Abstract
An energy gap is, in principle, a dominant parameter in superconductivity. However, this view has been challenged for the case of highT_{c} cuprates, because anisotropic evolution of a dwavelike superconducting gap with underdoping has been difficult to formulate along with a critical temperature T_{c}. Here we show that a nodalgap energy 2Δ_{N} closely follows 8.5 k_{B}T_{c} with underdoping and is also proportional to the product of an antinodal gap energy Δ^{*} and a squareroot superfluid density √P_{s} for Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}, using lowenergy synchrotronradiation angleresolved photoemission. The quantitative relations imply that the distinction between the nodal and antinodal gaps stems from the separation of the condensation and formation of electron pairs, and that the nodalgap suppression represents the substantial phase incoherence inherent in a strongcoupling superconducting state. These simple gapbased formulae reasonably describe a crucial part of the unconventional mechanism governing T_{c}.
Introduction
In conventional Bardeen–Cooper–Schrieffer (BCS) theory, a superconducting critical temperature T_{c} is proportional to an energy gap, which opens in the electronic excitation spectrum and stands for both the electronpairing energy and the superconducting order parameter^{1}. This proportionality has been considered to break down in highT_{c} cuprates. As hole concentration p decreases from the optimum, T_{c} decreases, even though the amplitude of dwavelike gap Δ^{*} increases^{2,3}. Instead, superfluid density ρ_{s} decreases along with such underdoping, and thus attracted interest as another key parameter for T_{c}^{4,5,6,7,8,9,10}. Recently, a variety of experimental gap energies in the superconducting state have been classified into two groups: those that increase like Δ^{*} and those that decrease like T_{c} with underdoping^{11,12,13}. The former energy, Δ^{*}, remains as a pseudogap above T_{c} for the underdoped cuprates, as typically seen in the electronic excitation spectra around an antinode^{14}. In the close vicinity of a node, by contrast, the gap closes right above T_{c}, as highlighted by Lee et al.^{15} and Pushp et al.^{16} The further experimental evidences suggest that the lowenergy nearnodal excitations are more relevant to T_{c} than the antinodal ones^{16,17}. Nevertheless, the gap energies defined under a dichotomy between nearnodal and antinodal regions have been difficult to formulate along with the doping dependences of T_{c} and ρ_{s}^{17,18}. The unformulated behaviours of the nodal and antinodal gaps posed severe challenges not only to their mutual relationship but also to their standard role in electron pairing. The identities of these gaps have thus been at the center of controversy over a unified picture of the unconventional pairing state in the highT_{c} cuprates, and intensively examined for clues to the principles underlying the peculiar behaviour of T_{c}.
Here we report our finding of simple formulae,
based on new experimental data of the nodal gap, Δ_{N}, and the antinodal gap, Δ^{*}, for Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} (Bi2212). Clearcut gap images over the entire Fermi surface have been obtained from lowenergy synchrotronradiation angleresolved photoemission (ARPES)^{19,20}, and allowed us to extract the nodal limit of the gap slope beyond the momentumspace dichotomy. As a result, the nodalgap energy Δ_{N} is reconciled with T_{c}, indicating its relevance to the condensation of the electron pairs. A high plateau value of Δ_{N}/T_{c} ratio exhibits sufficiently strong electron coupling for the underdoped Bi2212. Furthermore, the nodal and antinodal gaps are quantitatively related by a factor of the squareroot superfluid density, , in a wide holeconcentration range. This reduction factor is identical to the theoretical prediction for the effect of incoherent pair excitations^{21,22}, implying that the antinodal gap energy is relevant to the formation of the electron pairs. Thus, we argue that the strong coupling makes a critical difference to the pairing state, and that a substantial number of electrons remain paired, despite dropping out of the coherent superfluid in the underdoped superconducting cuprates. The gapbased formulae shed light on a crucial part of the mechanism governing T_{c} through parameterization with the pairing energy Δ^{*} and the surviving superfluid density ρ_{s}^{4,5}.
Results
ARPES data
Figure 1 demonstrates that welldefined electronic dispersions were observed throughout the Brillouin zone, and exemplifies extraction of the gap images from a seamless body of ARPES data. The Fermi surfaces were rigorously determined from the minimumgap loci as marked in Fig. 1b, because both inaccuracy and broadening of the cutting path result in an overestimate of the gap energy. Figure 1c shows that the Fermi surfaces (blue curves) of the bonding band (BB) and antibonding band (AB) of the CuO_{2} bilayer are clearly resolved in the momentum space. The spectra were collected along the two Fermi surfaces from the node through to the antinode, and displayed in Fig. 1e as image plots of energy versus offnode angle θ (Fig. 1a). The distinct energy splitting seen in Fig. 1e indicates that the BB and AB gaps are resolved in our experiment.
The comprehensive highresolution gap study has been supported by lowenergy synchrotron radiation^{19,20}. First, the use of lowenergy photons in ARPES experiments facilitates improvement in energy and momentum resolution in compensation for the narrowing of the momentum field of view^{15,18,19,20,23}. Second, the tunability of synchrotron radiation allows us to optimize the excitationphoton energy hν^{19,20}. We found that the observability in the offnodal region is dramatically improved by slightly increasing hν from those used in laserbased ARPES studies, hν≤7 eV^{15,18,23}. As shown in Fig. 2, using hν=8.5 eV, the spectral intensities of the BB and AB are sufficient even far off the node, θ≳25°, whereas using hν=7.0 eV, the spectral intensity concentrates only around the node of the AB. Beside the observability, Fig. 2d shows that the spectral peaks for hν=8.5 and 7.0 eV are equally sharp, and that the difference between the BB and AB gaps is insignificant. In the present study, we performed the extensive dopingdependent gap measurements with both hν=8.5 and 7.0 eV, and ruled out unexpected effects of transition matrix elements.
Anisotropic gap evolution
Deviation from the dwave gap is directly imaged in the energyversussin2θ plots of the ARPES spectra. Figure 3a–d reveals that, as hole concentration p decreases from the optimum, the linearity in sin 2θ becomes severely distorted, making clear a departure from the standard dwave form, Δ(θ)sin2θ, for the underdoped superconducting Bi2212. For indepth analysis, the gap energies Δ(θ) were determined by fitting energydistribution curves (EDCs), and overlaid as small circles in Fig. 3a–d (see Methods). The high fitting quality is presented in Fig. 3e–h, along with typical EDCs (also see Supplementary Fig. S1). Here the artifact of the antinodal peak smearing is ruled out of the causes for this deviation^{24}, because the curvature in sin 2θ is established within the region, sin 2θ0.75, where the welldefined sharp and intense peak is observed for an underdoped sample with T_{c}=66 K (UD66), as shown in Fig. 3b.
A key finding from the raw spectral images in Fig. 3a is that the gap deviation penetrates to the close vicinity of the node. The appreciable curvature provides evidence that the gap slope versus sin2θ smoothly and asymptotically decreases on approaching the nodal limit, θ→0. So far, it has often been assumed that there is some discontinuity between the nodal and antinodal spectra or a certain region with pure dwave gap^{15,17,18,23,25}. However, it is difficult to define them with the allround seamless images in Fig. 3a–d, because the spectral features, such as peak energy, peak width and their derivatives along Fermi surface, gradually change from the antinode to the node^{26}. Probably, the nature beyond the simple dichotomy between the nearnodal and antinodal regions comes to light in Bi2212, whose superconducting gap is much larger than that of La_{2−x}Sr_{x}CuO_{4}^{25}. The deviation from standard gap behaviour is also deduced from the temperature below which the gap persists, despite the loss of superconductivity. The gapclosing temperature gradually decreases to T_{c} on approaching the nodal limit, θ→0, in a way similar to the deviation from the dwave gap^{15,27}.
For parameterization of the anisotropic superconducting gap, two methodologies are recognized. One is based on the presence of the pure dwave region, and to deduce the nearnodalgap energy from the dwave fit within such a bounded region^{15,17,18,23,25}. However, the nearnodal curvatures seen in Fig. 3a hindered us from applying this method to our data, because it would introduce an averaging effect and an extra uncertainty arising from the illdefined region boundary for the case of the underdoped Bi2212. On this basis, we aimed at the nodal limit of the gap slope in quest of an intrinsic parameter, and adopted the nexthigherharmonic fit over the unbounded region after Mesot et al.^{28} and Kohsaka et al.^{29} We defined the nodal and antinodal gap energies as and Δ^{*}=Δ(θ)_{θ=45°}, respectively, so that Δ_{N}/Δ^{*}=1 is satisfied for the ideal dwave gap as depicted in Fig. 3j. Hence, the fitting function is expressed as
where the first term is solely responsible for the nodalgap slope, the second term models the gap deviation without adjustable angle parameter and its asymptotic behaviour is consistent with the empirical indistinctness of the pure dwave region. Consequently, as overlaid in Fig. 3a–d, this function well captures the curved gap profiles, Δ(θ) (solid curves), and their nodal tangents, Δ_{N} sin2θ (blue lines), for all the doping levels of Bi2212.
The shift in focus provides a new perspective on the nodal gap. Dividing the momentum space largely into two half regions, the gap extending over the nearnodal half is less sensitive to underdoping than that in the antinodal half, as seen from Fig. 3a–d. This general trend is consistent with previous reports^{16,17,18}. Combining this trend with the appreciable curvatures observed in Fig. 3a, it follows that the nodal limit of the gap slope, Δ_{N}, decreases with underdoping from OP91 to UD66, and then to UD42, as indicated by blue tangent lines in Fig. 3a–c. The asymptotic behaviour of the gap deviation suggests that the parameter would be purified by taking the nodal limit θ→0.
Relation to critical temperature
The superconductinggap energies at various dopings are scaled by T_{c}, and put together in Fig. 4a, which reveals that the nodal limit of Δ(θ)/T_{c} remains unchanged with underdoping in contrast to the antinodal limit. Figure 4a also shows that the energies of the BB and AB gaps determined with hν=8.5 and 7.0 eV are all consistent with the single fittedcurve for each doping level. The two gap parameters, Δ_{N} and Δ^{*}, determined from the nexthigherharmonic fit are plotted as a function of hole concentration in Fig. 4b. This shows that the nodal gap Δ_{N} closely follows the decrease in T_{c} with underdoping, departing from the antinodal gap Δ^{*}. The gaptoT_{c} ratios plotted in Fig. 4c are worth noting. As hole concentration p decreases from the overdoped limit, both the nodal and antinodal ones increase with keeping a constant proportion, Δ_{N}=0.87Δ^{*}. This is the canonical behaviour expected when the coupling is getting stronger. A further decrease in p leads to a plateau of 2Δ_{N}/k_{B}T_{c}=8.5, which is about twice the meanfield prediction 4.3 for dwave weakcoupling superconductors, and meanwhile to a continuing increase in 2Δ^{*}/k_{B}T_{c}^{2,3,13}. These features are beyond the scenario of the standard weakcoupling theory. Figure 4d shows that our data for Bi2212 finely converge on the line of 2Δ_{N}/k_{B}T_{c}=8.5 in particular from the optimum to a heavily underdoped level, and that similar values of 2Δ_{N}/k_{B}T_{c} have been reported for optimally doped singlelayer cuprates^{23,25}.
The proportional relation, 2Δ_{N}=8.5 k_{B}T_{c}, reconciles the critical temperature with the nodal slope of the distorted dwave superconducting gap. It seems reasonable that T_{c}, in effect, depends on Δ_{N} rather than Δ^{*}, because thermal quasiparticle excitations concentrate in the vicinity of the node and hardly occur around the antinode in particular for the strongcoupling case, 2Δ^{*}/k_{B}T_{c}4.3. The association between the nodal excitations and T_{c} has been proposed in various ways^{12,13,15,16,17}. In particular, the decrease in Δ_{N} with underdoping has been deduced from the lowenergy slope of B_{2g}Raman spectra^{12} and the quasiparticle interference in scanningtunnelling images^{29}. Besides, in possible relation to Δ_{N}, the characteristic energies having a p dependence similar to T_{c} have been detected by Andreevreflection, B_{2g}Raman, breakjunctiontunnelling and inelasticneutronscattering experiments in the superconducting state^{11}.
Relation to superfluid density
An insight comes from the analogy between the doping p and temperature T dependences. The pdependent distortion of the superconducting gap is presented with normalization to Δ^{*} in Fig. 4e, whereas the T dependence has been reported by Lee et al.^{15} With both increasing p and decreasing T, the superconducting gap approaches the ideal dwave form. As p decreases conversely, Δ_{N} is suppressed relative to Δ*, and decreases towards zero on approaching the disappearance of the superconductivity (Fig. 4e). This is analogous to what is observed with increasing T (ref. 15). Noting that the superfluid density ρ_{s} decreases towards zero on approaching the critical temperature, T=T_{c}, irrespective of the pseudogap temperature^{4,5,7,10}, one may correlate the nodalgap suppression, Δ_{N}/Δ^{*}, with the decrease in ρ_{s}.
In this way, we found another simple relation, . Figure 4f–h compare the square of nodaltoantinodal gap ratio, (Δ_{N}/Δ^{*})^{2}, with the superfluid density, ρ_{s}, deduced from the superconductingpeak ratio in antinodal ARPES spectra^{6}, and from alternatingcurrentsusceptibility and heatcapacity data^{7,10}. All of them increase linearly in p with an onset at p~0.07, showing a saturation point at p~0.19 (refs 6, 7, 10). Such a doping dependence of ρ_{s} is known to be universal for the cuprates, and the critical doping of p~0.19 is not only evident from the superfluid data but has also been identified in the electricalresistivity and tunnelling data^{7,13,30}. The intimate relationship between the nodal and antinodal gaps is in accord with the continuity between the nodal and antinodal spectra (Fig. 3a–d).
Discussion
The squareroot dependence on ρ_{s} is usually characteristic of the order parameter, as expected from Ginzburg–Landau theory^{1}. More specifically, a general form of has been theoretically predicted for the superfluid reduction due to the incoherent pair excitations inherent in the strongcoupling superconductors, where the degeneracy between the order parameter Δ_{sc} and the pairformation energy Δ is split, and the former is manifested in the energy of the nearnodal spectral peak as Δ_{N}, whereas the latter dominates the antinodal peak energy Δ^{*}, as shown by taking into account the lifetime effect^{21,22}. Here the superfluid density in the weakcoupling Bardeen–Cooper–Schrieffer model, , can be regarded as an approximate constant, because the pdependences of the Fermisurface perimeter and the normalstate Fermi velocity are relatively small^{31}. Therefore, the strongcoupling scenario well explains our empirical relation, , with μm^{−2}, and is also consistent with the observations of the high gaptoT_{c} ratios (Fig. 4c) and the strong renormalization of dispersion upon the superconducting transition^{31}. Under this scenario for the nodalgap suppression, the phase fluctuation persisting down to a temperature below T_{c} is responsible for the decrease in superfluid density. However, the intrinsic phase fluctuation arising solely from the strong coupling diminishes at temperatures much lower than T_{c}^{21,22}. Hence, in conjunction with the strong coupling, some extra source generating the incoherent pair excitations at low temperatures is assumed to be present, on the basis of the phenomenology among Δ_{N}, Δ^{*} and ρ_{s}. The possible candidates for this scattering source are the orders competing with the superconductivity^{22}. In particular, it has indeed been observed by scanningtunnelling experiments that the nanoscale spatial domains of densitywavelike modulation spread over the underdoped Bi2212 (ref. 32). Perhaps, the limit of 2Δ_{N}/k_{B}T_{c}≤8.5 may imply that such competing orders are practically inevitable for the strongcoupling superconductivity.
Within the weakcoupling scenario, by contrast, there seem no schemes to relate the nodal and antinodal gap energies, to our knowledge. To reconcile 2Δ_{N}/k_{B}T_{c}=8.5 with the weakcoupling constant 4.3, one needs a pindependent reduction factor of ~0.5. The length of ‘Fermi arc’ in the normal state (Fig. 3i) seems irrelevant, because of its approximately linear p dependence, θ_{arc}p (see Supplementary Note 1 and Supplementary Fig. S2)^{15,33}, although its experimental uncertainty obscures the behaviour of arcendpoint gap, Δ_{arc}≡Δ_{N} sinθ_{arc} (Fig. 4c). Furthermore, once in the superconducting state, one cannot distinguish between the spectra at angles θ inside (bold labels) and outside (italic labels) the Fermi arc, as shown in Fig. 3e–h, nor define the boundary of the momentum region with a coherent peak for Bi2212, as pointed out by Vishik et al.^{26}
Combining the two relations, we obtain . This can be a gapbased formulation of the longstanding phasefluctuation paradigm that associate T_{c} with both ρ_{s} and the pairformation energy, (refs 4, 5). Notably, recent investigations have revealed that the rising exponent of T_{c} as a power of ρ_{s} is generally less than 1, and that specifically is satisfied near ρ_{s}=0 (refs 8, 9). This behaviour is compatible with , as far as the variation of Δ^{*} is negligible. Regardless of interpretation, the present phenomenological formulae put strong constraints on existing theories, and provide simple bases for future approaches to the highT_{c} superconductivity.
Methods
Samples
Highquality single crystals of Bi2212 were grown by travelingsolvent floatingzone method and subjected to a postannealing procedure for regulation of doping level. An overdoped sample of T_{c}=80 K (OD80), an optimally doped sample of T_{c}=91 K (OP91) and five underdoped samples of T_{c}=81, 77, 73, 66 and 62 K (UD81, UD77, UD73, UD66 and UD62, respectively) were prepared from the crystal whose nominal composition is Bi_{2.1}Sr_{1.8}CaCu_{2}O_{8+δ}. Two heavily underdoped samples of T_{c}=42 and 30 K (UD42 and UD30, respectively) were prepared from the Dysubstituted crystal of Bi_{2.2}Sr_{1.8}Ca_{0.8}Dy_{0.2}Cu_{2}O_{8+δ}. Two heavily overdoped samples of T_{c}=73 and 63 K (OD73 and OD63, respectively) were prepared from the Pbsubstituted crystal of Bi_{1.54}Pb_{0.6}Sr_{1.88}CaCu_{2}O_{8+δ}. Details of sample preparation are described elsewhere^{34}. Hole concentrations p have been deduced from the samples’ T_{c}, using a phenomenological relation, , from Presland et al.^{35} with K.
ARPES measurement
ARPES experiments were performed on a helical undulator beamline, BL9A of the Hiroshima Synchrotron Radiation Center, using a SCIENTA R4000 analyzer. Total instrumental energy resolution was set at 5 meV. Clean surfaces were obtained by cleaving the samples in situ, and all the ARPES spectra were collected under ultrahigh vacuum better than 5 × 10^{−11} Torr. Energies were calibrated with reference to the intermittently monitored Fermi edge of polycrystalline gold.
Fitting EDCs
The superconductinggap energies have been determined by fitting EDCs. For the spectral function A(ω) at a minimumgap locus, we adopted a widely used phenomenological form introduced by Norman et al.^{36},
where the peak position and width are given by the superconductinggap energy Δ and the singleparticle scattering rate Γ, respectively. In accordance with the experimental incoherent spectral weight increasing towards higher energies, we added a background linear in energy, a+bω, to A(ω), and then applied the multiplication by the Fermi–Dirac distribution function f_{T}(ω) as
As the experimental antinodal peaks are more asymmetric than the above model function I(ω), we additionally used the integraltype background for practical evaluation of the peak position, and applied convolution with the Gaussian representing instrumental resolution. As a result, the fit over a wide energy range of ω≥−100 meV has been achieved all along the Fermi surface for all the doping levels. The EDCs and their fits are shown in Supplementary Fig. S1, and those after symmetrization are presented in Fig. 3e–h. As shown by the small circles in Fig. 3a–d, the result of this procedure consistently tracks the spectral peak at the superconductinggap energy with precision.
Additional information
How to cite this article: Anzai, H. et al. Relation between the nodal and antinodal gap and critical temperature in superconducting Bi2212. Nat. Commun. 4:1815 doi: 10.1038/ncomms2805 (2013).
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Acknowledgements
We thank Z.X. Shen and A. Fujimori for their useful discussions, and T. Fujita and Y. Nakashima for their help with the experimental study. H.A. acknowledges financial support from JSPS as a research fellow. This work was supported by KAKENHI (20740199). The ARPES experiments were performed under the approval of HRSC (Proposal No. 09A11 and 10A24).
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H.A. and A.I. designed the experiment, analysed the data and wrote the manuscript with support from M.T. The ARPES data were acquired by H.A. with support from M.A. and H.N. The highquality single crystals were grown by M.I., K.F., S.I. and S.U. All authors discussed the results and commented on the manuscript.
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Supplementary Figures S1 and S2, Supplementary Note S1 and Supplementary References (PDF 493 kb)
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Anzai, H., Ino, A., Arita, M. et al. Relation between the nodal and antinodal gap and critical temperature in superconducting Bi2212. Nat Commun 4, 1815 (2013). https://doi.org/10.1038/ncomms2805
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DOI: https://doi.org/10.1038/ncomms2805
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