Abstract
The relationship between the cuprate pseudogap (Δ_{p}) and superconducting gap (Δ_{s}) remains an unsolved mystery. Here, we present a temperature and dopingdependent tunneling study of submicron Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} intrinsic Josephson junctions, which provides a clear evidence that Δ_{s} closes at a temperature T_{c}_{0} well above the superconducting transition temperature T_{c} but far below the pseudogap opening temperature T*. We show that the superconducting pairing first occurs predominantly on a limited Fermi surface near the node below T_{c}_{0}, accompanied by a Fermi arc due to the lifetime effects of quasiparticles and Cooper pairs. The arc length has a linear temperature dependence, and as temperature decreases below T_{c} it reduces to zero while pairing spreads to the antinodal region of the pseudogap leading to a dwave superconducting gap on the entire Fermi surface at lower temperatures.
Introduction
The properties of the pseudogap and its relation to the superconducting gap are among the central issues in the search for the cuprate pairing mechanism. A number of spectroscopic studies such as scanning tunneling microscopy (STM) and angleresolved photoemission spectroscopy (ARPES) have been reported^{1,2,3,4,5,6,7,8,9,10,11,12,13,14}. Some experiments indicate that the pseudogap may arise fully from precursor superconductivity (singlegap picture)^{1,2,3,4,5}, while others suggest an origin that is unrelated to superconductivity (twogap picture)^{6,7,8,9,10,11,12,13,14}. In the latter case, uncertainty exists as precursor pairing in certain temperature range above the superconducting transition temperature T_{c} is reported in some experiments^{8,13,14}, which contrast with other experiments in which the superconducting gap Δ_{s} is found to close at T_{c}^{6,9,10,12}. In this paper, we address the issue using the temperature and dopingdependent tunneling spectroscopy of submicron Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} intrinsic Josephson junctions.
For conventional BardeenCooperSchrieffer (BCS) superconductors, Giaever's planartype tunnel junctions^{15} provided decisive measurements of the superconducting gap, the electronic density of states (DOS), the quasiparticle scattering rate, and the effective spectrum of phonons that mediate pairing^{16,18,17}. Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} intrinsic Josephson junctions^{19} are the similar planartype junctions with the best quality one may have for cuprate superconductors. As is shown in the inset of Fig. 1, these junctions are formed within the crystal with CuO_{2} doublelayers as superconducting electrodes and BiO/SrO interlayers as the tunnel barrier. Such superconductorinsulatorsuperconductor (SIS) junctions avoid all kinds of extrinsic uncertainties during experiment and can offer stable and reproducible temperaturedependent measurements. Earlier spectroscopic studies using these junctions suffered from sample's selfheating that severely distorts the tunneling spectra and many efforts were made to solve the problem^{20,21,22,23,24,25,26,27}. One effort involved optimizing the surfacelayer contact and reducing the junction size well below 1 μm, which are shown to suppress heating sufficiently in the case of near optimally doped samples^{23,24,25,26}. The data presented below were based on these works and extended to samples with different doping strength.
The present work demonstrates that the superconducting gap Δ_{s} closes at a temperature T_{c}_{0} well above T_{c} but far below the pseudogap opening temperature T*, which supports a twogap picture with superconducting pairing persisting up to T_{c}_{0}. The pairing is found to occur first on a limited Fermi surface near the node below T_{c}_{0}, accompanied by a Fermi arc due to finite quasiparticle scattering rate and pair decay rate. The arc length has a linear temperature dependence, and as temperature decreases below T_{c} it reduces to zero while pairing spreads to the antinodal region of the pseudogap leading to a dwave superconducting gap on the entire Fermi surface at lower temperatures.
Results
Experimental spectra
In Fig. 1, we show the tunneling conductance σ(V, T) at typical temperatures for four samples from underdoped (UD) to overdoped (OD) with T_{c} = 71, 80, 89 and 79 K, respectively (see Methods and Supplementary Information for details). The data are normalized to σ(V, T*) with T* = 310, 280, 230 and 140 K, above which the spectra become gapless. At low temperatures they exhibit the familiar peakdiphump structure with the superconducting coherence peak height and position, the peakdip separation all evolving systematically with the doping strength. The dip feature, caused possibly by electron coupling to a Boson spectrum with energy linked to the peakdip separation^{28}, disappears gradually as temperature approaches T_{c}.
In Fig. 2 ad, we plot half the conductance peak position in meV versus temperature (squares), which represents Δ_{s} approximately at low temperatures. We see that for all samples the value decreases slightly with increasing temperature toward T_{c}, similar to the BCS gap versus temperature dependence. Near and above T_{c}, however, it differs substantially for different doping samples, which should result from the increasing roles played by the pseudogap and by the lifetime effects of the quasiparticles and Cooper pairs.
Temperature dependence of the superconducting gap
A key feature one expects for superconductors is that Δ_{s} follows the BCSlike gap equation and closes at a temperature, possibly higher than T_{c}^{29}, where Cooper pairs vanish. To clarify the situation, we fitted our experimental spectra with a DOS that is widely used in tunneling experiment for both BCS superconductors^{18,30} and cuprates^{1,26,31}:where a dwave gap is considered and the subscript s denotes the superconducting part. θ and γ_{s} are the angle of inplane momentum measured from (π,0) (see Fig. 2e) and the parameter characterizing the lifetime effects, respectively. The DOS was first proposed by Dynes et al.^{30} and recently shown^{26} to be related to a phenomenological selfenergy developed for the pseudogap discussion in which Δ_{s} extends to the precursor pairing regime above T_{c}^{32}. Taking the UD89K data as an example, we replot half the peak position below T_{c} in Fig. 2f, in which lines are the BCS dwave gap that closes at T* (dashed), T_{c} (dotted) and T_{c}_{0} = 140 K to be discussed below (solid). In the singlegap picture with pairing starting at T*, Δ_{s} should vary along the dashed line near and above T_{c} if the lifetime effects are taken into account. Δ_{s} obtained from fit to the normalized spectra σ(V, T)/σ(V, T*) using N_{s}(θ, ω) over the whole Fermi surface^{26} is shown in Fig. 2f as open squares (γ_{s} not shown for clarity). It is seen that the result deviates significantly from the dashed line, which means that the singlegap picture does not lead to an appropriate description.
In an STM experiment on Bi_{2}Sr_{2}CuO_{6+δ} superconductors, Boyer et al.^{6} found that when seemingly irregular experimental spectra are normalized to the one slightly above T_{c}, they reveal a homogeneous superconducting gap that closes at T_{c}. This treatment eliminates the effect of the pseudogap that already exists above T_{c}. (Such treatment was also applied for Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} SIStype junctions^{28} and BCS SIStype junctions where effects unrelated to superconductivity are successfully removed^{16}). In the related twogap picture, one may view the two phases as coexisting and being anticorrelated on the Fermi surface with different spectral weights, and there is a boundary θ_{p} below and above which they dominate respectively^{11}. Fig. 2e shows a simple cutoff presentation as used in STM experiments^{7,8}. In the present work, we fitted the normalized spectra σ(V, T)/σ(V, T_{c}), considering consistently N_{s}(θ, ω) for θ > θ_{p} only on the Fermi surface so the pseudogapdominant region was excluded in the tunneling current calculation (see Methods). The resulting Δ_{s} taking θ_{p} = 12°, a value close to the STM observation^{7}, is shown as open circles in Fig. 2f for the UD89K sample. It can be seen that the fit is again unsatisfactory when compared to the BCS curve (dotted line).
A satisfactory fit was nevertheless obtained when it was performed with respect to σ(V, T)/σ(V, T_{c}_{0}) where T_{c} < T_{c}_{0} < T*, for which pair formation starting at T_{c}_{0} should be assumed. Precursor pairing above T_{c} has been suggested previously in some experiments^{8,13,14,33,34}. We note that the half peak position in Fig. 2c (squares) shows an obvious turning near 140 K. In Fig. 2f (also in c), Δ_{s} from the fit considering T_{c}_{0} = 140 K and excluding the pseudogap region of θ < θ_{p} = 12° is shown as solid uptriangles. We see that Δ_{s} follows nicely the BCS curve (solid line) above T_{c} in this case. As temperature decreases below T_{c}, however, it deviates increasingly with decreasing temperature. To understand this behavior, we also plot Δ_{s} obtained with θ_{p} = 0 as downtriangles in Fig. 2c, which shows a clear tendency of approaching the BCS solid line below T_{c}. These two results can be naturally explained if, as temperature decreases below T_{c}, the superconducting pairing gradually spreads to the antinode on the Fermi surface with θ < θ_{p}, which is predominantly occupied by the pseudogap phase above T_{c}.
Similar results were obtained for other samples and they are displayed in Fig. 2ad together with the fitted γ_{s} shown as open uptriangles. T_{c}_{0} from the best fit for the four samples is 150, 130, 140 and 100 K, respectively. For the more underdoped UD71K sample, pair spreading into the antinodal region is seen in a more limited temperature range below T_{c} since a tunneling dip quickly develops, which is beyond the simple description using N_{s}(θ, ω)^{28}. On the other hand, all the data above T_{c} show a compelling evidence that the superconducting gap Δ_{s} closes at T_{c}_{0}. They demonstrate that the superconducting phase grows out from the pseudogap phase with T_{c}_{0} as the Cooper pair formation temperature, which supports a twogap picture with precursor pairing extending from above T_{c} up to T_{c}_{0}.
Temperature dependence of the zerobias conductance
Additional evidence that the superconducting gap Δ_{s} closes (or opens if we look with decreasing temperature) at T_{c}_{0} came from the direct experimental data of the zerobias conductance σ(0, T), which should be largely related to the density of states near the Fermi level and thus is sensitive to the formation of an energy gap. In Fig. 3, we plot the measured σ(0, T)/σ(0, T*) for the four samples in the temperature range from 4.2 K to T*. The up and down arrows indicate T_{c}_{0} and T_{c}, respectively. It can be seen that with lowering temperature an accelerated decrease occurs starting from T_{c}_{0} for all samples, which corresponds to the fast decrease of the density of states resulting from the opening of the superconducting gap Δ_{s}. Also, such a decrease is seen to continue farther below T_{c} for higher doping samples. This can be explained considering that for higher doping samples, T_{c} is closer to T_{c}_{0} so Δ_{s} will increase more below T_{c} before reaching the lowtemperature value (see Fig. 2), which leads to the further reduction of σ(0, T).
Parameters of the superconducting and pseudogap phases
We emphasize that our fit based on σ(V, T)/σ(V, T_{c}_{0}) assumes a temperatureindependent pseudogap. As is discussed by Boyer et al.^{6} this should be a reasonable approximation. In many experiments such as STM^{1} the pseudogap peak position is found nearly temperature independent and it disappears by “fillingup” as temperature approaches T*. If we take the half peak position at T_{c}_{0} in Fig. 2 ad to characterize the pseudogap Δ_{p}, it shows a distinct doping dependence as that of Δ_{s}. In Fig. 4a, we plot T*, T_{c}_{0} and T_{c} against the doping level p, while Δ_{s} and Δ_{p} are shown in Fig. 4b and the resulting 2Δ_{s}/kT_{c}_{0} in the inset. In Fig. 4b, Δ_{p} is seen to have a fast increase as p reduces to the more underdoped level, as observed in ARPES experiments^{35}. On the overdoped side, it continues to decrease to a value below Δ_{s}.
Fermi arcs derived from lifetime parameters
The lifetime effects play an important role in the precursor pairing regime from around T_{c} up to T_{c}_{0} due to increasing γ_{s}. One of the consequences is the appearance of a Fermi arc near the node with θ > θ_{0} (see Fig. 2e), which is defined through the peak separation of the spectral function A(k, ω) around Fermi surface in ARPES experiments^{36,37}. The abovementioned selfenergy model^{32}, from which N_{s}(θ, ω) can be derived^{26}, contains three parameters: the quasiparticle scattering rate Γ, the pair decay rate Γ_{Δ}, and Δ_{s}, with γ_{s} = (Γ + Γ_{Δ})/2. Assuming a linear temperature dependence of Γ, we inferred both Γ and Γ_{Δ} from the fitted parameters γ_{s} in Fig. 2ad. With known Δ_{s}, Γ and Γ_{Δ}, A(k_{F}, ω) was determined and the arc length l_{arc} = 1− (4/π)θ_{0} was calculated^{26,36,37} (see Methods). In Fig. 5, we show the calculated l_{arc} versus temperature for the four samples. The results display an approximate linear temperature dependence, which is quite general as discussed in the ARPES data analysis using the same selfenergy in a simplified situation of Γ = Γ_{Δ}^{36,37}.
Discussion
We have shown that for the four Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} crystals with different doping levels the superconducting gap Δ_{s} closes at a temperature T_{c}_{0} well above the superconducting transition temperature T_{c} but far below the pseudogap opening temperature T* thus an extensive precursor pairing regime between T_{c} and T_{c}_{0} is demonstrated. In the Methods section, we present an alternative fitting procedure considering both the superconducting part (Δ_{s}, γ_{s}) and the pseudogap part (Δ_{p}, γ_{p}), which leads to the same conclusion as using the conventional approach of normalizing out the pseudogap contribution described above. It is shown that Δ_{p} is nearly constant from slightly below T_{c} up to T* while γ_{p} experiences a continuous increase, which is consistent with the fillingup character of the pseudogap as temperature approaches T* from below.
So far the STM and ARPES results supporting the twogap scenario alone for the Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} materials are still diverse and controversial. Some results suggest that below T_{c} the superconducting gap would coexist with the pseudogap at the antinode^{12,13} while others indicate that they reside at the nodal and antinodal regions separately^{7}. Above T_{c}, precursor pairing is demonstrated in some experiments^{8,13} whereas a superconducting gap closing at T_{c} is also observed^{9}. The present tunneling results clearly support the precursor pairing view in the temperature range from T_{c} to T_{c}_{0}, which is similar to the results in Refs. 8 and 13. In this temperature range, the superconducting gap and the pseudogap locate predominantly at the node and the antinode, respectively. We note that both the results of Δ_{s} presented as solid up and downtriangles in Fig. 2c are obtained by fitting to the spectra that are normalized to the one at T_{c}_{0}. In this case, the pseudogap contribution is not considered in the fits but it still exists. Therefore the result that the superconducting pairing spreads into the antinodal region below T_{c} means that the two components coexist at the antinode. Since all the data of Δ_{s} in Fig. 2a–d (solid uptriangles) show similar upturns as temperature decreases below T_{c}, we believe the coexisting nature to be true for all samples. On the other hand, for the UD71K sample we see from Fig. 2a (squares) that the pseudogap spectral peak quickly diminishes and switches to the superconducting peak below T_{c}. This may indicate that the spectral weight of the pseudogap becomes small compared to that of the superconducting gap below T_{c} for this sample which is still not deep enough into underdoping, or the pseudogap structure is obscured by the growth of the dip structure in the tunneling spectra. For higher doping samples, uncertainty arises from the fact that the superconducting gap and pseudogap scales becomes similar (see Fig. 2).
Fermi arcs in the ARPES experiments often show a relatively large size just above T_{c}, which collapse as temperature decreases below T_{c}^{3,9,38}. In the twogap scenario, the collapse results from the opening of the superconducting gap on the arc at T_{c}^{9}. Our results are similar to those in a sense that T_{c}_{0} is in the place of T_{c} and the arc region is defined from θ_{p} to π/2  θ_{p} in Fig. 2e in the pseudogap state. As mentioned above, the lifetime effects in the superconducting state can be successfully used to explain the linear temperature dependence of the arc length l_{arc}^{36,37}. It is interesting to note that in the singlegap picture l_{arc} will exhibit a faster rise as temperature increases across T_{c} and therefore has a larger value compared to those in Fig. 5 just above T_{c}^{26}, which is consistent with the results observed in ARPES experiments^{3,38}. On the other hand, our present results, including the development of the superconducting gap at T_{c}_{0} > T_{c} on an arc spanned in the pseudogap state and the temperature dependence of the arc resulting from the lifetime effects in the superconducting state, as depicted in Fig. 5, bear a close resemblance to the STM observations^{8,29}. The differences and similarities in these ARPES and tunneling experiments remain to be explained in the future.
Open questions that are of further interest are the nature of the pseudogap and whether the superconducting and pseudogap phases are formed from the same underlying physics. Recent experiments suggest that the pseudogap phase can result from various densitywave and other states, which may compete^{11} or have an intimate relationship with the superconducting state^{14}. Our results indicate that the pseudogap Δ_{p} has a distinct temperature and doping dependence compared to Δ_{s}, which may not be in favor of the view that they have a common microscopic origin. In the classical BCS superconductors, the strong Coulomb and phonon interactions between electrons in the normal state lead to an average correlation energy in the order of eV, which is much larger than the pairbinding energy of meV. The strong interactions are later removed in Landau's Fermiliquid theory with quasiparticles replacing the bare electrons. Consideration of the interaction neglected in Landau's approximation leads to the coupling between quasiparticles and formation of Cooper pairs^{17}. In the present case of cuprate superconductors, however, the situation is different and is more complicated as we see that the pseudogap size can be larger, comparable, and smaller than the superconducting gap when doping increases.
Methods
Experimental details
Mesatype intrinsic Josephson junctions (IJJs)^{19,20,21,22,23,24,25,26,27} were used in this work with their geometry shown schematically in Fig. S1. Details of the sample fabrication have been described elsewhere^{23,24}. To reduce samples selfheating, a notorious problem in IJJs studies, we took special care to reduce the contact resistance between Au films and Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} crystals which results in the surface layers with good properties^{23,25}. In addition, mesa sizes were reduced well below 1 μm as it was demonstrated that heating can be largely neglected in this case^{24}. Other methods to reduce heating include using IJJs made of HgBr_{2} intercalated Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} crystals^{20,27} and adopting shortpulse measurements^{21}, which are discussed extensively recently^{27}. These studies demonstrate tunneling spectra with moderate sharpness of the conductance peak and clear presence of the dip feature after the reduction of heating, as achieved in the present experiment shown in Fig. 1. (See Supplementary Information for further details.)
Spectra fit separating the pseudogap contribution
The IV characteristics of a superconductorinsulatorsuperconductor (SIS) junction can be calculated from^{18}:where R_{N} is junction's normalstate resistance, n(ω) is the DOS of two identical Selectrodes and f(ω) is the Fermi function. Our results were obtained by fitting the normalized experimental spectra using σ = dI/dV from equation (1) with the following normalized DOS for n(ω):where cos^{2}(2θ) comes from the directional tunneling matrix element, which is found to improve the description for the intrinsic tunneling process within Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} crystals^{26,27,28}. In equation (2), integration is performed from θ_{p} to π/4 with θ_{p} ≥ 0 due to symmetry. If the superconducting phase is considered on the entire Fermi surface, we have θ_{p} = 0. As discussed in the paper, our central results in Fig. 2a–d were obtained from fitting σ(V,T)/σ(V, T_{c}_{0}) with a nonzero θ_{p} to exclude the pseudogapdominant region on the Fermi surface.
The θ_{p} parameters obtained from fits to the four samples used in this work are listed in Table SI. For samples from UD89K to OD79K, θ_{p} decreases from 12° to 10°. This trend is consistent with the STM observations^{7}. However, for UD80K and UD71K samples, θ_{p} is 10° and 11°. The slight inconsistency could be caused by the fact that the UD80K and UD71K samples were yttrium doped, which were different from the oxygen doped UD89K and OD79K samples and might have altered crystalline arrangement resulting in reduced pseudogap expansion in momentum space. The satisfactoriness of our fit using these parameters can be seen in Fig. S4.
Using normalized spectra to get rid of the effects unrelated to superconductivity is a common practice in tunneling experiments for both BCS superconductors^{16,18} and cuprates^{6,28} in both SIN (N being a normal metal)^{6,18} and SIS^{16,28} type tunnel junctions. For example, McMillan and Rowell studied the SIS type Pb junctions^{16}. By normalizing the data below T_{c} to the one above T_{c}, additional structures in the measured spectra resulting from tunnel barrier phonons are successfully removed. The phonon spectra extracted from the data are exactly the same as those obtained from the SIN type Pb junctions. Below we further justify this approach for the present experiment by considering both the superconducting and pseudogap contributions in the fitting procedure.
According to the twogap scenario, from T_{c}_{0} up to T* there is only the pseudogap phase located predominantly near the antinode with θ < θ_{p} and one has N_{s}(θ, ω) = 1 for θ > θ_{p}. Below T_{c}_{0} down to at least T_{c} the superconducting and pseudogap phases exist predominantly above and below θ_{p}, respectively (see Fig. 2e). If we use the same form of dwave DOS to model the pseudogap phase that may come from various densitywave states, now denoted by N_{p}(θ, ω) with two parameters Δ_{p} and γ_{p}, we can write the following DOS for n(ω) in equation (1) for T > T_{c}_{0}:where C_{S} is a constant from the ungapped part on the Fermi surface. For T < T_{c}_{0} we have The IV curve can be calculated above T_{c}_{0} from and below T_{c}_{0} from In Fig. S5, we show the results from fits using I_{>}(V) and I_{<}(V) to the normalized experimental spectra σ(V, T)/σ(V, T*) (note that T* is used as normalization temperature instead of T_{c}_{0}), taking also the UD89K IJJs as the example. Uptriangles are replotted Δ_{s} and γ_{s} from Fig. 2c and 2f. Above T_{c}_{0}, only the pseudogap is concerned, the parameters Δ_{p} and γ_{p} are thus directly determined using I_{>}(V), which are shown as downtriangles above T_{c}_{0}. The downtriangles shown in the figure below T_{c}_{0} are obtained using I_{<}(V) and the replotted Δ_{s} and γ_{s} parameters. In other words, if these Δ_{p} and γ_{p} are used, the two fitting approaches would produce the same Δ_{s} and γ_{s}. For comparison, squares in Fig. S5 show the Δ_{s} and γ_{s} when Δ_{p} and γ_{p} values at 150 K are used for temperatures below T_{c}_{0}. These data show nearly the same Δ_{s} but slightly different γ_{s}.
These results confirm our central conclusion that the superconducting gap Δ_{s} closes at T_{c}_{0}. We note that Δ_{p} in Fig. S5 is nearly constant while γ_{p} increases with increasing temperature all the way up to T*, which demonstrate that the pseudogap disappears by “fillingup” as temperature approaches T*. Since a continuing decrease of γ_{p} down to T_{c} seems reasonable, both Δ_{s} and γ_{s} parameters obtained from the simple fitting approach using σ(V, T)/σ(V, T_{c}_{0}) and equations (1) and (2) should be a good approximation.
Fermi arc calculation
For the discussion of the cuprate pseudogap in ARPES experiments, Norman et al. proposed a phenomenological selfenergy taking account of the lifetime effects^{32}:
where _{k} is the energy of bare electrons relative to the value at the Fermi surface. From equation (7) it can be shown that the Green's function G(k, ω) = 1/[ω − _{k} − Σ(k, ω)] has the form
The spectral function on the Fermi surface A(k_{F}, ω) = −(1/π)ImG(k_{F}, ω), assuming Δ_{k} = Δ_{s} cos(2θ), is given by^{26} In the ARPES experiments, it is considered to be gapped if A(k_{F}, ω) has maxima at ω = ±ω_{p} ≠ 0, while Fermi arc appears at places where A(k_{F}, ω) has maximum only at ω = 0. Thus ω_{p} can be found by setting the first derivative of equation (9) to zero:where . By setting the second derivative to zero, the angle θ_{0} at which the arc starts is found to be The relative arc length l_{arc} is defined by In the present work, the quasiparticle scattering rate Γ and pair decay rate Γ_{Δ} were estimated from the experimentally fitted parameter γ_{s} in Fig. 2a–d via the relation γ_{s} = (Γ + Γ_{Δ})/2^{26}. We assumed a linear temperature dependence of Γ and considered that Γ is larger than Γ_{Δ}, which should be reasonable from the basic physical considerations. In Fig. S6 the results of Γ and Γ_{Δ} for the four samples are shown, which were determined considering that Γ_{Δ} = 0 near T_{c} and the slope of Γ set close to that of γ_{s}. The corresponding l_{arc} versus T calculated are plotted in Fig. 5.
References
 1.
Fischer, ø., Kugler, M., MaggioAprile, I., Berthod, C. & Renner, C. Scanning tunneling spectroscopy of hightemperature superconductors. Rev. Mod. Phys. 79, 353–419 (2007).
 2.
Shi, M. et al. Coherent dwave superconducting gap in underdoped La_{2–x}Sr_{x}CuO_{4} by AngleResolved Photoemission Spectroscopy Phys. Rev. Lett. 101, 047002 (2008).
 3.
Nakayama, K. et al. Evolution of a pairinginduced pseudogap from the superconducting state gap of (Bi,Pb)_{2}Sr_{2}CuO_{6}. Phys. Rev. Lett. 102, 227006 (2009).
 4.
Meng, J. et al. Monotonic dwave superconducting gap of the optimally doped Bi_{2}Sr_{1.6}La_{0.4}CuO_{6} superconductor by laserbased angleresolved photoemission spectroscopy. Phys. Rev. B 79, 024514 (2009).
 5.
Chatterjee, U. et al. Observation of a dwave nodal liquid in highly underdoped Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}. Nat. Phys. 6, 99–103 (2010).
 6.
Boyer, M. C. et al. Imaging the two gaps of the hightemperature superconductor Bi_{2}Sr_{2}CuO_{6+δ}. Nat. Phys. 3, 802–806 (2007).
 7.
Kohsaka, Y. et al. How Cooper pairs vanish approaching the Mott insulator in Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}. Nature 454, 1072–1078 (2008).
 8.
Lee, J. et al. Spectroscopic fingerprint of phaseincoherent superconductivity in the underdoped Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}. Science 325, 1099–1103 (2009).
 9.
Lee, W. S. et al. Abrupt onset of a second energy gap at the superconducting transition of underdoped Bi2212. Nature 450, 81–84 (2007).
 10.
Ma, J.H. et al. Coexistence of competing orders with two energy gaps in real and momentum space in the high temperature superconductor Bi_{2}Sr_{2–x}La_{x}CuO_{6+δ}. Phys. Rev. Lett. 101, 207002 (2008).
 11.
Kondo, T. et al. Competition between the pseudogap and superconductivity in the highTc copper oxides. Nature 457, 296–300 (2009).
 12.
Vishik, I. M. et al. ARPES studies of cuprate Fermiology: superconductivity, pseudogap and quasiparticle dynamics. New J. Phys. 12, 105008 (2010).
 13.
Kondo, T. et al. Disentangling Cooperpair formation above the transition temperature from the pseudogap state in the cuprates. Nat. Phys. 7, 21–25 (2011).
 14.
RuiHuaet al. From a singleband metal to a hightemperature superconductor via two thermal phase transitions. Science 331, 1579–1583 (2011).
 15.
Giaever, I. Energy gap in superconductors measured by electron tunneling. Phys. Rev. Lett. 5, 147 (1960).
 16.
McMillan, W. L. & Rowell, J. M. Tunneling and strongcoupling superconductivity, in Superconductivity, edited by R. D. Parks (Marcel Dekker, New York, 1969).
 17.
Schrieffer, J. R. Theory of Superconductivity, (Benjamin, New York, 1964).
 18.
Wolf, E. L. Principles of Electron Tunneling Spectroscopy, (Oxford University Press, New York, 1985).
 19.
Kleiner, R. et al. Intrinsic Josephson effects in Bi_{2}Sr_{2}CaCu_{2}O_{8} single crystals. Phys. Rev. Lett. 68, 2394 (1992).
 20.
Yurgens, A. et al. Pseudogap features of intrinsic tunneling in (HgBr_{2})Bi2212 single crystals. Int. J. Mod. Phys. B 13, 3758–3763 (1999).
 21.
Anagawa, K. et al. 60 ns time scale short pulse interlayer tunneling spectroscopy for Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}. Appl. Phys. Lett. 83, 2381–2383 (2003).
 22.
Krasnov, V. M., Sandberg, M. & Zogaj, I. In situ measurement of selfheating in intrinsic tunneling spectroscopy. Phys. Rev. Lett. 94, 077003 (2005).
 23.
Zhao, S. P. et al. Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} intrinsic Josephson junctions: Surface layer characterization and control. Phys. Rev. B 72, 184511 (2005).
 24.
Zhu, X. B. et al. Intrinsic tunneling spectroscopy of Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}: The junctionsize dependence of selfheating. Phys. Rev. B 73, 224501 (2006).
 25.
Li, S. X. et al. Observation of macroscopic quantum tunneling in a single Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} surface intrinsic Josephson junction. Phys. Rev. Lett. 99, 037002 (2007).
 26.
Zhao, S. P., Zhu, X. B. & Tang, H. Tunneling spectra of submicron Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} intrinsic Josephson junctions: evolution from superconducting gap to pseudogap. Eur. Phys. J. B 71, 195–201 (2009).
 27.
Kurter, C. et al. Counterintuitive consequence of heating in stronglydriven intrinsic junctions of Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} mesas. Phys. Rev. B 81, 224518 (2010).
 28.
Zasadzinski, J. F. et al. Persistence of strong electron coupling to a narrow boson spectrum in overdoped Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} tunneling data. Phys. Rev. Lett. 96, 017004 (2006).
 29.
Pasupathy, A. N. et al. Electronic origin of the inhomogeneous pairing interaction in the highT_{c} superconductor Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}. Science 320, 196–201 (2008).
 30.
Dynes, R. C., Narayanamurti, V. & Garno, J. P. Direct measurement of quasiparticlelifetime broadening in a strongcoupled superconductor. Phys. Rev. Lett. 41, 1509 (1978).
 31.
Miyakawa, N. et al. Strong dependence of the superconducting gap on oxygen doping from tunneling measurements on Bi_{2}Sr_{2}CaCu_{2}O_{8–δ}. Phys. Rev. Lett. 80, 157 (1998).
 32.
Norman, M. R. et al. Phenomenology of the lowenergy spectral function in highT_{c} superconductors. Phys. Rev. B 57, R11093 (1998).
 33.
Wang, Y. et al. Fieldenhanced diamagnetism in the pseudogap state of the cuprate Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} superconductor in an intense magnetic field. Phys. Rev. Lett. 95, 247002 (2005).
 34.
Wen, H. H. et al. Specificheat measurement of a residual superconducting state in the normal state of underdoped Bi_{2}Sr_{2–x}La_{x}CuO_{6+δ} cuprate superconductors. Phys. Rev. Lett. 103, 067002 (2009).
 35.
Ideta, S. et al. Enhanced superconducting gaps in the trilayer hightemperature Bi_{2}Sr_{2}Ca_{2}Cu_{3}O_{10+δ} cuprate superconductor. Phys. Rev. Lett. 104, 227001 (2010).
 36.
Norman, M. R. et al. Modeling the Fermi arc in underdoped cuprates. Phys. Rev. B 76, 174501 (2007).
 37.
Chubukov, A. V. et al. Gapless pairing and the Fermi arc in the cuprates. Phys. Rev. B 76, R180501 (2007).
 38.
Kanigel, A. et al. Protected nodes and the collapse of Fermi arcs in highT_{c} cuprate superconductors. Phys. Rev. Lett. 99, 157001 (2007).
Acknowledgements
We thank N. P. Ong, Siyuan Han, X. J. Zhou and Z. Y. Weng for helpful discussions. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10974242 and 50825206) and the Ministry of Science and Technology of China (Grant No. 2011CBA00106).
Author information
Affiliations
Beijing National Laboratory for Condensed Matter Physics, Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China
 J. K. Ren
 , X. B. Zhu
 , H. F. Yu
 , Ye Tian
 , H. F. Yang
 , C. Z. Gu
 , N. L. Wang
 , Y. F. Ren
 & S. P. Zhao
Authors
Search for J. K. Ren in:
Search for X. B. Zhu in:
Search for H. F. Yu in:
Search for Ye Tian in:
Search for H. F. Yang in:
Search for C. Z. Gu in:
Search for N. L. Wang in:
Search for Y. F. Ren in:
Search for S. P. Zhao in:
Contributions
J.K.R., X.B.Z., Y.F.R., H.F.Yang and C.Z.G. prepared the mesa samples. J.K.R., X.B.Z., H.F.Yu and Ye T. did the measurement. J.K.R. and S.P.Z. performed the data analysis. N.L.W. provided and prepared single crystals for the UD71K, UD80K and OD79K samples. S.P.Z. designed the experiment and wrote the manuscript.
Competing interests
The authors declare no competing financial interests.
Corresponding author
Correspondence to S. P. Zhao.
Supplementary information
PDF files
 1.
Supplementary Information
Supplementary information
Rights and permissions
This work is licensed under a Creative Commons AttributionNonCommercialShareALike 3.0 Unported License. To view a copy of this license, visit http://creativecommons.org/licenses/byncsa/3.0/
To obtain permission to reuse content from this article visit RightsLink.
About this article
Further reading

SignReversing Hall Effect in Atomically Thin HighTemperature Bi2.1Sr1.9CaCu2.0O8+δ Superconductors
Physical Review Letters (2019)

Phase Diagram of Bi2Sr2CaCu2O8+d Constructed on Basis of Electronic Structure Investigations
Journal of the Physical Society of Japan (2019)

Simple analytical model of the effect of high pressure on the critical temperature and other thermodynamic properties of superconductors
Scientific Reports (2018)

Explanation of Nonlinear InPlane Resistivity and Hall Coefficient in the Normal State of Cuprates: Polaronic Approach
Journal of Superconductivity and Novel Magnetism (2018)

Analytical assessment of some characteristic ratios for swave superconductors
Frontiers of Physics (2018)
Comments
By submitting a comment you agree to abide by our Terms and Community Guidelines. If you find something abusive or that does not comply with our terms or guidelines please flag it as inappropriate.