## Abstract

Identifying the mechanism of superconductivity in the high-temperature cuprate superconductors is one of the major outstanding problems in physics. We report local measurements of the onset of superconducting pairing in the high–transition temperature (*T*_{c}) superconductor Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} using a lattice-tracking spectroscopy technique with a scanning tunneling microscope. We can determine the temperature dependence of the pairing energy gaps, the electronic excitations in the absence of pairing, and the effect of the local coupling of electrons to bosonic excitations. Our measurements reveal that the strength of pairing is determined by the unusual electronic excitations of the normal state, suggesting that strong electron-electron interactions rather than low-energy (<0.1 volts) electron-boson interactions are responsible for superconductivity in the cuprates.

Central to the current debate on the mechanism underlying high-temperature superconductivity is the question of whether electron pairing in cuprates is caused by the exchange of bosonic excitations and can therefore be described by an extension of the Bardeen-Cooper-Schrieffer (BCS) theory, which has successfully explained phonon-mediated superconductivity in metals and alloys for the past 50 years (*1*, *2*). Alternatively, it has been argued that the large Coulomb interaction in doped Mott insulators can result in a fundamentally different mechanism for pairing that cannot be approximated by a retarded boson-mediated interaction between electrons (*3*–*5*). In copper oxides, candidates for a BCS-like bosonic glue for pair binding include a magnetic resonance mode (near 40 meV for hole-doped cuprates) (*6*–*8*), the spectrum of high-energy spin excitations (above 40 meV) (*9*–*12*), fluctuations around a quantum critical point (up to several hundred millielectron volts) (*13*), and phonons (e.g., those near 40 meV) (*14*, *15*). Various spectroscopic measurements have probed these bosonic excitations and their coupling to electronic states in the cuprates (*14*–*20*); however, the connection between these excitations and the pairing mechanism has remained elusive (*5*, *21*).

Particularly challenging is the fact that the pairing strength and the temperatures over which pairs form in cuprate samples such as Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} are spatially inhomogeneous (*22*–*24*), a behavior that complicates the interpretation of macroscopically averaged experiments. Our technique allows for quantitative characterization of the electron-boson coupling and its correlation with the inhomogeneous pairing on the nanometer scale in the high-*T*_{c} superconductor Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}. Our key finding is that although the coupling between electrons and bosons in the energy range of 20 to 120 meV influences the energy dependence of the pairing interaction, it does not control its local strength. We show instead that the local pairing strength is determined by asymmetric electron-hole excitations that are seen in the normal-state properties well above the temperature where pairs first form.

**Lattice-tracking tunneling spectroscopy of inhomogeneous superconductors.** Electron tunneling spectroscopy of superconductors is a powerful method for quantitative measurements of the onset of electron pairing and the boson exchange mechanism in superconductors (*25*). The pioneering measurements of McMillan and Rowell (*26*), and their analysis (*27*) based on the extension of the BCS theory by Eliashberg (*28*), provided unequivocal evidence for a phonon-mediated mechanism of superconductivity in metals and alloys. In their study, the dependence of the tunnel conductance on the voltage was used to extract the phonon density of states and the strength of the electron-phonon coupling. These two quantities determine the transition temperature as well as the energy dependence of the pairing gap.

The success of tunneling as a quantitative spectroscopic probe in conventional superconductors relied on performing measurements in both the superconducting and normal state (*29*) in order to exclude the complication due to tunneling matrix elements as well as inelastic tunneling processes. For an inhomogeneous superconductor such as Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}, in which electronic states and the superconducting energy gap vary on the nanometer scale, similar experiments pose a technical challenge. It is necessary to track specific atomic locations on the lattice of the sample from low temperatures to temperatures above *T*_{c} with a scanning tunneling microscope (STM) (*24*, *30*). To enable such measurements, we have developed a thermally compensated ultrahigh vacuum STM system that can maintain thermal stability to better than 10 mK during spectroscopy experiments up to temperatures of 110 K.

The lattice-tracking spectroscopy technique has been used to measure the evolution of tunneling conductance *dI*/*dV*(*r,V,T*) with temperature in samples of Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} that are overdoped (*T*_{c} = 68 K; Fig. 1, A and B) and optimally doped (*T*_{c} = 93 K; Fig. 1, D and E) at specific atomic sites. Spectra at different atomic sites show different energy gaps at low temperatures and evolve differently with increasing temperature (*24*). We find that all low-temperature spectra in highly overdoped samples (Fig. 1, A and B) evolve into spectra at high temperatures that are relatively featureless at low energies (<100 meV). In contrast, for optimally doped samples, we find that the spectra in most regions continue to have energy and temperature dependence at temperatures well above *T*_{c} (Fig. 1, D and E).

When the tunneling conductance at high temperatures is weakly dependent on temperature and energy, following previous work on conventional superconductors (*29*, *31*), we probe the effects of superconductivity by examining the ratio *R*(*r,V,T*) between the tunneling conductance in the superconducting and normal states measured under the same STM setup conditions (1) Here, *N*_{S}(*r,V,T*) and *N*_{N}(*r,V*) are the respective superconducting and normal density of electronic states at atomic site *r* as a function of energy (*eV*). This ratio, which is independent of the tunneling matrix element, will be used to extract the temperature dependence of the energy gap and features associated with strong coupling of electrons to bosonic modes. Although our experimental technique can be extended to study samples at any doping, we focus on the quantitative analysis of the local temperature dependence of electronic states on overdoped samples. Photoemission (*17*) and Raman spectroscopy (*32*) studies on Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} samples have shown the absence of pseudogap phenomena at similar doping levels, simplifying the analysis of the normalized spectra.

**Temperature dependence of the local pairing gap and quasiparticle lifetime.** The temperature evolution of the conductance ratio *R*(*r,V,T*), in overdoped Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} samples at two representative locations of the sample with different low-temperature energy gaps, is shown in Fig. 2, A and B. Motivated by the fact that the low-temperature ratio resembles that expected from a single energy gap in the spectrum, we compare these ratios with that expected from the thermally broadened density of states of a *d*-wave superconductor (2) Here, *i* is the square root of –1, Δ *r,T* is the local *d*-wave gap amplitude (considered energy-independent for low energies), Γ(*r,T*) corresponds to the local inverse lifetime of the quasiparticle excitations (*33*), and *f* (*E,T*) is the Fermi function. Such an analysis neglects the complication resulting from the momentum dependence of the band structure as well as higher-order angular terms in the superconducting gap.

Given that the experimental spectra in our sample display a gap above *T*_{c}, one can question the appropriateness of using a single gap parameter to describe the normalized spectrum (Eq. 2). In general, the value of the energy gap can be determined by two different methods: from the slope of the normalized spectra near zero energy and from the energy at which the conductance spectra show a peak. If there were two different gaps—for example, one dominating the nodal (near zero energy) and one dominating the antinodal (energy of the conductance peak) regions [as might be the case in underdoped Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} samples (*17*, *24*, *32*)]—the two procedures would yield two different gap values. We find that the gap values obtained from the two methods agree at all locations on the sample, which further justifies the use of a single gap (Eq. 2) to describe the spectra in our overdoped samples.

In general, we find that the model in Eq. 2 provides an excellent fit to the experimental data at low energies for all points on the overdoped samples (Fig. 2, A and B). Using this model, we can extract the local values of Δ(*r,T*) and Γ(*r,T*), showing that at each point the gaps decrease monotonically with increasing temperature and close at a local temperature *T*_{p}(*r*) > *T*_{c} (Fig. 2C). We find that Γ(*r*,*T*) is much smaller than the gaps at all locations at low temperatures (Fig. 2D).

With increasing temperature, we find that the smaller gaps close first, with the largest gaps surviving to temperatures well above *T*_{c}. Regions of the sample with smaller gaps also show *R*(*r,V,T*), which exceeds that predicted from the local *d*-wave model in Eq. 2 for *eV*∼Δ likely (*r,T*). This behavior is likely due to localization effects experienced by the quasiparticles in small gap regions that cannot penetrate the larger gap regions (*34*). The extracted values of Γ(*r,T*) also agree with this scenario: The regions with the smallest gaps show no lifetime broadening, whereas the larger gap regions have a small lifetime broadening at low temperatures (Fig. 2D). The variation inΓ(*r,T*) is a consequence of the fact that the excitations in the large gap regions can decay into nearby regions with smaller energy gaps, but not vice versa.

As the smaller gaps begin to close with increasing temperature, we find that the Γ(*r,T*) in the large gap regions begins to increase rapidly. Overall, the analysis of the experimental data for all regions on the sample demonstrates that the spatially averaged Γ(*T*) shows a dramatic increase at a temperature *T*∼*T*_{c}, when the sample loses long-range phase coherence (Fig. 2D, inset). This observation is in accordance with previous macroscopically averaged measurements (*35*) on samples in the overdoped regime; however, previous measurements did not correlate inhomogeneous behavior of the gaps and quasiparticle lifetimes.

**Coupling of electrons to bosonic modes.** Having established that our measurements can be examined within the context of a local *d*-wave gap model, we turn to examining the deviation of the conductance ratio from this model for *E* > Δ. Although other effects such as inelastic tunneling (*29*, *36*) can cause such deviations, only strong coupling of electrons to bosonic modes is known to cause the superconducting-state tunneling conductance to dip strongly below the normal state (*27*, *29*). As illustrated in Fig. 3A, all points on the samples show a voltage range (around 50 to 80 meV) in which the conductance ratio *R*(*r,V,T*) is reduced below 1 and show systematic deviations from the local *d*-wave model. Analogous to previous work on conventional superconductors (*2*), these deviations provide a quantitative method to determine the strength of electron-boson coupling. The analysis of the spectra based on the features of *R*(*r,V,T*) instead of the bare *dI/dV*(*r,V,T*) or *d*^{2}*I/dV*^{2}(*r,V,T*) avoids complications due to the spatial variation of normal-state features and tunneling matrix element variations (*21*). Although many previous studies, including those that used an STM (*15*, *37*), have examined electron-boson features in the 20- to 120-meV range, a quantitative comparison of the electron-boson coupling at different locations of the sample with different pairing gaps has not been accomplished. The comparison of *R*(*r,V,T*) at different locations allows us to quantitatively evaluate the role of bosonic features in the development of the pairing gaps and their inhomogeneity.

To study the relative strength of electron-boson coupling at different locations on the sample, we consider that the strong coupling to a bosonic mode at energy Ω in a superconductor results in features in the conductance ratio at *eV* =Δ+Ω (*15*, *29*, *38*). As the pairing gap is locally varying, we plot the *R*(*r,V,T*) as a function of *eV* –Δ(*r*) for different atomic sites on the sample with low-temperature Δ (*r*) ranging between 15 and 32 meV in Fig. 3B. This figure demonstrates that different locations on the sample show similar *R*(*r,V,T*) curves in magnitude and shape, once we take into account their varying pairing gaps. The only significant difference between the spectra occurs at low energies, where lifetime broadening effects, as well as the angle dependence of the electron-boson coupling, play a role. A measure of the energy Ω of the bosonic mode is the energy at which the dip occurs in the spectrum. We find that the average energy of this dip is 35 ± 3 mV. A quantitative measure of the strength of the local coupling constant is the root mean square deviation (RMSD) of *R*(*r,V,T*) from the weak-coupling *d*-wave model (Eq. 2) in the energy range 20 to 120 meV beyond the gap. These deviations show no correlation [for both positive and negative biases (Fig. 3C)] with the size of the local gap within our experimental error.

For boson-mediated pairing, variation of the pairing gap can be caused by changes in either the local boson energy or the local coupling between the boson and electrons (*2*). Such changes are reflected directly in the size and energy range of the strong-coupling features in the conductance ratio. Indeed, in metallic alloy systems (*39*, *40*) where pairing is controlled by strong electron-phonon coupling, the magnitude of strong-coupling features in the conductance ratio scales with the gap size [see supporting online material (SOM) text S1]. Because both the energy scale of the boson modes and the local electron-boson coupling do not correlate strongly with the magnitude of the local pairing gap in our samples, we conclude that the coupling to bosons in the range of 20 to 120 meV cannot be responsible for these inhomogeneous pairing gaps.

Although bosons may not be critical to pairing in Bi_{2}Sr_{2}CaCu_{2}O_{8+δ}, the *R*(*r,V,T*) curves clearly show that these boson modes give a strong energy dependence to the gap function. Specifically, modification of Eq. 2 with a complex energy-dependent pairing gap Δ(*r*,ω) = Δ_{R}(*r*,ω)+*i*Δ_{I} (*r*,ω), where Δ_{R} and Δ_{I} are the real and imaginary part of the gap function at energy ω, can be used to capture the bosonic features in the conductance ratio at higher energies. Within such a model, we estimate that the interaction with bosonic excitations (in the range of 20 to 120 meV) results in a substantial imaginary component of the pairing interaction (about 25 meV in magnitude at Ε = 40 meV).

**Spatial structure of normal-state excitations and inhomogeneous pairing interaction.** In search of the origin of the inhomogeneity in the pairing interaction in the cuprates, we focus on spectroscopic measurements of the electronic excitations in the normal state and their correlation with the inhomogeneity in the superconducting gaps. To reach the normal state, the temperature has to be high enough such that all the local pairing gaps have collapsed. For overdoped Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} samples (hole-doping *x* = 0.24, *T*_{c} = 62 K), less than 1% of the sample shows a gap at 90 K. In the intermediate temperature between *T*_{c} and 90 K, these samples show a mixture of ungapped and partially gapped spectra, as previously reported (*24*). Above 90 K, the tunneling spectra (Fig. 4A) are gapless at all locations on the sample but show asymmetric behavior for electron and hole tunneling. Careful examination of these spectra, over a wide range of energies, shows that electronic excitations in the sample are still spatially inhomogeneous at temperatures well above that when pairs first form in the sample. The spatial inhomogeneity of the normal state's electronic excitations can be measured by means of conductance (*dI*/*dV*) maps at various voltages, which show variations on the length scale of order 20 Å (Fig. 4, B and D to I). The magnitude of the variations is strongest for the conductance map obtained at the Fermi level (Fig. 4B), but such variations persist up to a few hundred millielectron volts.

The spatial variations of the normal-state conductance can be compared with the low-temperature variations of the gap by means of our lattice-tracking technique. Shown in Fig. 4C is a gap map measured at 50 K over the exact same area of the sample as in Fig. 4B (*41*). We can see a marked similarity between this gap map and the conductance map at the Fermi energy (Fig. 4B): Regions with a lower normal-state conductance at the Fermi level nucleate superconducting gaps at higher temperatures, resulting in larger low-temperature gaps. Quantifying these correlations in Fig. 5A, we show that the normal-state conductance map and the low-temperature gap map are strongly anticorrelated (–0.75). Further, both these maps have very similar autocorrelation lengths, indicating that the spatial variation of the normal state is intimately linked to that of the low-temperature gap. These measurements show that the variation of the superconducting state in Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} samples, which for typical superconductors is characterized by a temperature-dependent superconducting coherence length, appears to be primarily determined by the spatial variation of the normal state.

Because our measurements show that the spatial variation of the normal-state conductance and the low-temperature pairing gap maps are intimately connected, we can associate an average normal-state spectrum with a low-temperature gap value. We thus average together normal-state spectra of regions of the sample that show identical low-temperature gaps and plot these average spectra in Fig. 5B. Systematic differences in the normal-state spectra foreshadow the eventual variation of the gap in the superconducting state. In particular, the systematic shift of a “hump” in the normal-state tunneling spectra at negative bias, in the range of –150 to –300 meV (Fig. 5C), as well as the value of tunneling conductance at the Fermi energy (Fig. 5B, inset), tracks the size of the superconducting gap observed at low temperatures. Whereas both the tunneling matrix element and the density of states of the tip influence the shape of the tunneling conductance in the normal state, the features of the normal state and the correlation (Fig. 5) have been observed in measurements with several different microtips (*42*).

It is important to compare our measurements of the normal state with those obtained from other spectroscopic techniques. In both angle-resolved photoemission and optical spectroscopy, strong renormalization of the single-particle excitations has been observed over an energy range of ∼200 to 400 meV below the Fermi energy in Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} samples (*17*–*20*, *38*, *43*–*45*). Such effects have been interpreted either as a result of the coupling of electrons with a spectrum of bosonic excitations (such as spin fluctuations) or as a consequence of the energy band structure (such as the bilayer splitting) of this compound (*19*, *46*). Although strong electron-boson coupling can be expected to modify the shape of the normal-state spectrum, it is difficult to associate the normal-state features that we measure with coupling to bosonic excitations because of their strong electron-hole asymmetry. Although we cannot rule out bosonic excitations as the origin of these features, candidate bosons would have to couple very asymmetrically to the tunneling of electrons and holes. Assigning these features to effects calculated from a simple non-interacting band structure can also be questioned, given both the strong spatial variation at the atomic scale of the normal-state spectra reported here and the strong renormalization of single-particle states at similar energies in other spectroscopic studies of the normal state. Instead, these features might be the excitations of a doped Mott insulator where the electron and hole excitations are naturally asymmetric as a consequence of the strong Coulomb interaction (*47*–*49*). Some recent calculations indeed produce a hump in hole-like excitations that correlate with the strength of pairing (*47*, *49*). Although there is no clear consensus on the right model for these excitations, our experiments show that the spectroscopic features of this state are indeed the origin of the nanoscale variation of the pairing strength in the superconducting state. Further experiments in samples at different hole-doping levels at higher temperatures will be required to provide the detailed evolution of these atomic-scale spectroscopic features of the normal state across the phase diagram.

**Concluding remarks.** From a broader perspective, we have used the spatial variation of the pairing gaps, which gives rise to a range of pairing temperatures in nanoscale regions of our samples, as a diagnostic tool to find clues to the underlying mechanism of superconductivity. Temperature-dependent lattice-tracking spectroscopy has allowed us to demonstrate that electron-boson coupling in the 20- to 120-meV range does not cause the variation of pairing gaps and onset temperatures in our samples. In contrast, we find that the high-energy (up to ∼ 400 mV) hole-like excitations of the normal state are a direct predictor of the strength of pairing and its spatial variation. The anticorrelation between the normal-state conductance at the Fermi level and local strength of pairing also runs contrary to a BCS-like pairing mechanism, where the coupling to bosons is proportional to the density of states at the Fermi energy (*2*).

Finally, we address the underlying cause of variations of the normal-state excitations in Bi_{2}Sr_{2}CaCu_{2}O_{8+δ} samples. Our analysis finds that both structural and electronic features of the samples contribute to such variations. We find that there are small correlations (about 10%) of the normal-state conductance maps with the structural supermodulation along the *b* axis in these samples. Similarly, we find that maps of electronic resonances around –900 meV, previously probed in similar samples with STM (*50*), are correlated with the normal-state conductance maps (about 30%) (SOM). Our measurements show that structural and chemical inhomogeneity affects both the excitations of the normal state and the superconducting gap. As is common to several correlated systems, many structural and electronic features can influence the onset and strength of collective phenomena (*51*). Our ability to correlate nanoscale excitation spectra between two distinct electronic states at the same atomic site provides the capacity to study correlated phenomena in compounds with heterogeneous chemical and structural properties.

**Supporting Online Material**

www.sciencemag.org/cgi/content/full/320/5873/196/DC1

SOM Text S1 and S2

Figs. S1 and S2

References