Abstract
A central issue in high-Tc superconductivity is the nature of the normal-state gap (pseudogap)1 in the underdoped regime and its relationship with superconductivity. Despite persistent efforts, theoretical ideas for the pseudogap evolve around fluctuating superconductivity2, competing order3,4,5,6,7,8 and spectral weight suppression due to many-body effects9. Recently, although some experiments in the superconducting state indicate a distinction between the superconducting gap and pseudogap 10,11,12,13,14, others in the normal state, either by extrapolation from high-temperature data15 or directly from La1.875Ba0.125CuO4 (LBCO-1/8) at low temperature16, suggest the ground-state pseudogap is a single gap of d-wave17 form. Here, we report angle-resolved photoemission data from LBCO-1/8, collected with improved experimental conditions, that reveal the ground-state pseudogap has a pronounced deviation from the simple d-wave form. It contains two distinct components: a d-wave component within an extended region around the node and the other abruptly enhanced close to the antinode, pointing to a dual nature of the pseudogap in this failed high-Tc superconductor that involves a possible precursor-pairing energy scale around the node and another of different but unknown origin near the antinode.
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The first high-Tc superconductor discovered, La2−xBaxCuO4 (LBCO), holds a unique position in the field because of an anomalously strong bulk Tc suppression near x=1/8. Right around this ‘magic’ doping level, scattering experiments by neutrons18,19 and X-rays20 find a static spin and charge (stripe) order. By itself, this observation raises a series of intriguing questions: whether the stripe order is a competing order that suppresses the superconductivity in LBCO-1/8; if the answer is positive, which aspect, the pairing strength or the phase coherence, is involved in the Tc suppression and how this mechanism applies to other dopings or families. For our investigation of the ‘ground-state’ pseudogap, as defined in ref. 16, which ignores the residual superconductivity, its sufficiently high doping yet extremely low bulk Tc (∼4 K) makes LBCO-1/8 an ideal system: especially for the small-gap measurement near the node, difficulties due to either the unscreened disorder potential, a problem for extremely low doping, or trivial thermal broadening, a problem above Tc for higher doping, are circumvented.
Whereas thermal effects require an extrapolation from high-temperature data to obtain the ground-state physics15, a direct measurement on LBCO-1/8 at low temperature has been made16. With experimental resolutions compromised to obtain sufficient signal to noise, a simple d-wave gap function is reported with no discernible nodal quasi-particles found. Given the importance of this issue, we have carried out an angle-resolved photoemission21 study of LBCO-1/8 at T>Tc with much improved resolutions in a measurement geometry favourable for the detection of nodal quasi-particles (see Fig. 1 and Supplementary Information, Section SI). Our data provide two important new insights. First, there is a well-defined nodal quasi-particle peak, suggesting nodal quasi-particles can exist in the stripe-ordered state. Second, there is a rich gap structure, suggesting the pseudogap physics is more elaborate than the simple d-wave version. In particular, a new kind of pseudogap, which is not smoothly connected to the usual one tied to the antinodal region, can exist in the nodal region when superconductivity is suppressed owing to the loss of phase coherence.
As shown in Fig. 1, there exists a well-defined quasi-particle peak in the energy distribution curves (EDCs) at the Fermi crossing points (kF) around the node. On dispersion towards the antinode, the line shape quickly becomes incoherent. Data taken with different photon energies (h ν) in different Brillouin zones show consistent results (see Supplementary Information, Fig. S1), providing the first unambiguous piece of direct evidence that nodal quasi-particles survive in the stripe-ordered state. This suggests that these two seemingly very different aspects of cuprate phenomenology can be compatible with each other22.
The observation of quasi-particle peaks in the EDCs gives a firm foundation for the gap analysis around the node. However, given the small gap size as well as a relatively small peak–background intensity ratio and large quasi-particle peak linewidth (Γ) (Fig. 1d), quantitative determination of the gap is non-trivial in LBCO-1/8. In the following, we will present an analysis that addresses two important aspects of our data: (1) the gap measured by the leading edge gap (LEG) method, the same as that used in ref. 16, is shown to have two components and cannot fit a simple d-wave form (Fig. 2); (2) the gap around the node, measured either by the LEG method (Fig. 2) or by a curve fitting procedure commonly used in the field (Fig. 3), is shown to be d-wave-like with a finite gap slope.
In Fig. 2a, from the systematic shift of the leading-edge midpoint (LEM) along the energy axis, we notice that the LEG keeps increasing from the node towards the antinode with the rate of increase getting larger close to the antinode. This is more clearly shown in Fig. 2b, where the momentum dependence of the extracted LEG from two selected samples is plotted. Consistent results are obtained at different times after sample cleaving, showing no signs of sample ageing, which could affect small-gap measurements, and in different Brillouin zones with different h ν on another sample. They unanimously point to a pronounced deviation of the gap function from the simple d-wave form. As detailed in Supplementary Information, Section SIIIB,C,SIV, this observation goes beyond the interpretation based on either a simple d-wave gap under the influence of experimental resolutions (see Supplementary Information, Fig. S10) and the line-shape decoherence towards the antinode (see Supplementary Information, Fig. S11) or a single pairing gap with the inclusion of higher d-wave gap harmonics23 (see Supplementary Information, Fig. S7). It naturally reveals a striking characteristic: this normal-state gap has two distinct components, with the one around the node (the nodal gap) exhibiting a simple d-wave form, that is, a linear dependence with respect to [cos(kx)−cos(ky)]/2, along the underlying Fermi surface over a significant momentum range and the other setting in near the antinode (the antinodal gap), which deviates sharply from its nodal counterpart.
Although it is clear that the LEG opens in a d-wave fashion around the node (Fig. 2a, inset), it is too crude to conclude that the real gap (Δ) function of the system is also simple d-wave-like because the leading edge shift in principle can be due to the variation in Γ even if Δ is fixed24. This alternative has to be explored especially for LBCO-1/8, where Γ≫Δ near the node. Thus, we fit the EF-symmetrized EDCs at kF to a phenomenological model25 that assumes a self-energy, Σ(kF,ω)=−i Γ+(Δ2/ω) (Fit Model), where Γ and Δ are subject to the fit (Fig. 3a). Because of the small peak–background ratio, the fitting results are affected by the background subtraction. In Fig. 3b, we show the results of Δ without any background subtraction, or with a momentum distribution curve (MDC) constant background or an EDC integral background subtraction (see Supplementary Information, Fig. S5). Two general trends of the results are: (1) regardless of the methods for background subtraction, an initial gap slope is robustly defined close to the node where quasi-particle peaks are clearly present in e1–e3; (2) as the peak feature weakens, a large background dependence appears starting from e4. For e4, with more background subtracted from the high-binding-energy side of the peak feature, Δ decreases and tends to fall onto the initial gap slope. For the completeness of our analysis, we have also shown in Supplementary Information, Fig. S6 fitting results based on another model that fits the data less well. Although the quantitative gap values are model dependent, its d-wave form remains robust (see Supplementary Information, Section SIID).
Summarizing the above, both the model-independent (the LEG method) and model-dependent (the curve fitting procedure) gap analysis suggest that the nodal gap opens in a d-wave fashion with a finite slope. Despite its presence in the normal state, it is highly suggestive that this nodal gap is of a pairing origin based on the following reasons. (1) Its d-wave form is consistent with the generic pairing symmetry of cuprate superconductors, particularly in La-based cuprates, where a similar gap in the nodal region is found to be related to superconductivity12. (2) Its gap slope strikingly coincides within error bars with those of La2−xSrxCuO4 (LSCO) x=0.110 (ref. 26) and LBCO x=0.083 (ref. 27) (Tc=26 K and 23 K, respectively, see Supplementary Information, Fig. S8 for raw data), which are both in the superconducting state (Fig. 4a). Compared with the one analysed using Fit Model in Bi2212 of a similar doping level well below Tc (Fig. 3c in ref. 11), it exhibits a factor of ∼2 reduction, reminiscent of the optimal Tc difference between these two families. (3) Its susceptibility to thermal smearing in contrast to its antinodal counterpart (Fig. 4b) is reminiscent of the cases in other superconducting cuprates11,12. The existence of precursor pairing in the normal state of LBCO-1/8 is further supported by recent transport measurements28. A precipitous drop in the in-plane resistivity at T2D∼40 K, where the concurrent stripe (as a density wave) formation would often result in an increase of resistivity, implies an onset of superconducting fluctuations. Although the coincidence of its partial closing with the simultaneous resistivity increase at T=T2D on heating (Fig. 4b) suggests the precursor-pairing origin of the nodal gap rather than the conventional density-wave type, understanding the relationship between these two coexisting orders in the system at T<T2D still poses a challenge.
On the other hand, the pairing strength, as reflected by the slope of the nodal pairing gap, is comparable in LBCO-1/8 and its neighbouring compounds of much higher bulk Tc values (Fig. 4a). Hence, a natural explanation for the bulk Tc suppression in LBCO-1/8 is its lack of a global superconducting phase coherence. Interestingly, this loss of superconducting phase coherence coincides with the stabilization of the stripe order at x∼1/8. Intrigued by this, theorists have proposed that the global superconducting phase coherence can be prohibited through the dynamical interlayer decoupling in the system where superconductivity is modulated by the stripe order of some specific configurations29,30. Nevertheless, the modulated superconductivity of a non-zero wave vector generally does not produce a d-wave gap with a point node as observed here. Our finding would put a strong constraint on future theoretical attempts to resolve the microscopic mechanism for the failure of high-Tc superconductivity in LBCO-1/8.
As suggested by its different momentum dependence from its nodal counterpart, the antinodal pseudogap may have a different origin, which has been the subject for intense discussions in the literature2,3,4,5,6,7,8,9. Generally, as inferred from Supplementary Information, Fig. S11, a similar antinodal phenomenology as we observed can be achieved either by (1) a true gap opening together with a strong quasi-particle scattering (the Δ–Γ physics) or by (2) a great suppression of the quasi-particle spectral weight (the Z physics). Whereas attempts of scheme (1) working in the weak coupling approach can produce a qualitatively similar quasi-particle gap structure by considering the competition between superconductivity and the charge6,7- or/and spin8,29-density-wave order, scheme (2) demands a strong-coupling route (for example, ref. 9) where the extended quasi-particle analysis as presented above might break down owing to its incapability of capturing the lowest-lying excitations of a vanishing spectral weight. In any case, new physics other than the nodal-type precursor pairing alone is required to fully capture the essence of the antinodal pseudogap.
In contrast to the notion of a simple d-wave nodal liquid as the pseudogap ground state that is directly derived from the antinodal pseudogap15,16, our observation of an apparent break-up in the gap function suggests a very different picture. It reveals a much richer pseudogap physics with its two aspects manifesting differently in distinct momentum regions, that is, the nodal precursor pairing and the antinodal pseudogap of different but unknown origin. These two aspects might be emphasized differently by different experimental probes in the normal state, which has led to the two extremes of ideas for the pseudogap, in particular, whether it has a direct relationship with pairing. Our results suggest a plausible reconciliation between them.
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Acknowledgements
The authors would like to acknowledge helpful discussions with H. Yao, W.-S. Lee, E. Berg, T. P. Devereaux, S. A. Kivelson, X. J. Zhou and T. Xiang and revision of the manuscript by R. G. Moore. R.-H.H. thanks G. Yu for kind assistance in the sample preparation and the SGF for financial support. The work at the ALS is supported by the DOE Office of Basic Energy Science under Contract No. DE-AC02-05CH11231. This work at Stanford is supported by the DOE Office of Science, Division of Materials Science and Engineering, under Contract No. DE-FG03-01ER45929-A001.
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T.S., M.F. and K.Y., T.A. and Y.K. provided and prepared the LSCO x=0.110, LBCO-1/8 and LBCO x=0.083 samples, respectively. S.-K.M., K.T. and N.M. maintained the experimental endstation and ensured its high performance. R.-H.H. and K.T. carried out the experiment with the assistance of S.-K.M., N.M. and T.S. R.-H.H. did and K.T. repeated the data analysis and simulations. R.-H.H. wrote the paper with helpful suggestions and comments by S.-K.M. Z.H. and Z.-X.S. are responsible for project direction, planning and infrastructure.
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He, RH., Tanaka, K., Mo, SK. et al. Energy gaps in the failed high-Tc superconductor La1.875Ba0.125CuO4. Nature Phys 5, 119–123 (2009). https://doi.org/10.1038/nphys1159
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DOI: https://doi.org/10.1038/nphys1159
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