SUPPLEMENTARY INFORMATION

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1 doi: /nature06219 SUPPLEMENTARY INFORMATION Abrupt Onset of Second Energy Gap at Superconducting Transition of Underdoped Bi2212 Wei-Sheng Lee 1, I. M. Vishik 1, K. Tanaka 1,2, D. H. Lu 1, T. Sasagawa 1, N. Nagaosa 3, T. P. Devereaux, Z. Hussain 2, & Z. -X. Shen 1 Samples and Experimental method High quality single crystals of underdoped Bi 2 Sr 2 CaCu 2 O 8+δ with T C = 75 K (UD75K and 92K (UD92K, and an overdoped sample with T C = 86 K (OD86K were selected for the experiments. The carrier concentrations of the samples were carefully adjusted by a post annealing procedure. Slightly underdoped samples (UD92K were prepared by a heattreatment of the crystals in air at 800 C for 200 hours, followed by rapid quenching to room temperature. The onset temperature of superconducting transition, T on C, determined by SQUID magnetometry, was 92 K with a transition width less than 1 K. Heavily underdoped samples (UD75K, T on C = 75 K, were obtained by additional heat treatment of the UD92K crystals: they were sealed in an evacuated quartz-tube with P(O 2 < torr, annealed at 550 C for 200 hours, followed by rapid quenching to room temperature. Overdoped samples (OD86K, T on C = 86 K, were also obtained by additional heat treatment of the UD92K crystals; they were sealed in an quartz-tube with P(O 2 ~ 2 atm, annealed at 500 C for 200 hours, followed by rapid quenching to room temperature. Angle-resolved photoemission spectroscopy measurements were performed at beamline 5- of the Stanford Synchrotron Radiation Laboratory (SSRL with a SCIENTA R000 electron analyzer. All data shown in this paper were taken using 22.7 ev photons to excite the photo-electrons, except for the data shown in Fig. 2(d-f, in which low energy photons 1

2 (7 ev were used. Total energy resolution was set to 7 mev for the data shown in Fig. 1, Fig. 3(b and 3(c, while the data shown in Fig. 2 (a-c and Fig. 3(d were measured with an energy resolution of 5 mev. Taking the advantage of the low photon energy (7 ev, the data shown in Fig. 2(d-f were measured with an energy resolution of 3.2 mev. The measurements were performed in the Γ-Y quadrant in the 1st Brillouin zone, where the main Fermi surface near the nodal region can be well separated from the replica Fermi surface arising from the photo-electron diffraction by the super-modulation of the crystal structure. The temperature fluctuation during our measurement was less than 0.1 K. To maintain a clean surface, the sample were cleaved in situ and measured in an ultra high vacuum chamber with a base pressure of better than Torr. ARPES spectrum represents the occupied part of the single-particle spectral function, which is distorted near E F because of the Fermi-Dirac (FD function cut-off 2. An approximate way to remove this effect is to divide ARPES spectrum by an effective Fermi- Dirac function, which is generated from the convolution of the FD function at the sample temperature with the energy resolution function. This procedure allows us to trace the band dispersion above E F, where thermal population leads to appreciable spectral weight at higher temperatures. This method was applied in Fig. 1. We also note that at 22.7 ev both the bonding band and antibonding band can be resolved in our data. In the antinodal region, the antibonding band and the replica band due to crystal super-modulation mix up and can not be distinguished, so we traced the gap along the Fermi surface of bonding band, which was used to make Fig. 3 in our paper. 2

3 The issue about the resolvable gap size by ARPES at a given resolution It is a common perception that photoemission spectrum cannot resolve a gap smaller than its energy resolution. Whether photoemission spectrum can resolve the gap not only depends on the gap size, but also the details of the spectrum, such as its lineshape and width. In Fig. S1(a and S1(b, we demonstrate situations when there are two Lorentzian curves with their peaks positioned at +/- 3 mev to simulate the symmetrized EDC with 3 mev gap. The width of the Lorentzian curves is deliberately set to different values between Fig. S1(a and S1(b. As can be seen, after applying a Gaussian convolution of 5 mev FWHM to imitate the effect of energy resolution, the two peaks can still be resolved in (a, but not in (b. Although the nominal gap sizes are 3 mev in both cases, the spectrum width determines whether this gap can be resolved by 5 mev instrumental resolution. To demonstrate the resolution effects on spectrum with more realistic lineshape, the phenomenological model proposed by M. R. Norman et al. (see next section is used to generate simulated EDCs shown in Fig. S1(c. The width of the simulated EDCs is set to be comparable to that of experimental data near the nodal region at a temperature close to T C. As can be seen, 2 mev gap can be resolved even with 5 mev energy resolution. These simulations demonstrate that the error bar of our gap size measurement should not be set as the face value of the energy resolution. Instead, we estimate our error bar by taking account the uncertainty from the fitting procedure (±0.5 mev, the uncertainty of Fermi energy (±0.5 mev, and additional 100% margin, as described in the text of the paper and in the next section. 3

4 Intensity (arb. unit (a Two Lorentizans (b Two Lorentizans (c at +/- 3 mev FWHM = 2 mev ΔE=0 mev ΔE=5 mev at +/- 3 mev FWHM = 5 mev ΔE=0 mev ΔE=5 mev Model (M. Norman et al. Δ = 2 mev, Γ=20 mev ΔE= 0 mev ΔE= 5 mev ΔE=10 mev E - E F (ev E - E F (ev E - E F (ev Figure S1 Demonstrations of the resolution effects in symmetrized ARPES spectrum. (a Two Lorentzian curves with their peak positions located at +/- 3 mev and a FWHM of 2 mev to mimic the symmetrized EDC with a gap of 3 mev. A Gaussian convolution with FWHM 5 mev is applied to simulate the effect of the instrumental resolution. In this case, the two peaks (i.e. the gap can still be resolved. (b Same as (a, except for the width of the Lorentzian curve is broader (FWHM = 5 mev. In this case, the gap can not be resolved by 5 mev resolution, even though the gap size is the same as in (a. (c The phenomenological model proposed by M. R. Norman et al. is used to generate the EDCs that are comparable to the measured data. This is the same model we used for the gap fitting. The gap size (Δ is set to 2 mev and the width (Γ is set to 20 mev. The two peaks (i.e. the gap can still be resolved by 5 mev instrumental resolution (black curve, while it can not be resolved with a 10 mev instrumental resolution (blue curve. Fitting the gap size To determine the gap size, we used the phenomenological model described in Ref. 23 to fit our data. The self-energy at Fermi crossing point k F in the superconducting state may be expressed as 2 Δ ( k F, ω = iγ +, ω

5 where Γ is the life time of the quasi-particle related to the width of the spectrum and Δ represents the gap at Fermi crossing point. The spectral function is then calculated according to A( k F 1 1 1, ω = ImG( kf, ω = Im. ω Σ( k, ω F In addition, a Gaussian convolution representing the instrumental resolution was applied to the spectral function. Finally, we fit the symmetrized ARPES spectra with this convoluted spectral function to obtain Δ and Γ. In each cut, all available EDCs near k F were fitted; the gap size was determined by averaging over EDCs which are in the proximity of the EDC(s with the smallest fitted gap size. The fitted Γ of the data shown in lower panel (7 ev data of Fig. 2(c is summarized in Fig. S2(b. Other data shown in Fig. 2 also exhibit a similar temperature dependence of Γ. We note that this model fits reasonably well to those data with a clear peak in the spectrum; thus, all superconducting state data shown in the paper can be fit fairly well by this model. The results shown in this paper were obtained by fitting the symmetrized data in the energy range of ±70 mev. The error bar is determined in the same manner described earlier. To obtain the gap size for the pseudogap in C5-C7, we also used the same phenomenological model. We found that the fitting is not as robust as in the superconducting gap, especially for the data taken at locations near the antinodal region, such as C7. This is because there is no clear peak in the spectrum, as illustrated in the Fig. 1(c. The fitted Δ exhibits larger fluctuation with different choice of the fitting range, yielding a larger error bar. 5

6 The collapse of the gap when temperature approaches T C In Fig. 2(a, we show the raw EDC near the Fermi crossing point to demonstrate the temperature dependence of the thermally-populated Bogoliubov quasiparticle band, which is seen to move toward E F with increasing temperature and vanishes above T C. Since this behavior can be seen directly in the raw spectrum shown in Fig. 2(a, we consider it to be strong evidence that the superconducting gap reduces and collapses across the superconducting transition. However, one might ask whether these data can be alternatively interpreted in a situation that the gap remains the same for all temperatures, but becomes irresolvable just because of the thermal broadening effect from both the Fermi function and quasiparticle life time. In this section, simulated EDCs under such scenario are demonstrated to argue that this scenario is inconsistent with our data. Starting from a simplest case, the thermal broadening effect contributed entirely from the Fermi function is illustrated in Fig. S2(a. The EDCs are first generated by the aforementioned model with parameters Δ = 15 mev, and Γ = 13.5 mev fixed for all temperatures, and then multiplied by the Fermi function at corresponding temperature. This set of parameters is consistent with the gap size of the 7 ev data shown in Fig. 2(c at 10 K, and the line width at 70K. Gaussian convolution with FWHM=3.2 mev is also applied to simulate the resolution effect. As can be seen, the peak position of the thermally-populated Bogoliubov dispersion remains at about the same energy for these temperatures, which is inconsistent with our data. Next, we demonstrate a more realistic case in which both temperature dependent Fermi function and quasiparticle lifetime are considered. In Fig. S2(c, the simulated EDCs are generated in the same procedure described in Fig. S2(a except that a temperature dependent quasiparticle lifetime obtained from our data were used (Fig. S2(b. As shown, 6

7 there is no obvious shift of the peak position of the thermally-populated Bogoliubov band within this range of Γ. This is also inconsistent with the experimental data shown in Fig. 2(a and (e. Thus, the disagreement between the simulations and experimental data strongly suggests that the gap indeed reduces and vanishes across the superconducting transition. (a (b (c Simulation Fixed Γ Pt. C Γ corresponding to 7 ev Data shown in Fig. 2(c Simulation Realistic Γ 70 K 82 K 87 K 92 K 97 K E - E F (ev Γ(meV T C Temperature (K 70 K 82 K 87 K 92 K 97 K E - E F (ev Figure S2 Simulated EDCs for the case of temperature independent gap. (a Simulated EDCs at various temperatures, which are generated by multiplying spectra created from the phenomenological model with Δ =15 mev, and Γ = 13.5 mev to the Fermi-Dirac function with corresponding temperatures. The dashed line serves as a guide-to-the-eye, which indicates the peak position of the thermally-populated Bogoliubov band. (b Fitted quasiparticle lifetime (Γ of the 7 ev data shown in Fig. 2(c. (c EDCs generated by the same procedure as those shown in (a, but with a temperature dependent Γ as graphed in (b. The peak position of the thermally-populated Bogoliubov band remains unchanged as indicated by the dashed line. 7

8 Finally, as a remark, the coherence factor of the Bogoliubov quasiparticle dispersion in cuprates has been quantitatively analyzed, which is consistent with the BCS picture 31. Simulations of temperature dependent STM spectrum One of our important findings described in this paper is the observation of a temperature dependent evolution of the gap function. An interesting question is how this temperature dependent gap function affects STM spectra at corresponding temperatures. To gain some insight into this issue, we adopted the simple model described in the Supplementary Information of Ref. 15 to simulate temperature dependent STM spectra. For simplicity, the lifetime broadening Γ is set to zero for this simulation and the linear normal state background is neglected. The gap functions used for calculating STM spectra were obtained by fitting to our data. The fitted curves are described in the following: [ ] (0 2 cos ( ( cos(6 0.1 cos( ( cos(2 10 ( C C C C k k k K T K T K T < < + Δ > > = = Δ + = Δ = Δ = Δ = Δ, where Δ 0 is set to 37 mev for the simulations of 10K and 82K, and 2 mev for that of 102K. C is set to be 0.1, describing the boundary of the gapless region on the Fermi surface at 102K. We note that the gap function for the pseudogap (T = 102K described above only valid between 0 and /. To extend to other octants of the Brillouin zone, 8

9 symmetrization and sign change is needed to maintain the d x 2 -y 2 symmetry. The gap functions are plotted in Fig. S3(a as the solid curves. (a K 82 K 102 K (b 102 K 82 K Δ (mev k (, di/dv 10 K Γ FS angle (degree V (mev Figure S3 Gap functions and simulations of temperature dependent STM spectra. (a The gap functions at different temperature are plotted versus Fermi surface angle as defined in the inset. This is the same set of data shown in Fig. 3(b. The solid curves are the fitted gap functions as described in the text. (b Simulated STM spectra with the gap functions at corresponding temperatures shown in (a. The simulated STM spectra are demonstrated in Fig. S3(b. As can be seen, the closing of the gap near the nodal region manifests mostly near the zero bias of the spectrum, while the apparent gap is dominated by the temperature independent pseudogap in the antinodal region. This is consistent with the existing temperature dependence STM data. Furthermore, it has also been mentioned in the Supplementary Information of Ref. 15 that the simple d-wave form can only fit to the low temperature STM data. This is consistent with our observation that the gap function at higher temperature (82 K is no longer a 9

10 simple d-wave form. We also note that since the nodal behavior is obscured by the larger antinodal region contribution in STM spectra, it is hard to see that the superconducting gap closes at T C, unless a proper normalization process has been applied, as demonstrated in Ref. 16. Supplementary Notes: 29. H. Matsui et al.,bcs-like Bogoliubov Quasiparticles in High-T C Superconductors Observed by Angle-Resolved Photoemission Spectroscopy. Phys. Rev. Lett. 90, (

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