Introduction

The microscopic mechanism of high critical temperature (Tc) superconductivity in cuprates has been a mystery for the last four decades. While various exotic phases, such as pseudogap, charge order, and strange metal phases, have been found in over 200 compounds, one of the most fundamental features in doping−temperature (T) phase diagrams of cuprates is that Tc forms a dome-like region near the antiferromagnetic (AF) order phase boundary1,2,3. This proximity of superconductivity and AF order makes AF spin fluctuation a compelling candidate as the pairing glue that mediates superconductivity. Theoretical studies in support of this conjecture have well elucidated the unconventional d-wave pairing symmetry observed in curates using electron-AF spin fluctuation coupling4,5. On the experimental side, quasiparticles and their renormalization attributed to many-body interactions have been directly observed by angle-resolved photoemission spectroscopy (ARPES)6,7. Although substantial progress has been made in understanding the electron-boson scattering, demonstrating how the spin fluctuation coupling specifically dresses the electrons and manipulates the superconductivity remains a spectroscopic challenge because the complex phase diagram and strong electron correlation complicate the interpretation of quasiparticle spectra.

For the interaction of electrons with the AF order, one of the most widely accepted features demonstrated by early ARPES and concomitant mean-field studies is that the emergence of a long-range (LR)-AF super-lattice potential with under-electron doping results in the folding of the quasiparticle band and reconstructs the Fermi surface from a large circle (L-circle) to small hole (h-) and electron (e-) pockets (Fig. 1 and Supplementary Fig. 1)6,7,8,9. In addition to this mean-field scheme with the LR order, further low-energy spectral analysis also found traces of the reconstructed Fermi pockets in the doping range of the superconducting (SC) phase beyond the LR-AF order phase boundary10,11,12, pointing to the possible contributions from the short-range (SR)-AF fluctuations or topological orders10. Moreover, it has been even argued that both folded AF and unfolded pristine components are detected simultaneously in the single-particle ARPES spectrum7,13, which is subject to the strong electron correlation. To date, a complete understanding of the exotic folding behavior is still lacking, calling for more sophisticated spectral analysis.

Fig. 1: Electronic structure and antiferromagnetic (AF) band reconstruction of electron-doped cuprates.
Fig. 1: Electronic structure and antiferromagnetic (AF) band reconstruction of electron-doped cuprates.
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a Fermi surface (FS) and nodal band ARPES spectra of PLCCO as a function of electron doping (n). Non-SC, UD17, and OD11 indicate non-superconducting, underdoping, and overdoping, respectively, where the number indicates Tc in Kelvins. b Copper oxide plane with antiferromagnetically ordered spins. The black lines show the pristine unit cell, and the pink-shaded square is the AF unit cell. c Schematics of pristine (left) and AF pocket (right) FSs of electron-doped cuprates. The middle panel shows the band-folding process, attributed to the AF unit cell doubling shown in b. The pink-shaded area is the AF Brillouin zone.

Although the folding effect is still difficult to define in the SC region, experimental evidence indicating the possible correlation between the superconductivity and one of the AF pockets has been reported through magneto-transport studies. Previous magnetoresistance and Shubnikov–de Haas quantum oscillation measurements on electron-doped Nd2−xCexCuO4±δ observed clues of the h-pocket contribution to the conductivity only in the doping range of the SC region11,12, suggesting that spin-coupled quasiparticles on the h-pocket play a crucial role in driving the superconductivity. However, it is still debated whether the external magnetic field applied for those magneto-transport measurements not only suppresses the superconductivity but also reinforces the AF order and pins the pockets to the system. Therefore, at least two important, related questions remain to be answered regarding the h-pocket-driven superconductivity: is the h-pocket tied to the superconductivity even in ambient conditions without an external magnetic field? and if this is the case, which magnetic ground state gets along with the pocket state?

To resolve this puzzle, we carried out ARPES and zero-field muon spin rotation/relaxation (μSR) measurements on the electron-doped cuprate Pr1−xLaCexCuO4±δ (PLCCO), followed by spectral simulations with the Hubbard model. By scrutinizing the low-energy photoemission spectra near the Fermi energy (EF), we found that the single-particle spectrum of the nodal quasiparticle band of PLCCO consists of both folded AF and unfolded pristine components, as also shown by our numerical calculations based on cluster perturbation theory (CPT). Tracking the doping evolution of the folded AF branch, we reveal that the superconductivity develops as the folded hole band component incipiently crosses the EF in the SR-AF ground state region around a putative quantum critical point (QCP) where the static-to-dynamic AF quantum phase transition occurs. Furthermore, we find that the zero-energy quasiparticle weight of the hole band Zhole is in proportion to Tc, while the gap from the band top to EF is negatively correlated with Tc. These results suggest that the incipient hole band that forms the underlying h-pocket is driven by electron-spin fluctuation interaction and plays a crucial role in the emergence of superconductivity.

Results

Correlation between superconductivity and hole pocket

Figure 2a shows the doping-dependent ARPES spectra of the low-energy quasiparticle state dispersing along the nodal direction, the so-called “nodal band”, after dividing the Fermi–Dirac distribution (also see Supplementary Fig. 2a). The most under-doped (UD) sample with n ≈ 0.06, referred to as Non-SC, exhibits a clear energy gap that seems to close gradually as the electron doping n increases. According to the mean-filed picture, the gap originates from the splitting of the quasiparticle band into upper and lower bands due to effective Q = (π, π) AF scattering, and the gap closing signals a dissipation of AF super-lattice potential (Fig. 1 and Supplementary Fig. 1)7,8,9. Although the AF band reconstruction is expected to result in the band folding, the folded AF branch is invisible in the raw spectra (Fig. 2a) because its intensity is weak due to the matrix-element effect6. The shadow AF branch indeed comes into view with a two-dimensional stacking of energy distribution curves (EDCs) when the folding effect is strong enough. For example, in the EDC plots of the nodal band spectra in Supplementary Fig. 3, the flattened band top and the back bending of the band are well-defined for the heavily under-doped samples (n ≈ 0.06 and 0.11). In addition, we get even better visibility of the shadow branch by normalizing the maximum intensity of the EDCs at each momentum, as displayed in Fig. 2b (for details of the analysis, see Supplementary Fig. 4)14. Accordingly, the total hole band dispersion with the folding center at the AF zone boundary (pink dashed line) is obviously seen in the normalized spectra of those under-doped samples (n ≈ 0.06 and 0.11 in Fig. 2b).

Fig. 2: Doping evolution of the nodal band in PLCCO.
Fig. 2: Doping evolution of the nodal band in PLCCO.
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a ARPES spectra of the nodal band after Fermi–Dirac correction for various doping samples. OP indicates optimal doping. b Nodal band spectra with the maximum energy distribution curve (EDC) that brightens the “folded shadow band” forbidden in (a) (see Supplementary Fig. 3). c Schematic spectra of the FS at the nodal point and hot spot. The black peak at the nodal point indicates the nodal band spectrum. The red and green peaks show the hole (h-) pocket and large (L-) circle components, respectively. d Superposed FS consisting of electron (e-) pocket, h-pocket, and L-circle. The pink dashed lines mark the AF zone boundary.

On the other hand, the spectrum of the AF branch becomes scattered and less well-defined with optimal and overdoping even in the analyzed spectra (to the right of the pink dashed line for n ≈ 0.135 and above in Fig. 2b). At the same time, the spectrum around the folding center is significantly broadened, and the band top is no longer immediately discernible due not only to the weakening of folding effect but also to the strong enhancement of in-gap spectral intensity (see the schematic nodal spectrum in Fig. 2c). In the electron-doped cuprates, the enhancement of the in-gap spectral weight has been considered to result from the growth of the unfolded pristine state15,16,17. As such, the simultaneous observation of the AF branch and the in-gap spectrum signals that PLCCO virtually has a superposed Fermi surface consisting of the AF pockets and the pristine L-circle components13 (Fig. 2d). From the theoretical perspective, such a superposition is beyond the conventional mean-field picture but is associated with the many-body effect in the fluctuating correlated electron system11,13,18,19,20. From a phenomenological standpoint, it represents the coexistence of spin-interacting and spin-non-interacting states in a single-particle spectral function. Considering that the spin-interacting electrons form the h-pockets, extracting the AF components from the nodal band spectra would be a starting point for understanding the relationship between the h-pocket and superconductivity. However, it is a technically challenging issue.

To address this challenge, we attempted a phenomenological analysis based on the variation in the spectral distributions along the Fermi surface. Specifically, we paid attention to the spectrum measured at the position of the hot spot as it also undergoes a similar in-gap state filling to the nodal band spectrum but has the folding band at much deeper binding energy (Supplementary Fig. 2)15,16,17. Figure 3a displays the doping-dependent EDCs obtained by integrating the spectrum within a small momentum window around the folding center of the nodal and hot spot band spectra, marked as “nodal” and “hot spot”. For the Non-SC sample with n ≈ 0.06, both the nodal and hot spot EDCs show clear AF gaps with peak energy ~30 and ~100 meV, respectively, with almost negligible in-gap spectral intensity at EF. With increasing doping, the zero-energy intensity dramatically enhances at both points (see doping evolution of black dashed and green dotted lines in Fig. 3a). Because of its subtle gap, a strong intermixing of the folded and unfolded components is expected in the in-gap zero-energy states of the nodal EDC. (see the nodal spectra of the SC samples in Fig. 2c). On the other hand, due to a relatively large gap size, the in-gap zero-energy state at the hot spot is considered to be dominated by the pristine component with a tiny fraction of the AF component (see the hot-spot spectrum in Fig. 2c)16. In this framework, assuming that the pristine Fermi surface has a nearly momentum-invariant spectral weight distribution along the L-circle contour7,8,9,21, the difference in the zero-energy spectral intensity between the nodal and hot spot approximately provides the quasiparticle weight of the folded band Zhole_phn that contributes to the formation of the underlying h-pocket (see red arrows in Fig. 3a).

Fig. 3: Correlation between the Fermi hole pocket and superconductivity.
Fig. 3: Correlation between the Fermi hole pocket and superconductivity.
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a EDCs at (π/2, π/2) and hot-spot point for various dopings. Black dashed and green dotted lines guide the eye to the zero-energy intensity of each EDC. b EDCs at (π/2, π/2) with Gaussian function fitting results. Red, green, and blue solid lines correspond to AF peak, in-gap pristine peak, and background, respectively. The red vertical arrows in a, b denote Zhole obtained by the phenomenological method (Zhole_phn) and the Gaussian fitting (Zhole_fit). c, d, e Tc, Zhole_phn, Zhole_fit, and Δhole as a function of n, where Δhole is a binding energy of AF peak obtained by the Gaussian fitting. f Zhole_phn, Zhole_fit, and Δhole as a function of Tc. Error bars of Zhole_phn represent signal-to-noise error. Error bars of Zhole_fit and Δhole are determined based on the fitting error of the EDC peak.

Based on this phenomenological perspective, we obtained Zhole_phn from all measured samples and plotted it in Fig. 3d as a function of n. At a glance, it shows dramatic doping evolution with the maximum around the optimal doping, contradicting the monotonic increase of the in-gap pristine spectral weight (green dotted line in Fig. 3a). Interestingly, Zhole_phn exhibits a dome-shaped doping dependence, which resembles the Tc evolution (Fig. 3c, d). This similar doping dependence between Zhole_phn and Tc is shown as a proportional relationship in Fig. 3f; Zhole_phn linearly increases with increasing Tc. This result provides a couple of valuable insights into the SC nature associated with the h-pocket. Firstly, given that the Zhole_phn measures the occupancy of the hole band state at EF, the correlation between Zhole_phn and Tc suggests that the density of state (DOS) of the h-pocket is a key determinant of Tc. Simultaneously, it also highlights that the pair interaction between the holes at the pockets plays a crucial role in mediating the superconductivity. From this aspect, the holes are viewed as quasiparticles dressed by AF fluctuations, which have been theoretically known to induce pairing correlations in both the perturbative22 and strongly correlated regime23. These interpretations are supported by our numerical calculations as well as μSR analysis, shown in the following sections.

In addition to the phenomenological analysis, we carried out a function-fit analysis on the nodal band spectra without involving the hot-spot spectra. The nodal EDCs could be well fitted with three Gaussians: AF peak, in-gap pristine peak, and high-binding energy background (see the red, green, and blue lines, respectively, in Fig. 3b), where the AF peak represents the leading peak of the hole band and its spectral weight at EF corresponds to the h-pocket quasiparticle weight (see Zhole_fit with red arrows in Fig. 3b). The resulting quasiparticle weights Zhole_phn and Zhole_fit, obtained by the phenomenological method and the Gaussian fitting, respectively, show quite similar doping-dependent behaviors. Furthermore, we carried out a Lucy-Richardson deconvolution24 to rule out the experimental resolution effect (see Supplementary Note 2). In consequence, we obtained a consistent result from the deconvoluted data (Supplementary Fig. 5). This additional analysis confirms that the proportionality between Zhole and Tc is a robust result regardless of the analysis method employed. Meanwhile, electron-phonon coupling also causes renormalization of the electronic structure. However, its effect on the zero-energy quasiparticle weight was found to be insignificant.

We highlight that the function fitting also allows us to determine the AF peak position Δhole even in the overdoping region where the peak shape is ambiguous due to strong in-gap pristine intensity. We define the energy gap Δhole as the binding energy of the hole band top. Figure 3e displays Δhole with an inverted axis as a function of n. Intriguingly, Δhole shows a doping evolution similar to Zhole and Tc (Fig. 3c–e), suggesting a negative correlation between Δhole and Tc (see Fig. 3f); Tc increases with decreasing Δhole. It is noteworthy that the hole band retains a shallow gap Δhole = ~13 meV even at the optimal doping n  ≈ 0.15 (Fig. 3e), indicative of the gap over the entire SC region. A recent high-resolution ARPES study also observed a similar AF gap in the nodal band of optimally electron-doped Nd2−xCexCuO4±δ (x = 0.15)13. On the other hand, Δhole larger than 26 meV completely rules out the superconductivity (see the linear fitting result in the Supplementary Note 2 and Supplementary Fig. 6). The shallow (Δhole < 26 meV) and large gap (Δhole > 26 meV) in the SC and non-SC samples are reminiscent of incipient band pairing and its pairing cut-off energy, respectively25.

Numerical simulations of the hole pocket

We argue that the unconventional reconstruction of the Fermi surface, characterized by the coexisting L-circle and pockets, originates from strong correlations instead of static disorder. To demonstrate this, we simulated the single-band Hubbard model, a many-body model that describes the physics of cuprates, using CPT26,27. The CPT spectral simulation, invoking both electron correlations and SR-AF fluctuations, allows us to trace the evolution of the AF branch. Figure 4a, b show the resulting electronic structure as a function of n (see Methods and Supplementary Fig. 7 for details of the simulation). Unlike the single-band folding predicted by mean-field theory, the simulated spectral functions of this strongly correlated system show the coexistence of two peaks near the nodal Fermi surface. Although the two peaks are sharper and more separated in the simulation obtained from the clean model, the overall trend is consistent with the experimental findings that show an increase in the in-gap spectral weight at the hot spot, resulting in a peak-dip-hump-like feature in the EDC of OD samples (see Fig. 3a)16,17.

Fig. 4: Numerically calculated electronic structures of PLCCO based on cluster perturbation theory (CPT).
Fig. 4: Numerically calculated electronic structures of PLCCO based on cluster perturbation theory (CPT).
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a, b Simulated A(k, ω) at high symmetry cut: (0, π) to (π/2, π/2) and Γ to (π, π), respectively. The white dashed boxes highlight the nodal hole band top, where the spectral weight is integrated to get Ztheory. c Zhole, Ztheory, and Tc, as a function of n. Error bars of Zhole_fit are determined based on the fitting error of the EDC peak shown in Fig. 3.

By integrating the simulated single-particle spectrum around the folding center (π/2, π/2) of the hole band (inside the white dashed boxes in Fig. 4a, b), we further estimate the theoretical quasiparticle weight Ztheory. Figure 4c presents Ztheory, along with Zhole_fit and Tc, as a function of n. The overlap of Zhole_fit and the SC domes again shows the scaling relation between Zhole and Tc. The concurrent dome-shaped doping dependence of Ztheory further indicates the role of the marked interplay of the electron correlation and AF fluctuations in shaping the observed Zhole dome. If the AF fluctuations act as the pairing glue21, the dome structure of Zhole, and consequently Tc, can be interpreted as the result of gradually developed quasiparticles with AF fluctuations. The quantitative difference between Zhole and Ztheory is mainly due to two factors. First, the simplicity of the single-band Hubbard model and finite-size simulation neglects the impact of orbital fluctuations and LR interactions, which affect the self-energy. Second, while Zhole is obtained by integrating the experimental spectral weight in the immediate vicinity of EF, a comprehensive description of quasiparticles should involve a wider energy window.

Hole pocket with antiferromagnetic quantum phase fluctuations

We now turn to the magnetic correlations of PLCCO, measured by zero-field μSR spectroscopy. Figure 5a presents the time-differential μSR spectra of the SC compounds at selected dopings and temperatures. For all samples, the muon spin polarization Pz(t) shows faster relaxation (drop) as the temperature T decreases, indicating an increase in the local magnetic fields. The lack of an oscillatory signal down to the base T = 2 K implies the absence of LR order in the SC phase28 (See Supplementary Note 4). For UD sample UD15, Pz(t) relaxes to 1/3 of its initial value below 20 K, suggesting the formation of SR clustered static order, i.e., a “spin-freezing phase28. Notably, with the higher dopings, the low-T Pz(t) relaxes to values smaller than 1/3 of the initial asymmetry, while retaining its relatively fast relaxation. This implies that with increasing doping, the SR static order evolves to more dynamically fluctuating Cu spins28.

Fig. 5: Quantum critical-like transition with dissolution of static order near OP.
Fig. 5: Quantum critical-like transition with dissolution of static order near OP.
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a T-dependent zero-field muon spin rotation/relaxation spectra Pz(t) for samples UD15, UD20, OP24, and OD18. Solid lines show the fit to the data described in the “Methods”. Error bars are statistical errors. b, c T-dependent relaxation rates λ1(T) for UD15, UD20, OP24, and OD18. Solid and dashed arrows indicate the transition temperatures for the spin-freezing phase Tf and fluctuating short-range-order phase T*. The solid lines are power-law fits, λp(T) ~ T−α (see α in Supplementary Fig. 9). The difference between the power-law fit and data, Δλ(T) = λp(T) − λ1(T), in the low-T range denotes the fraction of frozen spins in the inset of b. The muon spin relaxation rate λ1 for UD15 is scaled by a factor of 1/8 for clarity. Error bars reflect the fitting error of the Pz(t) shown in (a).

In addition to the qualitative understanding of the muon spin relaxation, quantitative analysis of the relaxation rate λ shines more light on the quantum phase transition across the optimal doping. From the zero-field μSR spectra Pz(t)’s, we extracted the transverse and longitudinal components of the muon spin relaxation rate, referred to λ1 and λ2, respectively (see Supplementary Note 4). Specifically, we focus on doping and T-dependent λ1, since it clearly shows the magnetic phase transition feature. Figure 5b, c display the λ1(T) with various doping concentrations. For UD samples UD15 and UD20, two characteristic temperatures, T* and Tf, are identified as pointed by solid and dashed arrows, respectively. Between T* and Tf, λ1(T) follows power-law dependence, λp(T) ~ T−α (solid lines in Fig. 5b with the exponent α in Supplementary Fig. 8e), reflecting a critical slowing of the Cu spin fluctuations with decreasing T. Below Tf, the deviation of λ1(T) from the extrapolated λp(T), as shown in Fig. 5b, demonstrates the build-up of the spin-freezing phase29,30,31,32,33. As evident from the inset of Fig. 5b, the frozen spin contribution, Δλ(T) = λp(T) − λ1(T), exhibits an order parameter-like increase with decreasing T. In comparison, for OD18, the absence of Tf leads to a levelling-off of λ1(T) below T*, indicating the dominance of truly dynamic AF fluctuations with overdoping (Fig. 5c)30. More significantly, λ1(T) of OP24 shows a single power-law behavior below T* (see the solid line in Fig. 5c), which means that quantum critical-like AF fluctuations dominate down to the low-T limit at optimal doping. Given that the single power law is a signature of critical behavior near a ferromagnetic or AF QCP34,35,36, the doping dependence of λ1(T) indicates that the optimal doping of PLCCO occurs near a putative AF QCP between the SR static order and dynamically fluctuating phases37.

Summarizing the ARPES and μSR results in an n–T phase diagram (Fig. 6), we have firmly established the relationship between Zhole and antiferromagnetism. Both the mutual exclusion of the Zhole-dome and LR-AF phase (See Supplementary Note 4) and the inverse proportion between the Tf and Zhole signal competition between the h-pocket and static AF order. Recall that a similar anti-correlation between the nodal quasiparticle weight and AF order was suggested in t-t′-J model calculations38. The LR-AF phase boundary around n ~ 0.07 also explains why experimentally obtained Zhole_fit emerges at larger doping than Ztheory (Fig. 4c) because the simulation excluded the LR interactions. Conversely, the dynamic SR-AF phase seems conducive to the formation of the h-pocket, which is supported by the proportionality of T* and Zhole in the over-doped regime where static order is lacking. Consequently, the Zhole dome, corresponding to the SC dome, is centered around the putative AF QCP, where the ground state of the SR-AF phase changes from the spin-freezing to the dynamic state. Note that near this putative QCP, we also observed a sign inversion of the Hall coefficient RH, which has been proposed as evidence for the QCP in the electron-doped cuprates39,40,41 (dashed vertical line in Fig. 6 and Supplementary Fig. 10).

Fig. 6: Electronic, magnetic, and SC phase diagram of PLCCO.
Fig. 6: Electronic, magnetic, and SC phase diagram of PLCCO.
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TN, Tf, and T* indicate Neel, partial spin-freezing, and fluctuating short-range antiferromagnetic (SR-AF) phase transition temperatures, respectively. LR-AF represents the long-range antiferromagnetic order phase. The schematic plot of SC and Zhole dome is based on the data shown in Fig. 3. The sign change in the Hall coefficient (RH) at low-T with overdoping is shown in Supplementary Fig. 10. Error bars are attributed to the ambiguity in determining the transition temperatures from λi(T) (i = 1, 2) (see Fig. 5 and Supplementary Fig. 9).

Discussion

The present phase diagram established by our comprehensive ARPES and μSR experiments demonstrates that the h-pocket with the incipient band stems from the inhomogeneous magnetic ground state with persistent SR-AF fluctuations near the putative QCP. Specifically, regarding Zhole as a measure of zero-energy DOS of the hole band at the folding center (±π/2, ±π/2) where |∇k (ω)| vanishes, the marked enhancement of Zhole resembles the impact of a van Hove singularity (vHS) 42,43,44,45, namely a band-edge vHS, on the emergence of Fermi surface instability. From this perspective, the putative QCP can be interpreted as the vHS in the vicinity of EF, enhancing the electronic susceptibility around 2kF = Q(π, π) (Fig. 1c) and triggering the instability in spin susceptibility ascribed to quantum critical fluctuations42,43,44. Indeed, Zhole is maximized as Δhole approaches 13 meV, which is in good agreement with the spin resonance energy of PLCCO46 (Fig. 3d, e). Furthermore, vHS has generally been considered a key to strengthening superconductivity, not only in hole-doped cuprates42,43,44 but also in other novel superconductors, including recently discovered twisted bilayer graphene47. Therefore, these observations suggest that in the electron-doped cuprates, AF instability and superconductivity are intertwined through the hole band-edge vHS.

A remaining important question is why superconductivity is predominantly dictated by the formation of small h-pockets instead of relatively large e-pockets? It has been thought that doped holes in cuprates enter the Cu 3d-O 2p-hybridized state and cause spin frustrations, while doped electrons reside in the Cu 3d state and cancel the spins6. If the holes created by the formation of h-pocket play the same role as the nominally doped holes, our results signal the importance of the oxygen state or the spin frustrations for the superconductivity48. On the other hand, referring to a previous study combining ARPES with t-t′-t′′-J model calculations for Nd2−xCexCuO4±δ49, electrons in the e-pocket may be favorably coupled to the static AF order, which competes with superconductivity. Note that the h-pocket in five-layered hole-doped Ba2Ca4Cu5O10(F, O)2 previously observed by ARPES similarly implies that superconductivity can be induced by the h-pocket alone, without a contribution from the anti-nodal region50. Further systematic investigations of the orbital character of the h-pocket may provide clues regarding the origin of hole superconductivity in the electron-doped cuprates.

Methods

Materials

Single crystals of PLCCO with x = 0.10, 0.15, and 0.18 were grown by the traveling-solvent floating-zone method. All of the crystal rods were cut into small pieces along the CuO2 plane, and annealed in a high-purity N2 gas atmosphere at 920–930 °C and vacuum with ~10−5 Torr at 790 °C for 10–24 h. Subsequent air annealing was performed on some of the samples, at a temperature between 500 and 800 °C for 5–10 h. We characterized the Tc of each sample within 10% of the SC shielding volume fraction by measuring the magnetic susceptibility with a magnetic property measurement system (MPMS; Quantum Design, San Diego, CA, USA). (see Supplementary Fig. 11) Hall resistivity was measured with a physical property measurement system (PPMS; Quantum Design).

ARPES measurements

ARPES experiments were performed at beamlines 5-2 and 5-4 of the Stanford Synchrotron Radiation Lightsource (SSRL, Menlo Park, CA, USA) and TPS 39 A Nano-ARPES beamline of the National Synchrotron Radiation Research Center (NSRRC, Hsinchu City, Taiwan). Samples were cleaved in situ, and experiments were performed at temperatures below 30 K, at a pressure around 4 × 10−11 Torr. We used linearly polarized 16.5 eV photons with an overall energy resolution of ~12–15 meV for the doping dependence study described in the main text, while using 50, 55, and 85 eV photons for the complementary studies shown in the Supplementary Information (see Supplementary Note 3 and Supplementary Fig. 13). The 16.5 eV irradiation and linear polarization aligned with the c-axis component of the sample are suitable for the present study, as this combination tends to brighten the nodal band spectrum and enhance hot-spot features18.

μSR measurements

The μSR measurements were performed using the M20 beamline at the TRIUMF facility (Vancouver, British Columbia, Canada). A dozen pieces of sliced PLCCO crystal (typical area: 1 cm2) were wrapped with silver foil and attached to the sample holder. Zero-field μSR (ZF-μSR) measurements were carried out over the temperature range of 2–200 K. The physical quantity measured was the evolution of the muon depolarization Pz(t) = [NB(t) − αNF(t)] = [NB(t) + αNF(t)], where NF(t) and NB(t) are the number of positrons counted at detectors antiparallel and parallel to the incident muon spin direction, respectively. α is the efficiency ratio between the forward and backward detectors. Pz(t) conveys information about the local magnetic field distribution at the muon-stopping sites. All of the data were analyzed using the free musrfit software package51. All the μSR spectra in Fig. 5a were obtained by subtracting the temperature-independent constant background from the raw data and then normalizing it with the theoretical initial asymmetry Pz(t = 0) estimated by the fittings. Details of the analysis process are described in Supplementary Information (see Supplementary Note 4).

CPT spectral simulations

Cluster perturbation theory (CPT) is designed to be an efficient method to estimate the \(A\left({{{\bf{k}}}},\omega \right)\) of strongly correlated systems26,27. When dividing the infinite plane into clusters, the Hamiltonian could be split into \(H={H}_{c}+{H}_{{{\mathrm{int}}}}\), where \({H}_{c}\) contains the (open-boundary) intra-cluster operators and \({H}_{{{\mathrm{int}}}}\) contains the operators with inter-cluster indices (hopping terms for the Hubbard model). Restricting ourselves to zero temperature, we use exact diagonalization (ED) to exactly solve the cluster Green’s function \({G}_{c}({{{\rm{\omega }}}})\) associated with the intra-cluster Hamiltonian \({H}_{c}\). Then the CPT method estimates Green’s function by treating \({H}_{{{\mathrm{int}}}}\) perturbatively, giving

$$G({{{\bf{k}}}},\omega )=\frac{{G}_{c}({{{\rm{\omega }}}})}{1-V({{{\bf{k}}}}){G}_{c}({{{\rm{\omega }}}})}$$

Here \(V({{{\bf{k}}}})\,=\,{\sum }_{{{{\bf{R}}}}}{H}_{{{\mathrm{int}}}}{e}^{i{{{\bf{k}}}}{{{\boldsymbol{\cdot }}}}{{{\bf{R}}}}}\) is the inter-cluster interactions projected to the intra-cluster coordinates. Taking the long-wavelength limit, we obtain the spectral function

$${A\left({{{\bf{k}}}},\omega \right)}_{{CPT}}=-\frac{1}{\pi N}{Im}{\sum}_{\sigma }{G}_{a,b}({{{\bf{k}}}},\omega ){e}^{i{{{\bf{k}}}}{{{\boldsymbol{\cdot }}}}({{{{\bf{r}}}}}_{{{{\boldsymbol{a}}}}}-{{{{\bf{r}}}}}_{{{{\boldsymbol{b}}}}})}$$

with a,b are intra-cluster site indices. In this paper, we employ a \(4\times 4\) cluster as the exact spectral solver. We further use an \(8\times 8\) superclusters to obtain finer doping intervals.