mm3/C (see Fig. 3) corresponds to a ratio of mobilities μh/μe ≈ 3, a reasonable value owing to the disorder-protected nature of the IP compared to the OPs. In short, this interpretation allows explaining both the QO spectrum and the value of RH at low temperatures. We cannot exclude that the strong magnetic field influences the ground state of Hg1223 as it is the case for the 3D charge order in underdoped YBCO19. However, this is clearly not the case in the 5-layer cuprates, where both ARPES at zero fields and QOs at high field give the same Fermi surface for the two inequivalent inner planes5.

Fig. 4: Sketch of the Fermi surface.
figure 4

a Crystallographic structure of tri-layer Hg1223. We sketch the presence of AFM order in the IP (purple arrow) and charge order (orange wave) in the OPs. b Corresponding reconstructed Fermi surface in presence of AFM order in the IP leading to a hole pocket (purple, F2 QO frequency) and CO order in the OP leading to an electron pocket (orange, F3 QO frequency). Both pockets are located in the nodal region of the quarter of the first Brillouin zone. Magnetic breakdown tunnelling between the pockets leads to an additional frequency F1 ≈ F3 − F2.

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Our interpretation based on scenario (4) implies that, in the cuprate where Tc is maximum among all superconductors, a metallic AFM state extends deep inside the SC phase. This is reminiscent of a quantum critical point scenario observed in other unconventional superconductors, where spin fluctuations extend away from the AFM ordered state. The dispersion of such magnetic excitations has first been measured in YBCO using inelastic neutron scattering35. Resonant inelastic x-ray scattering (RIXS) experiments have subsequently extracted the dispersion of these magnetic excitations, called paramagnons, up to high energy transfer in different cuprate families and over a large doping range36. Interestingly, a recent RIXS study on the two first members of the Hg-family of cuprates shows that the energy scale of the paramagnon spectra matches the ratio of Tc37. All of the above considerations strongly suggest a magnetic pairing mechanism for cuprates. In Hg1223, the clean nature and the absence of buckling of the inner CuO2 plane support the idea that the antiferromagnetic interaction J is large, leading to higher Tc25,37. Could the presence of charge order in the OPs be a consequence of charge imbalance and / or of out-of-plane disorder? In YBCO, charge order competes both with SC and AFM order. This could explain why Tc further increases in optimally doped Hg1223 by applying pressure38 which destabilizes charge order39.

Methods

Samples

Single crystals of the tri-layer cuprate HgBa2Ca2Cu3O8+δ have been synthesized using a self-flux growth technique as described in ref. 10. Using adequate heat treatment, Hg1223 can be largely underdoped and its doping level controlled. The doping p has been deduced from the empirical relation 1 − Tc/Tc,max = 82.6(p − 0.16)2, where Tc is the onset superconducting transition measured by SQUID (see Supplementary Fig. 1) and Tc,max = 133 K.

TDO measurements

Quantum oscillations have been measured using a contactless tunnel diode oscillator-based technique40 in two samples of Hg1223 at doping level p = 8% and p = 8.7%. Typical sample dimensions are 500 × 500 × 100 μm3. The experimental setup consists of a LC-tank circuit powered by a tunnelling diode oscillator biased in the negative resistance region of the current-voltage characteristic. The sample is placed in a compensated 8-shape coil (diameter and length of the coil are adapted for each sample to optimize the filling factor). The fundamental resonant frequency f0 of the whole circuit is around 25 MHz. The RF signal is amplified and demodulated down to a frequency of about 1 MHz using a heterodyne circuit. A high-speed acquisition system is used to digitize the signal. The data are post-analyzed using software to extract the field dependence of the resonance frequency fTDO, which is sensitive to the resistivity through the change in skin depth.

Hall effect measurements

The Hall effect was measured in the third sample of Hg1223 at doping level p = 8.8%. The dimensions of the sample are 700 × 450 × 90 μm3. Gold contacts were sputtered onto the surface of the sample before a heat treatment leading to contact resistances of a few ohms at room temperature and below 1 Ω at low temperature. The magnetic field H was applied along the c-axis of the tetragonal structure, perpendicular to the CuO2 planes in both polarities of the field. The high-temperature measurements were performed in a conventional pulsed magnet up to 68 T down to 10 K. At lower temperature, higher magnetic fields were required to quench superconductivity, so we extended our measurements up to 88 T, using a dual coil magnet. The pulsed-field measurements were performed using a conventional 4-point configuration with a current excitation of 5 mA at a frequency of 60 kHz. A high-speed acquisition system was used to digitize the reference signal (current) and the voltage drop across the sample at a frequency of 500 kHz. The data were post-analyzed with software to perform the phase comparison.