Unveiling phase diagram of the lightly doped high-$T_{c}$ cuprate superconductors with disorder removed

Abstract

The currently established electronic phase diagram of cuprates is based on a study of single- and double-layered compounds. These CuO₂ planes, however, are directly contacted with dopant layers, thus inevitably disordered with an inhomogeneous electronic state. Here, we solve this issue by investigating a 6-layered Ba₂Ca₅Cu₆O₁₂(F,O)₂ with inner CuO₂ layers, which are clean with the extremely low disorder, by angle-resolved photoemission spectroscopy (ARPES) and quantum oscillation measurements. We find a tiny Fermi pocket with a doping level less than 1 $\%$ to exhibit well-defined quasiparticle peaks which surprisingly lack the polaronic feature. This provides the first evidence that the slightest amount of carriers is enough to turn a Mott insulating state into a metallic state with long-lived quasiparticles. By tuning hole carriers, we also find an unexpected phase transition from the superconducting to metallic states at 4 $\%$. Our results are distinct from the nodal liquid state with polaronic features proposed as an anomaly of the heavily underdoped cuprates.

Introduction

Over 30 years of research on the cuprates has led to a “unified” form of the phase diagram, supposed to be applicable to various cuprates Keimer2015 . According to it, the Mott insulator with the antiferromagnetic (AF) order persists up to about 5 $\%$ of carrier doping ($p\sim 0.05$), followed by a dome-shaped superconducting (SC) phase; these two phases are clearly separated without the slightest overlap. In the underdoped region, the pseudogap Yasuoka ; Timusk and charge-density-wave (CDW) Wise_CDW states competes with superconductivity Kondo ; Chang ; Damascelli_CDW ; Hashimoto ; Ghiringhelli . These states develop most significantly around the antinode [or ($\pi,0$) region], leaving only arc-like segments of the Fermi surface (FS) even above $T_{c}$ Norman . As the hole doping decreases, the Fermi arc shrinks and eventually becomes point nodes (or nodal liquid state) at the edge of the Mott insulating phase Kanigel2006 ; Yoshida_LSCO ; Zhou_Bi2201 ; Shen2004 ; Chatterjee . In this state, the quasiparticle peak is tiny and accompanied by polaron-like broad spectra. In addition, this peak disappears with leaving off the node since very broad spectra severely damped by the pseudogap prevail everywhere in momentum space. At doping levels further less, a gap is opened even in the (0,0)-($\pi,\pi$) direction Zhou_Bi2201 ; Shen2004 ; Vishik , turning the nodal liquid state into a full gap state consisting only of polaronic broad spectra all around the Brillouin zone (BZ).

Notably, the above phase diagram is based on the data of single- and double-layered cuprates; it is different from the phase diagrams with a significant overlap of the AF and SC phases in multilayer systems with three or more CuO₂ planes per unit cell Mukuda ; Shimizu . In the single- and double-layered cuprates, the CuO₂ plane is affected by the random potential induced by the adjacent dopant layers, leading to an inhomogeneous electronic state as revealed by scanning tunneling microscopy (STM) STM ; Pan_STM . It is, therefore, possible that the phase diagram is relevant only for disordered CuO₂ planes, especially in the lightly doped region sensitive to disorder. This circumstance may have hindered a fair comparison of the data with the theory describing the doped Mott state, which usually supposes an ideal CuO₂ plane without disorder AFSC_RMP ; AFSC_PRL . It could, however, be solved by focusing as a research target on the inner planes of the multilayer cuprates, which are protected by the outer CuO₂ planes screening the disorder effect from the dopant layers. According to nuclear magnetic resonance (NMR) studies Mukuda ; Shimizu , the carrier doping in the inner planes is much more homogeneous than that in the CuO₂ planes of other compounds, including the single-layered HgBa₂CuO_4+δ (Hg1201) and double-layered YBa₂Cu₃O_6.5 (Y123) thought as to be clean systems. With this advantage, the small Fermi Pocket, which had been elusive while predicted in the doped Mott state, was recently observed in the inner plane of a 5-layer compound Kunisada2020 . The multilayer cuprates, therefore, provide an elleclent platform to unveil the genuine electronic properties of the lightly-doped region, which is key to elucidating the pairing mechanism in cuprates. Above all, since the highest achievable $T_{c}$ among the existing substances is obtained in one of the multilayer cuprates (the trilayer HgBa₂Ca₂Cu₃O_8+δ pressureNature ; Proust ; Yamamoto ), the current subject is crucial for the development of condensed matter physics.

In this article, we have selected the 6-layer Ba₂Ca₅Cu₆O₁₂(F,O)₂ (T_c = 69 K; Supplementary Fig. S1) for a study, where the effective carrier doping of the inner planes should be very low. The electronic properties of the clean CuO₂ planes are revealed over a wide range of hole doping which is controlled by varying the number of inner planes and the in situ potassium deposition on the sample surface. The spectra with well-defined quasiparticle peaks lacking the polaronic features are detected all over the closed Fermi surface (or a tiny Fermi pocket), even at the doping level extremely close to the half-filling; it is distinct from the nodal liquid state and the polaronic state established in the heavily underdoped cuprates with the inevitable disorder. Furthermore, we find that the superconducting pairing occurs at $\sim$4 $\%$ doping, almost the same critical doping as in single-layered cuprates with CuO₂ planes severely disordered. This doping level ($\sim$4 $\%$), therefore, is not the consequence of an increase by the disorder but should be the critical amount of carriers essential for the pair formation even in the ideally clean CuO₂ plane.

Results

Figure 1a plots the spectral intensities close to the Fermi level ($E_{F}$) measured by laser-ARPES at the lowest temperature (T = 5 K). We found three sheets of FSs: One exhibits an arc-like structure typical for the underdoped cuprates, and the other two show small pockets around ($\pi/2,\pi/2$) corresponding to the doped Mott states with the AF order Marino_2020 . To further validate the ARPES results, we also observed the de Haas-van Alphen (dHvA) effect by the torque measurement, a bulk-sensitive probe. We detected quantum oscillations (Fig. 1c) consisting of mainly two frequencies (arrows in Fig. 1f), corresponding to the FS areas covering 1.2 and 4.8 $\%$ of the Brillouin zone. These values almost perfectly agree with the ARPES results (1.0 and 4.3 $\%$). Since carriers are doped from the dopant layers (hatched by orange in Fig. 1f), the doping amount should become less toward the inner planes. It is, thus, expected that the small Fermi pocket, large Fermi pocket, and Fermi arc are each formed by the inner-most layer (IP₀), second-inner plane (IP₁), and outer plane (OP), respectively, as noted in Fig. 1a and Fig. 1f. The validity of this one-to-one correspondence between FSs and CuO₂ layers can be confirmed by comparing these results with those of the 5-layer compound Kunisada2020 , as demonstrated below.

In Fig. 1b, we overlay the FSs of the 5- and 6-layer compounds determined from the laser-ARPES data. Here, note that the samples we observed have similar T_c values (T_c = 65 K and 69 K for 5- and 6-layer compounds, respectively), and thus these two should have similar doping levels. We found that the small Fermi pocket gets much smaller than that of the 5-layer compound, while the other FSs (large Fermi pocket and Fermi arc) remain almost the same. We also obtained the results supporting this conclusion by synchrotron-ARPES (Supplementary Fig. S2) with higher photon energies more generally used in the cuprate research. As summarized in Fig. 1e, the area of the small Fermi pocket labeled as IP₀ is changed by adding one more CuO₂ plane in the unit cell from five to six, and importantly, the area of the 6-layer compound gets half that of the 5-layer compound (arrows in Fig. 1e). This indicates that the carriers in IP₀ are simply split into two for IP₀s doubled in the 6-layer compound (arrows in Fig. 1f) without affecting other planes (IP₁ and OP); this means that the wave function of IP₀ is independent of those of IP₁ and OP. It is further justified by our experimental result that the superconducting gap is observed on the Fermi pocket of IP₁ but not of IP₀ (Supplementary Fig. S4). The mixing of layers should produce superconducting gaps of similar magnitudes, so our data against it indicate that the two pockets derive each from different layers (IP₀ or IP₁) that are essentially electronically decoupled.

We perform a model calculation based on our ARPES results and demonstrate that the mixing of layers is, indeed, negligible (Supplementary Fig. S3). The hybridization among layers is prevented by a potential difference induced by the carrier distribution along the $c$-axis. The inclusion of an interlayer hopping parameter (V) in the calculations doubly splits the small Fermi pocket, similarly to the bi-layer splitting observed in double-layer cuprates Feng2001 . This is attributed to double IP₀s adjacent to each other and sensitive to the V parameter. In contrast, splitting does not appear in the large Fermi pocket even for a relatively large value of V since double IP₁s are structurally separated. Notably, splitting of the small Fermi pocket (IP₀) is not experimentally observed both in ARPES and dHvA. We also note that the peak of the fast Fourier transformation (FFT) spectrum of the dHvA effect for the small Fermi pocket is relatively sharp in width (at least, sharper than the spectra of Y123 YBCO_quantum and Hg1201 Hg_quantum ), and it is almost the same as that for the large Fermi pocket. It indicates that there is not even the slightest splitting in the small Fermi pocket, so the interlayer hopping should be negligibly small in our samples. This is a remarkable feature of a low doping state, and compatible with the observation in Bi2212 that the bilayer splitting energy gets smaller with decreasing carrier concentration Anzai2010 . Calculations with such a small V estimate the mixture of wave functions from different CuO₂ planes to be negligible (less than 3 $\%$; Supplementary Fig. S3d). This leads us to conclude that the three FSs we observed are independently formed by three different CuO₂ planes (IP₀, IP₁, and IP₂). We emphasize that this is a new aspect of multilayered cuprates that was clarified only through the measurement of the 6-layer compounds with double IP₀s.

The above argument allows us to estimate the carrier concentration of each CuO₂ plane directly from the area of each FS. Our data indicate that the inner-most CuO₂ plane (IP₀) is doped by only 1 $\%$ of hole carriers, which is extremely close to the half-filling. In Fig. 2b and 2c, we plot the energy distribution curves (EDCs) around the small Fermi pocket for IP₀ (orange circles in Fig. 2a) and those symmetrized about E_F to eliminate the Fermi cut-off effect, respectively. Surprisingly, we find very sharp peaks in the spectra even for a state with such low carrier density. Here, note that the superconducting gap is absent most likely because the state of 1 $\%$ doping is situated outside the superconducting dome in the phase diagram. The spectral peak width is estimated to be about 7.1 meV (arrows in Fig. 2c), which is comparable to or even smaller than that (blue curve in Fig. 2c) of the optimally-doped Bi₂Sr₂CaCu₂O_8+δ (Bi2212) Kondo_Bi2212 , the cuprate material most well-studied by ARPES. This means that quasiparticles as long-lived as those in the optimally doped state can develop with a tiny amount of carrier doping in an ideally clean CuO₂ plane.

Notably, the peak width is constant all around the Fermi pocket including hot spots at which the FS and the antiferromagnetic zone boundary (AFZB) cross Armitage_hotspot ; Shen_hotspot . This is very different from the anisotropic nature widely acknowledged for the underdoped cuprates Timusk . We also emphasize that the electronic state unveiled here is distinct from the following features illustrated for the single- and double-layered cuprates: the nodal liquid state, where only the nodal direction is metallic and the other $k_{F}$ points are dominated by broad spectra with the pseudogap Kanigel2006 ; Zhou_Bi2201 ; Shen2004 ; Yoshida_LSCO , and the polaronic state, where quasiparticle peaks are tiny and largely buried by a hump-shaped incoherent part. In stark contrast, our data for inner planes exhibit surprisingly simple metallic features, forming a closed Fermi pocket (instead of the nodal liquid state) with well-defined quasiparticles (instead of the polaronic state). Here, note that the quasiparticle in IP₀ is not a product of the superconducting proximity effect from OP. This is evidenced by the fact that quantum oscillations (signals of well-defined quasiparticles) have been observed under conditions that the superconductivity is completely suppressed. ARPES also confirmed that the quasiparticle peak persists even above $T_{c}$ although its width gets broadened due to the thermal broadening effect (Supplementary Fig. S10).

The bands of Fermi arc and pockets are mutually close in momentum space, so their ARPES signals could interfere at high binding energies. To examine single-particle spectra of IP₀, we conducted the band-selective measurements by utilizing the matrix element effect. In Fig. 2e, we map the FS by synchrotron-ARPES not only of the 1st BZ but also up to the 2nd BZ. The data in the 2nd BZ are enhanced in intensity, and moreover, separate the bands of Fermi arc (OP) and pockets (IP₀ and IP₁) in the upper and lower half of the 2nd BZ, respectively. This separation is further confirmed in Figs. 2f and 2g by plotting the ARPES dispersions each across nodal $k_{F}$ points of OP and IP₀ (dashed lines in Fig. 2e). Only the latter exhibits the folded bands about AFZB, which are also revealed by MDCs at $E_{F}$ (the upper panels of Figs. 2f and 2g). We extract the EDC for IP₀ at $k_{F}$ in Fig. 2h, and find a sharp peak accompanied by a tail with relatively low intensities up to the energy scale of the band width. This validates that the Fermi pocket possesses well-defined quasiparticles lacking polaronic features, even though the doping level is extremely small.

In order to explore a wider doping range, we have performed the in situ potassium deposition on the samples. This technique is commonly used in ARPES Ohta2006 including a study of cuprates Hossain2008 ; Damascelli_YBCO ; Zhang2016 . Figures 3a to 3c display the FS mapping before and after the potassium deposition for different times (0, 30, and 60 seconds, respectively). The Fermi pockets get smaller with deposition time. This variation is more clearly demonstrated in Fig. 3f by extracting the momentum distribution curves (MDCs) along the AFZB at E_F. The momentum distance between each paired $k_{F}$s becomes shorter in the two pockets, representing the reduction of hole carriers in both the inner planes. In Fig. 3d and 3e, we plot the large and small Fermi pockets for IP₁ and IP₀, respectively, at three different deposition times, determined from the ARPES spectra. A systematic shrinkage of the pockets is confirmed, as estimated in Fig. 3g from their areas (or carrier concentrations $p$s). The $p$ value decreases faster in the large pocket (IP₁) than in the small pocket (IP₀); this is expected since IP₁ lies closer to the surface dopant layer where potassium is deposited, so it should be more efficiently doped. Our experiments could reduce the doping level of the innermost plane (IP₀) down to 0.7 $\%$ ($p=0.007$), which is so small as to nearly reach the half-filled Mott state.

Here we examine the doping evolution of the ARPES spectra. Figure 4a plots the EDCs at $k_{F}$ on the AFZB for IP₀ (orange circle in the inset of Fig. 4a) in 1.0 $\%$, 0.9 $\%$, and 0.7 $\%$ of doping levels. In the right panel, the symmetrized EDCs are also plotted. We found that sharp quasiparticle peaks persist down to the lowest carrier concentration of 0.7 $\%$ ($p=0.007$). Although the peak intensity was slightly suppressed due to the sample surface deterioration, the spectral peaks have almost the same width (Supplementary Fig. S8), indicating that the scattering rate (or lifetime) of quasiparticles is hardly changed with going toward the half-filling. The results imply that a single hole doped into the Mott insulator can behave as a long-lived quasiparticle in a clean CuO₂ plane (Fig 4b). It is in stark contrast to the property of the single- and double-layered cuprates, in which a quasiparticle peak rapidly dies out at doping levels lower than $\sim$10 $\%$ Damascelli_YBCO ; Zhou_Bi2201 ; Shen2004 ; Vishik , and even if there is some spectral weight at $E_{F}$, it is rather broad and disappears immediately with going away of the nodal direction Yoshida_LSCO ; Devereaux .

IP₁ also realizes a very clean electronic system as confirmed by its low Dingle temperature (T_D, proportional to the scattering rate) obtained by the dHvA quantum oscillations: T_D of IP₁ is estimated to be 12.3 K (Supplementary Fig. S5), which is between those of Y123 (T_D = 6.2 K) YBCO_quantum ; Chan_NC and Hg1201 (T_D = 18 K) Hg_quantum ; Chan_NC . Note that the carrier concentration of IP₁ directly estimated from the Fermi pocket area is small, only to be 4.3 $\%$, which is less than half those ($\sim$10 $\%$) of Y123 and Hg1201 samples used for the quantum oscillation measurements. Hence, the T_D of IP₁ is rather small, considering that it is obtained in such a low carrier concentration with a poor screening effect.

The spectra of IP₁ before the potassium deposition exhibit the superconducting gap consistent with the $d$-wave symmetry, being the largest at the tip of the Fermi pocket (see Supplemental Fig. S4 for more details). This data indicates the coexistence of superconductivity and AF magnetic order, similarly to the observations in the 5-layer compound by NMR Mukuda ; Shimizu and ARPES Kunisada2020 . Figure 4f plots the spectra at the tip of the Fermi pocket for three different doping levels (4.3 $\%$, 4.0 $\%$, and 3.6 $\%$) controlled by the potassium deposition. The superconducting gap observed at the original doping level of 4.3 $\%$ closes at 4.0 $\%$ (This is further justified in Supplementary Note 9). This indicates that the electronic state in IP₁ has got out of the superconducting dome and entered the metallic phase, i.e. the same as that of IP₀, by decreasing the hole doping. Also, the relatively small superconducting gap observed for the pristine surface (magnitude of $\Delta_{\rm tip}$ = 5 meV at the tip of the Fermi pocket and the order parameter $\Delta_{0}=11$ meV determined by extrapolating it to the antinode) is attributed to the carrier doping level (4.3 $\%$) located almost at the edge of the superconducting dome. The superconducting to metal transition we observed occurs by closing a gap along the entire Fermi surface (or pocket) while maintaining well-defined quasiparticles all over it. This is very different from the phase transition for the underdoped cuprates with disordered CuO₂ planes, where the pseudogap accompanied by the damped broad spectra plays a significant role. Rather, our observation is quite similar to the doping-induced phase transition across the superconducting dome on the overdoped side with no indication of the pseudogap, although the Fermi surface size is apparently different, corresponding to $p$, not to 1+$p$.

In Fig .4g, we illustrate a phase diagram of lightly-doped cuprates with extremely clean CuO₂ planes unveiled via the direct observation of the electronic structure by ARPES. The Mott insulating state is realized only when the CuO₂ plane is non-doped, and is strictly half-filled; only the slightest amount of hole doping changes it to a metallic state forming a Fermi pocket with well-defined quasiparticles on the top of the lower Hubbard band (or charge transfer band). We find that the effective mass in the innermost layer (IP₀) and the 2nd inner layer (IP₁) with different carrier concentrations ($\sim$1 $\%$ and $\sim$4 $\%$, respectively) are almost the same ($\sim$0.6 $m_{0}$) by quantum oscillation measurements (Supplementary Fig. S7). This indicates a lack of a pronounced band narrowing when varying $p$ toward the half-filled Mott state. The transition to a Mott insulator most likely occurs by completely removing hole carriers from the lower Hubbard band until the perfect half-filling, rather than by controlling the band width. At 4 $\%$, the metal-to-superconductor transition occurs by opening the superconducting gap, and the system enters the phase where the AF order and superconductivity coexist. Intriguingly, this critical doping level is almost the same as that of some single- and double-layer compounds, such as La_2-xSr_xCuO₄ which is known to be severely disordered according to the NMR studies Takagi1992 ; Yamamoto2001 ; Groen1990 ; Liang2006 . This indicates that 4 $\%$ is the intrinsic critical doping level necessary to form the superconducting pairs even in the CuO₂ planes with the clean-limit condition. This scenario differs from theories that suggest that superconductivity occurs simultaneously with the appearance of metallicities by carrier doping to the ideal Mott insulator without disorder AFSC_RMP ; AFSC_PRL . Our results will provide crucial insight into understanding the intrinsic relationship between the Mott physics and the pairing mechanism in the lightly-doped CuO₂ planes, which has not been accessible for the single- and double-layered compounds mainly studied in the long history of cuprate research.

Methods

Samples: Single crystals of underdoped Ba₂Ca₅Cu₆O₁₂(F,O)₂ (see crystal structure in Fig. 1f) with $T_{c}$=69 K were grown at between 1100 ^∘C and 1200 ^∘C under a pressure of 4.5 GPa without an intentional flux. The starting composition for the crystal synthesis is Ba₂Ca₃Cu₄O_7.9F_2.1, which is known to be almost the same in the single crystals Tokiwa . We have conducted X-ray diffraction measurements along the $c$-axis for all the sample pieces and confirmed that they are single crystals, not the mixtures of crystal domains with different numbers of CuO₂ layers per unit cell. Magnetic susceptibilities for these crystals (Fig. S1) show a sharp superconducting transition with $\sim$4 K in width, indicative of high quality in our samples; signal-to-noise ratio is not so high owing to the small volume in our crystals ($\sim 300\times 300\times 50$ $\mu$m in crystal size). Laue image of the single crystal (Fig. S1b) shows a four-fold rotational symmetry with no indication of structural modulations.

ARPES measurements: Laser-based ARPES data were accumulated using a laboratory-based system consisting of a Scienta R4000 electron analyzer and a 6.994 eV laser (the 6th harmonic of Nd:YVO₄ quasi-continuous wave). The data presented are measured at 5 K. The overall energy resolution in the ARPES experiment was set to 1.4 meV. Synchrotron-based ARPES measurements were performed at high-resolution branch (HR-ARPES) of the beamline I05 in the Diamond Light Source, equipped with a MBS A-1 analyzer. The data presented are measured at the photon energy of 55 eV and at the temperature of 10 K. The overall energy resolution was set to $\sim$10 meV in our experiments. In both the laser- and synchrotron-ARPES measurements, a typical cleavage method was used to get a clean surface of the samples: a top post glued on the crystal is hit in situ to obtain a flat surface suitable for the ARPES measurements. The cleavage plane has been confirmed by STM to be along the F/O dopant layers Tokiwa .

Quantum oscillation measurements: Torque magnetometry experiments were performed with a commercial piezoresistive cantilever (SEIKO PRC-120) Torque in pulsed magnetic fields up to 60 T (36 ms pulse duration). The cantilever directly detects the magnetic torque ($\tau$) as the result of the anisotropic magnetization of the sample, $\tau=$$M$$\times$$H$, and the magnetic quantum oscillation known as the de Haas-van Alphen (dHvA) oscillation was observed. Figure 1c in the main paper shows the data after subtracting background, which was obtained by fitting a quadratic function to each curve of the raw data in the range of magnetic field between 26 and 60 T.

Acknowledgements
We thank M. Imada and Y. Yamaji for useful discussions, T. Yajima for technical assistance in the x-ray measurements performed using facilities of the Institute for Solid State Physics, University of Tokyo. We thank Diamond Light Source for access to beamline I05 under proposals SI30646, SI28930, and SI25416 that contributed to the results presented here. This work was supported by the JSPS KAKENHI (Grants Numbers. JP21H04439 and JP19H00651), by the Asahi Glass Foundation, by MEXT Q-LEAP (Grant No. JPMXS0118068681), and by MEXT as “Program for Promoting Researches on the Supercomputer Fugaku” (Basic Science for Emergence and Functionality in Quantum Matter Innovative Strongly-Correlated Electron Science by Integration of “Fugaku” and Frontier Experiments, JPMXP1020200104) (Project ID: hp200132/hp210163/hp220166).