Point nodes persisting far beyond Tc in Bi2212

In contrast to a complex feature of antinodal state, suffering from competing orders, the pairing gap of cuprates is obtained in the nodal region, which therefore holds the key to the superconducting mechanism. One of the biggest question is whether the point nodal state as a hallmark of d-wave pairing collapses at Tc like the BCS-type superconductors, or it instead survives above Tc turning into the preformed pair state. A difficulty in this issue comes from the small magnitude of the nodal gap, which has been preventing experimentalists from solving it. Here we use a laser ARPES capable of ultrahigh-energy resolution, and detect the point nodes surviving far beyond Tc in Bi2212. By tracking the temperature evolution of spectra, we reveal that the superconductivity occurs when the pair-breaking rate is suppressed smaller than the single-particle scattering rate on cooling, which governs the value of Tc in cuprates.

The "artificial" Fermi arc. The length of arc where the symmetrized EDCs have a single-peak (ε peak =0) for OP92K (a) and OD72K (b). The nodal liquid behavior, which has a T -linear fashion and is extrapolated to zero at T = 0, is superimposed in each panel.  (S1) for examining the effect of two different scattering rates (Γ single and Γpair) on the spectral shape. In all the curves presented, the gap magnitude is set to 10meV (∆ ≡ 10 meV). a, The Γ single dependence when Γpair ≡ 0meV. b, The Γpair dependence when Γ single ≡ 0meV. c, The spectral shape for various values of Γ single and Γpair, which are set to be equal. In each bottom panel of a, b, and c, all spectra in the upper panels are superimposed. While all the curves plotted are convoluted with a gaussian that has the width of the experimental energy resolution (∆ε = 1.4meV), the ∆ε value is so small that the difference in shape from original curves is negligible.
Energy (eV)    Figure 10: Signature for the gap existence in the single-peak spectra off the node. Symmetrized EDCs of OP92K at T > Tc measured off the node (bottom panel of a) and at the node (bottom panel of b). The kF points are marked with circles in the inset of a (φ = 13.5 • ) and b (φ = 0 • ). Two peaks in the off-nodal spectra merge to one peak at 103K (Tone). c, The spectral width (∆ε) at T > Tone are plotted as a function of temperature. The off-nodal ∆ε(T ) decreases with temperature and reaches the minimum at ∼ 135K (Tpair) before it eventually increases at elevated temperatures. The anomalous behavior contrasts to the nodal ∆ε(T ), which monotonically increases over the whole range of temperatures. In the upper panels of a and b, the spectra above Tone with a single-peak (rainbow colors) are extracted. In addition, the spectra at T > Tpair and T < Tpair are separated with an offset (black arrow). The insets show the magnified areas marked with the dashed squares. The black and magenta dashed arrows indicate the energy shifts of spectral tails with increasing temperature. The inset of c plots the energy positions of symmetrized EDCs (ε peak ) as a function of temperature. The green curve indicates the BCS-type gap function with the onset of 135K, which fits well the ε peak (T ) at the low temperatures.      Figure 11: Comparison of fitting results with Eq.(S1) for OP92K data between two different settings to the energy gap, ∆. a, The ∆ is defined to be equal to the peak position of spectra [∆(T ) ≡ ε peak (T )], shown in the top panel. b, The ∆ is defined to have a BCS-type gap function with an onset at Tpair=135K, shown in the top panel. The middle and bottom panels plot the obtained fitting parameters, Γ single and Γpair, at several kF points in the momentum region, where the gapless Fermi arc was previously claimed to emerge at Tc (see circles and an arrow in the inset of b). The values of Γpair at high temperatures are not plotted, since the spectral shape is not sensitive to the Γpair when ∆ is small or zero, and thus it is impossible to precisely determine the value.   Figure 12: Comparison of fitting results with Eq.(S1) for OD72K data between two different settings to the energy gap, ∆. a, The ∆ is defined to be equal to the peak position of spectra [∆(T ) ≡ ε peak (T )], shown in the top panel. b, The ∆ is defined to have a BCS-type gap function with an onset at Tpair=89K, shown in the top panel. The middle and bottom panels plot the obtained fitting parameters, Γ single and Γpair, at several kF points in the momentum region, where the gapless Fermi arc was previously claimed to emerge at Tc (see circles and an arrow in the inset of b). The values of Γpair at high temperatures are not plotted, since the spectral shape is not sensitive to the Γpair when ∆ is small or zero, and thus it is impossible to precisely determine the value.
Supplementary Figure 13: Isotropic scattering mechanism around the node. a, b, ARPES spectra of OP92K measured at various kF points over a wide φ angle measured far below Tc (T =10K) (a) and above Tpair (T =150K) (b). In order to remove the effect of Fermi cut-off, the curves in a are symmetrized about the Fermi level, and the ones in b are divided by the Fermi function at 150K. c, The Fermi surface. The kF points studied in Fig.5a and Supplementary Figure 14 are marked by colored circles. d, The peak width of spectra (or Γ single ) in a and b.
The orange arrows in c and d indicate the momentum region where the gapless Fermi arc was previously claimed to emerge at Tc. The almost isotropic Γ single is obtained in this momentum region.
Supplementary Figure 14: Fitting results for OP92K data with the model function of Eq.(S1). a, Same panel as Fig. 5a in the main paper: single-particle scattering rate (Γ single ) and pair breaking rate (Γpair) extracted by setting the energy gap (∆) to be the BCS-type gap function (a green curve) with the onset of 135K. b, ARPES spectra (black curves) and fitting results (red curves) are overlapped. Fermi angle (φ) is described in the top of each panel, and the corresponding kF point is marked with a circle in the inset of a. We added a small background linear in energy (∝ |ω|) to the fitting function A(kF , ω) for φ = 11.7 • and 13.5 • , in order to properly extract the scattering rates.

Supplementary Note 1: Samples and Experimental method
Optimally doped Bi 2 Sr 2 CaCu 2 O 8+δ (OP92K) and overdoped (Bi,Pb) 2 Sr 2 CaCu 2 O 8+δ (OD72K) single crystals with T c =92K and 72K, respectively, were grown by the conventional floating-zone (FZ) technique. Magnetic susceptibilities for these single crystals are shown in Supplementary Figure 1. Sharp superconducting transitions with ∼1 K (OP92K) and ∼3 K (OD72K), indicative of a high quality, are confirmed. 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 with a repetition rate of 240 MHz). The overall energy resolution in the ARPES experiment was set to 1.4 meV for all the measurements.
In order to accomplish the temperature scan of spectra at a high precision, we made a special sample stage; we use a heat switch to thermally isolate the sample holder with a heater and a temperature sensor (totally 20 × 30 × 10 mm size) from the rest of the system when sweeping the sample temperature. With this technique, the samples are efficiently heated up from 10K up to 300K while keeping the temperature of manipulator rod other than the sample stage lower than 35K. It minimizes the degassing and keeps the pressure in the measurement chamber better than 2 × 10 −11 torr during the entire temperature sweeping. The method of local heating also prevents the thermal expansion of the long manipulator rod, and thus the sample position becomes unchanged with temperature. It enabled us to take data in fine temperature steps with the automated measurement of temperature scan from precisely the same spot on the crystal surface, which were essential to achieve the aim of this study.
In Supplementary Figs. 2 and 3, we show the ARPES spectra of OP92K obtained at various k F points with sweeping the temperature, with and without an offset, respectively. These are the original data used for the analysis in the main paper. The ARPES technique is sensitive to the condition of sample surface, and the biggest challenge in it is to prevent the aging of sample surface with time. In Supplementary Figure 4, we carefully check the stability of sample surface in our experiment by comparing two sets of spectra measured before and after the temperature scan for all the momentum cuts we measured. While it took about 12 hours to complete each temperature sweeping, almost perfect reproducibility of spectral line shapes is confirmed for all the data. This ensures that our data measured with the new heating technique are reliable and suitable for conducing the detailed analysis presented in the paper. Supplementary Figure 5 shows the data set similar to those in Supplementary Figure 3, but measured for OD72K.

Supplementary Note 2: Further Evidence for absence of the Fermi arc at T c
In the main paper, we demonstrate that the d-wave gap with a point node persists at T c by plotting the symmetrized EDCs at various k F points around the node (see Fig. 3d). Here we present a further evidence that the conclusion is robust and it is not sensitive to the selection of k F value. Supplementary Figure 6a shows the Fermi-function divided band dispersions (same as Fig.3a in the main paper) measured at T c along several momentum cuts (color lines in Supplementary Figure 6c). The corresponding EDCs are plotted in Supplementary Figure 6b. Along the diagonal cut (the most left panel), the spectra with one peak crosse E F due to the gap node. In contrast, two peak structure (marked with arrows), signifying a gap existence, is observed at k F (painted with red) for the other directions. This is a direct evidence that the d-wave gap with a point node persists, thus the Fermi arc is absent at T c .

Supplementary Note 3: Temperature evolution of the "artificial" Fermi arc
The Fermi arc [1] is one of the most interested features in cuprates. Previous studies have proposed that the gapless Fermi arc abruptly emerges at T c , and expands linearly with a further increase of temperature [2][3][4]. The arc length was claimed to be extrapolated to zero at T = 0 [3,4]. The so-called "nodal liquid" behavior has been having a huge influence on the condensed matter community.
In the main paper, we argue that the tracking of peak positions of the symmetrized EDCs (ε peak s) underestimates the gap closing temperature, thus provides an "artificial" Fermi arc. Here we demonstrate that even such an artificial arc behaves quite differently from the previous expectation, when high resolution data by a laser ARPES are examined. In Supplementary Figure 7a and 7a, we estimate the length of arc with ε peak =0 as a function of temperature for OP92K and OD72K, respectively. In the same panels, the nodal liquid behaviors are superimposed. Our results are significantly off these lines. As discussed in the main paper, the spectral line-shape is determined not only by the energy gap (∆), but also by the mutual relation between the single-particle scattering rate (Γ single ) and the pair breaking rate (Γ pair ). Therefore, it is rather reasonable that the growth of arc with ε peak =0 becomes arbitrary.
Supplementary Note 4: Effect of scattering rates, Γ single and Γ pair , on the spectral line shape In the analysis, we used the spectral function with a phenomenological self-energy (ref. [5]) given by Here ∆ is the energy gap, and ε(k) the energy dispersion with ε(k F ) =0. Γ single and Γ pair are the single particle scattering rate and the pair breaking rate, respectively. In Supplementary Figure 8, we examine the effect of these scattering rates on the spectral line shape at k F with fixing ∆ to 10 meV. To make it realistic, all the curves plotted are convoluted with a Gaussian that has the width of the experimental energy resolution (∆ε = 1.4meV). Nevertheless, note that the ∆ε value is so small that the difference in shape from original curves is negligible. Figure 8a shows the Γ single dependence of the spectral function when the Γ pair is fixed to zero. One finds that the intensity at E F is always zero regardless of the Γ single value, whereas the spectral width gets broadened with increasing Γ single . This situation strongly contrasts to that in our ARPES data (Supplementary Figure 2), which clearly show the filling of spectral weight at E F with increasing temperature. The behavior of gap filling can be produced by inputting non-zero Γ pair values in Eq.(S1). It is demonstrated in Supplementary Figure 8b, where the spectra with several Γ pair values are calculated with fixing Γ single to zero. Importantly, the energy position of spectral peak (ε peak ) dramatically decreases with an increase of the Γ pair value as a consequence of the spectral filling around E F , and eventually it becomes zero. This simulation indicates that the peak energy underestimates the "real" energy gap (ε peak < ∆) when the magnitude of Γ pair becomes large.
Here we should point out that the following self-energy has been used to estimate the magnitude of ∆ by a fitting to the ARPRS spectra in the many previous reports, This is the same formula as Eq.(S1) setting Γ pair = 0, and thus it always provides one the spectral intensity of zero at E F as discussed above (see Supplementary Figure 8a). We emphasize that the spectral weight around E F observed in the ARPES data was previously attributed to the spectral broadening coming from the finite energy resolution in ARPES (for example, see the supplementary information of ref. [2]). The extremely high energy resolution in our equipment enabled us to reveal that the spectral weight at E F is not an artifact due to the experimental resolution, but an intrinsic signature of the pair breaking effect. This signifies that the Eq.(S1) is more suitable than Eq.(S2) as the model spectral function of cuprates, and that the effect of Γ pair must be taken into account to properly estimate the value of ∆ from the spectral shape. In Supplementary Figure 8c, we simulate a special case setting Γ pair = Γ single in Eq.(S1). The self-energy and the corresponding spectral function at k F under this condition are derived as follow; where Γ ≡ Γ single = Γ pair . This formula, consisting of double Lorentzian functions, is familiar as the phenomenological BCS spectral function [6][7][8]. The calculated spectra for various Γ values are plotted in Supplementary Figure 8c. One can confirm a couple of discrepancies in these from our data. First, the spectral peak in the ARPRS data has an asymmetric shape about the peak energy, differently from the Lorentzian shape. Secondly, the effect of gap filling in this simulation is quite sensitive to the magnitude of Γ, and it is far more drastic than that observed in our data at elevated temperatures. These lead us to conclude that the Eq.(S3) is not a proper formula of spectral function for cuprates either; that is, the assumption of Γ pair = Γ single is not suitable. We also stress that the Dynes function [9] given by, is the density-of-states version of Eq.(S3) with Γ ≡ Γ single = Γ pair , hence it is also not adequate for cuprates. This is understandable because there is no good physical reason that these two different scattering rates should be equal. Our results demonstrate that a proper combination of Γ single and Γ pair is required to reproduce the data.
In the previous reports [10][11][12], the spectrum integrated along a selected momentum cut was utilized to study the gap evolution with temperature and its relation with the electron scattering. Here we argue the necessity of investigating the one-particle spectrum, rather than its momentum integration, for such a study. In Supplementary Figure 9, we demonstrate that a momentum-integrated spectrum, I DOS (ω) = A(k, ω)dk for A(k, ω) of Eq.(S1), is equally sensitive to the Γ single value ( Supplementary  Figs. 9a and 9b) Figs. 9a, 9c, 9e, and 9g) signify that the momentum-integrated spectrum is not capable of extracting the mutual relationship between the two scattering rates (Γ single and Γ pair ).
In contrast to it, the effects of Γ single and Γ pair on the line shape of the momentum-resolved spectrum A(k, ω) are independent, broadening the width and filling the weight around E F , respectively. This circumstance is demonstrated in Supplementary Figure 8, and also seen in Supplementary Figure 9h, at which the spectra at k F are extracted from images in Supplementary Figs. 9b, 9d, and 9f. (Spectral colors of red, blue, and green are corresponding to those of thick dashed lines on the images.) Therefore, the precise investigation for the line shape of one-particle spectra is required to separate the three important parameters (Γ single , Γ pair , and ∆) in Eq.(S1). The main challenge in this task, which has been troubling the ARPES people, is to accomplish both of a sufficient energy resolution and an ultra-high statistics in the ARPES spectra with no noise at a time, which are usually conflicting with each other. We have overcome this difficulty by using a laser ARPES with a unique cold finger (mentioned in "Experimental Method"), which was constructed in our lab. We emphasize that the Eq.(S1) is a minimal model to reproduce the ARPES data, thus the relation among the three physical parameters, revealed in the present work, provides a key ingredient to formulate the mechanism of the high-T c superconductivity in cuprates.
Supplementary Note 5: Signature of a gap existence in the off-nodal, single-peak spectra above T c Here we demonstrate that the off-nodal spectra above T c exhibit a signature of the gap opening even after becoming one-peak structure. The bottom panels of Supplementary Figure 10a and 10b plot the symmetrized EDCs of OP92K above T c measured at the node (φ = 0 • ) and off the node (φ = 13.5 • ), respectively. Two peaks seen in the off-nodal spectra merge to a single peak at T one =103K (Supplementary Figure 10a). For clarity, the spectra above the T one are indicated with rainbow colors, whereas those at the lower temperatures with gray colors in the both of Supplementary Figure 10a and 10a.
We find that the single-peak spectra off the node (rainbow curves in Supplementary Figure 10a) become sharper with increasing temperature up to ∼ 135K (T pair ) before eventually broadened at higher temperatures. The anomalous feature is more clearly demonstrated in the upper panel of Supplementary  Figure 10a, where the spectra below and above the T pair are separated with an offset. As seen in the the magnified insets, the spectral tail shifts to lower energies in the range of T < T pair (a magenta arrow), while it to the higher energies in T > T pair (a black arrow). It strongly contrasts to the case at the node (Supplementary Figure 10b); the spectral tail keeps shifting toward higher energies at elevated temperatures in the both temperature ranges (black arrows). We summarize these results in Supplementary Figure 10c, by plotting the spectral width ∆ε (HWHM) at and off the node as a function of temperature. The anomalous decrease of ∆ε is seen only for the data off the node (orange circles). As discussed in the main paper, a plausible explanation for it is that the off-nodal spectra have small energy gaps, even though they exhibit single peak shapes, and they become sharper up to the temperate (T pair ) at which the energy gap completely closes.
Supplementary Note 6: Spectral fitting under the assumption of ∆(T ) ≡ ε peak (T ) In the main paper, we claim that the energy of spectral peak underestimates the energy gap (ε peak < ∆) close to the node at high temperatures, at which the spectral peak width becomes larger than the magnitude of the energy gap. Here we perform a spectral fitting to our ARPES data with Eq.(S1) [same as Eq.(1) in the main paper], assuming that these two values are identical [∆(T ) ≡ ε peak (T ), upper panel of Supplementary Figure 11a (OP92K) and Supplementary Figure 12a (OD72K)], and demonstrate that the extracted parameter has an unrealistic behavior, reflecting the irrelevance of ∆(T ) ≡ ε peak (T ). The middle and bottom panels of Supplementary Figure 11a plot the Γ single (T ) and Γ pair (T ) of OP92K, respectively, obtained by the fitting to the spectra in the momentum region, where the Fermi arc was previously claimed to appear at T c (orange arrow in the insets of Supplementary Figure 11b). For a comparison, we also plot in Supplementary Figure 11b (OP92K) the corresponding results for ∆(T ) of the BCS-like gap function, which are presented in the main paper (Fig. 5a). The difference between the two fitting-results is very clear; the former (Supplementary Figure 11a) shows an abnormal upturn in the Γ single (T ) curves on cooling in the gapped momentum region, and it contrasts to the latter result (Supplementary Figure 11b) showing almost identical curves in Γ single (T ) for all φs. In Supplementary  Figure 12, we present the same data set as in Supplementary Figure 11, but for OD72K. Similarly to the case of OP92K, an abnormal upturn is obtained in the Γ single (T ) when ∆(T ) ≡ ε peak (T ) (the middle panel of Supplementary Figure 12a), and it is corrected by setting ∆(T ) to be the BCS-like gap function (the middle panel of Supplementary Figure 12b).
The behavior of the abnormal upturn is interpreted as follows. An energy gap is still open above T c even though the spectrum has a single peak, and thus the spectral peak width overestimates the scattering rate. Since the degree of the overestimation gets higher with an increase of the energy gap on cooling, the upturn in the Γ single (T ) appears. The upturn seems to be more enhanced with getting away from the node (see middle panel of Supplementary Figure 11a and Supplementary Figure 12a). This is also expected as the overestimation becomes more serious toward the antinode with a larger energy gap. This scenario is further validated by the fact that the upturns start around T ∼ 130K and T ∼ 90K for OP92K and OD72K, respectively, which are consistent with the onset temperatures of pair formation determined in the main paper.
One might think that the cuprates are known to have a significantly anisotropic scattering mechanism, thus the situation in Supplementary Figure 11a (OP92K) and Supplementary Figure 12a (OD72K) with the strong momentum variation could be intrinsic. To argue against this, here we demonstrate that even the optimally doped Bi2212, in fact, has an isotropic scattering mechanism around the node. Supplementary Figures 13a and 13b show the spectra at various k F points over a wide φ angle measured at the lowest temperature (T =10K) and above the pairing temperature (T =150K), respectively. At these temperatures, the value of Γ single can be estimated simply from the peak width of spectra, since the magnitude of Γ pair is negligible at 10K, and the second term in Eq.(S1) with Γ pair is irrelevant above the pairing temperature. The values of Γ single estimated from the peak width of spectra in Supplementary  Figure 13a and 13b are plotted in Supplementary Figure 13d with blue and red circles, respectively. The magnitude is almost constant over a wide φ centered at the node for both the temperatures, which indicates that the scattering mechanism near the node is isotropic. This strongly supports our assertion that the anisotropic behavior of Γ single with an upturn on cooling is an artifact. Beyond |φ| ≈ 15 • , the scattering rate abruptly increases. This signifies the evolution of the competing pseudogap, which becomes dominant around the antinode [13][14][15][16].

Supplementary Note 7: Fitting curves reproducing the ARPES spectra
In the main paper, we demonstrate the spectral fitting with a minimal model of Eq.(S1) to the data of OP92K at φ = 13.5 • (Fig.5c), as a typical example. Here we present all of the fitting results for the ARPES spectra of OP92K and OD72K, which provide the physical parameters plotted in Fig. 5a  (Supplementary Figure 14a) and Fig. 5b (Supplementary Figure 15a), respectively. Supplementary  Figure 14b and 15b shows the ARPES spectra (black curves) and fitting results (red curves) for OP92K and OD72K, respectively. The data are almost perfectly reproduced by the fitting curves.
Supplementary Note 8: The pairing signature detected in the nodal Γ single (T ) The single-particle scattering rate (Γ single ) at the node is extracted simply from the spectral width, thus it can be precisely determined without any models. Supplementary Figure 16a and 16b plots the temperature dependence of EDC width ∆ε at the nodal k F point of OP92K and OD72K, respectively (same as Γ single (T ) at φ = 0 • in Figs. 5a and 5b). We find that the curves of ∆ε(T ) deviate from the T -linear behavior (∆ε linear (T )) in the both samples. It is demonstrated more clearly in the bottom panels by plotting the amount of divination [∆ε linear (T ) − ∆ε data (T )]. The onset temperatures of the deviation are estimated to be ∼ 135K and ∼ 89K for OP92K and OD72K, respectively (yellow circles). The suppression of the scattering rate should be tied to the electron pairing. We found that the onset temperatures are indeed coincident with the T pair values (gap opening temperatures) estimated from the off-nodal spectra in the main paper.