Formation of electron pairs is essential to superconductivity. For conventional superconductors, tunnelling spectroscopy has established that pairing is mediated by bosonic modes (phonons); a peak in the second derivative of tunnel current d2I/dV2 corresponds to each phonon mode1,2,3. For high-transition-temperature (high-Tc) superconductivity, however, no boson mediating electron pairing has been identified. One explanation could be that electron pair formation4 and related electron–boson interactions are heterogeneous at the atomic scale and therefore challenging to characterize. However, with the latest advances in d2I/dV2 spectroscopy using scanning tunnelling microscopy, it has become possible to study bosonic modes directly at the atomic scale5. Here we report d2I/dV2 imaging6,7,8 studies of the high-Tc superconductor Bi2Sr2CaCu2O8+δ. We find intense disorder of electron–boson interaction energies at the nanometre scale, along with the expected modulations in d2I/dV2 (refs 9, 10). Changing the density of holes has minimal effects on both the average mode energies and the modulations, indicating that the bosonic modes are unrelated to electronic or magnetic structure. Instead, the modes appear to be local lattice vibrations, as substitution of 18O for 16O throughout the material reduces the average mode energy by approximately 6 per cent—the expected effect of this isotope substitution on lattice vibration frequencies5. Significantly, the mode energies are always spatially anticorrelated with the superconducting pairing-gap energies, suggesting an interplay between these lattice vibration modes and the superconductivity.
Strong-coupling superconductivity theory1,2,3,11, allowed the impact of electron–phonon interactions on the superconducting density of states, DOS(E), at energy E = Δ + Ω to be predicted (Δ is the superconducting energy gap and Ω the phonon energy). McMillan and Rowell used d2I/dV2 measurements on planar normal-insulator-superconductor tunnel junctions1 to reveal these signatures at boson energies Ω = E - Δ, identifying them with independently determined phonons of energy Ω. The electron–phonon spectral function measured from these d2I/dV2 spectra yielded the correct superconducting transition temperature—a milestone of twentieth-century physics.
By contrast, no consensus exists on the electron pairing mechanism of high-Tc superconductivity. One reason is that the extensive studies of bosonic modes12,13,14,15,16,17 and related electronic self-energy changes18,19,20,21,22,23,24,25,26 have not resulted in the unambiguous identification of a boson mediating the pairing. Effects of electron–boson interactions (EBI) on electronic self-energies are most widely studied18,19,20,21,22,23,24,25,26 via angle resolved photoemission. In the Δ = 0 or nodal direction of momentum space k∥(π/a0,π/a0), where a0 is the unit cell dimension, sudden changes or ‘kinks’ in quasiparticle dispersion E(k) occur between 50 meV and 80 meV below the Fermi energy18,19,20,21. Several studies have addressed the issue of whether these ‘kinks’ might be due to magnetic interactions19,20. On the other hand, it has been proposed that, because of the doping independence of their energies, the ‘kinks’ are due to electron–phonon interactions21. In the antinodal directions k ≈ (π/a0,0) where Δ is maximal, self energy changes22,23,24,25 also occur between 50 meV and 90 meV below the Fermi energy. But here the effects are thought to occur at E = Δ + Ω. These modes have been discussed in terms of both magnetic interactions22,23,24 and electron–phonon25 interactions. Effects of 16O/18O isotope substitution have been found primarily on states near k ≈ (π/a0,0), providing evidence for electron interactions with lattice modes26 (to be discussed in more detail below). Another EBI study technique is superconductor–insulator–superconductor (SIS) conductance measurement in break junctions27. Despite the challenge of interpreting SIS spectra in terms of the absolute value of Ω (they are not a direct measure of the superconducting DOS28), clear EBI features are detected; they are analysed in terms of magnetic modes. A final probe of copper oxide EBI is optical spectroscopy29,30; it reveals self-energy changes that have been ascribed to magnetic interactions via a sharp mode29 or a broad continuum30. Evidently, a definite conclusion for the identity of any pairing related EBI has proven elusive.
One reason for this situation might be that some key element of the EBI phenomenology has, so far, gone undetected. For example, if electron pairing4 and any related EBI were disordered at the atomic scale, the techniques described above could not yield a complete description of the relevant nanoscale EBI phenomena because they average over space. For such reasons, atomic-resolution d2I/dV2 spectroscopy of copper oxides has recently become the focus of considerable theoretical interest6,7,8,9,10. Three of the proposed applications are: (1) if the position (r) dependence of energy-gap disorder Δ(r) were due to atomic-scale pair-potential disorder4, pairing-related EBI could be examined directly at that scale; (2) studies of specific local bosonic modes (unrelated to pairing) at defects or impurity/dopant atoms6,7,8 could improve our understanding of EBI in copper oxides; and (3) Fourier transform d2I/dV2 imaging (dubbed FT-IETS)10 may help distinguish between different bosons involved in EBI. However, until now, severe technical constraints (see Supplementary Information) have prevented implementation of d2I/dV2 imaging of the necessary spatial/energetic precision.
Here we report atomic-resolution d2I/dV2-imaging studies of a high-Tc superconductor Bi2Sr2CaCu2O8+δ (Bi-2212). We use floating-zone-grown single crystals cleaved between the BiO planes in cryogenic ultrahigh vacuum and immediately inserted into the scanning tunnelling microscope head at 4.2 K. Figure 1a shows a topographic image of a typical BiO surface, while the inset shows the typical local density of states (LDOS) measured via a differential conductance dI/dV(E = eV) spectrum. Here we focus on the nanoscale spatial/energetic properties of the ubiquitous features at |E| > Δ in the LDOS (for example, see arrows in Fig. 1a inset).
Figure 1b gives typical examples of dI/dV(E) spectra measured at different locations of Fig. 1a. The vast majority of these spectra exhibit peaks in d2I/dV2(E) occurring at the point of maximum slope in dI/dV for |E| > Δ (arrows in Fig. 1b). Examples of the directly measured magnitude of the features in d2I/dV2 are shown in Fig. 1b, where the horizontal axis has first been converted to ω = E - Δ (where Δ is the distinct local gap magnitude for each spectrum). We provisionally consider these features as possible strong-coupling superconductivity1,2,3 signatures of EBI (see Supplementary Information).
The dI/dV(r,E) and d2I/dV2(r,E) are simultaneously imaged with atomic resolution and register. From the former, the gap-map Δ(r) is derived (Fig. 1c). From the latter, the energies Π(r)at which peaks in d2I/dV2(r,E) occur are measured. Within the context of strong-coupling superconductivity theory1,2,3,9, the local boson interaction energies would then be given by Ω(r) = Π(r) - Δ(r). Some systematic uncertainties may exist in the precision with which this process yields the absolute mode energy (especially for complex band structures). Nevertheless, the mean mode energy is estimated from the Ω(r) to be with a statistical spread of ± 8 meV. This is within the range of mode energies for antinodal EBI reported in photoemission studies22,23,24,25. However, it is from analysis of the Ω(r) images (Fig. 1d, Fig. 3d–f) that a very different perspective on the EBI of Bi-2212 emerges. We see immediately that boson energies Ω(r) are heterogeneous at the ∼2 nm scale with 40 meV < Ω(r) < 65 meV, spanning the range of antinodal mode energies from photoemission22,23,24,25. The necessity of atomic-resolution d2I/dV2(r,E)-imaging studies to fully explore EBI signatures in high-Tc superconductors becomes manifest here.
In theory9,10, spatial modulations of d2I/dV2(r,E) nearE = Ω + Δ can contain key information about the EBI. But we detect no spatially periodic structure in unprocessed d2I/dV2(E) images in this energy range. This can be understood, however, because Δ(r) is so strongly disordered (Fig. 1c) that any EBI10 effects in d2I/dV2(r,Ω + Δ(r)) would be spatially scrambled. To search for these effects, each d2I/dV2(E) must therefore be shifted to its bosonic energy scale ω(r) = E - Δ(r), a process we refer to as ‘gap referencing’. This converts the unprocessed d2I/dV2(r,E); |E| > Δ data into a new series of d2I/dV2(r,ω) images. These vary little (except in intensity) within the energy range 40 meV < ω < 65 meV where peaks in d2I/dV2(ω) are detectable (see Supplementary Fig. 1). Remarkably, they all exhibit the same spatial modulations, which did not exist in unprocessed d2I/dV2(r,E) data before gap referencing (see Supplementary Fig. 1).
To enhance their spatial contrast, we sum the d2I/dV2(r,ω) images over the boson energy range where EBI are detected: . This energy average produces lower spatial noise compared to a single energy map. A typical resulting Γ(r) is shown in Fig. 2a; the modulations are parallel to the Cu–O bond directions and have wavelengths ∼5a0 with correlation length of ∼50 Å. We then use Γ(q), the Fourier transform of Γ(r), in Fig. 2b to determine that the modulation wavevectors are p1 ≈ 2π/a0[(0.2,0);(0,0.2)] ± 15%. Within the theoretical models9,10, such d2I/dV2(r,ω) modulations are created when electronic states, renormalized by EBI, are scattered by disorder. When scattering is modelled as due to atomic-scale variations in the pair potential4, the predictions for d2I/dV2(r,ω) modulations10 are qualitatively consistent with data in Fig. 2, if the boson is a lattice vibration mode.
Next we study these d2I/dV2 signatures of EBI at a sequence of different hole densities per CuO2, p: p ≈ 0.12 → p ≈ 0.24. In Fig. 3a–c we show that the average superconducting energy gap decreases from ∼60 meV to ∼20 meV with increasing doping, as expected. In strong contrast, Fig. 3d–f shows that, although changes occur in spatial correlations of Ω(r), no change is detectable in the average boson energy . In fact, histograms of Δ and Ω measured on five samples at different dopings (Fig. 4) reveal that, while the distributions of Δ evolve rapidly with doping, those of Ω appear unchanged: for all dopings. We emphasize that the doping independence of is not because the average energy of the d2I/dV2 peak is unchanged with doping; changes from ≅ ± 105 meV to over the doping range. It is the difference, , between and which remains constant. Furthermore, we find the d2I/dV2-modulation wavevectors p1 are the same for all gap-referenced Γ(r)at all dopings (see Fig. 2c).
Thus, both d2I/dV2(r,ω) modulations and are independent of doped hole-density. Which boson13,14,15,16,17 could exhibit such doping-independent EBI characteristics? The ‘resonant’ spin-1 magnetic excitation mode15 appears inconsistent with the doping independence of mode energies because its energy is 43 meV in Bi-2212 but, more importantly, is believed to be strongly doping dependent. The incommensurate, dispersive, spin density wave modes16,17 also appear inconsistent because of their characteristic strong energy- or doping-dependences. By contrast, because energies of lattice-vibration modes change little with doping, they are logical candidates for the boson detected by d2I/dV2 imaging.
Indeed, electron–lattice interactions are well known to influence copper oxide superconductivity. For example, the energy of a phonon mode at momentum transfer Q ≈ 2π/a0[(0.25,0);(0,0.25)] diminishes rapidly towards 50 meV upon cooling into the superconducting state12,13,14, as might be expected for strong phonon interactions with superconducting quasiparticles. Furthermore, studies have revealed an unusual 16O/18O isotope substitution effect26 on the electronic structure; the data point to maximum influence of lattice modes on the high energy (E ≈ - 250 meV) electronic structure near the antinodes, with a much weaker impact above Tc and at low energy. Although these results are not consistent with the simplest Eliashberg picture of electron–phonon interactions26, they do represent evidence for interactions between states near k ≈ (π/a0,0) and lattice vibrational modes. Furthermore, there have been detailed photoemission studies25 of antinodal quasiparticles coupling to a bosonic mode—ascribed to a B1 g Cu–O bond-buckling phonon from theoretical analysis. Taken in combination, these studies point to interactions between antinodal quasiparticles and Cu–O-related lattice vibrations as influencing high temperature superconductivity—although the precise implications for electron pairing mechanism remain uncertain.
To test the hypothesis that bosons detectable by d2I/dV2-imaging techniques in Bi2Sr2CaCu2O8+δ are lattice vibration modes, we prepared crystals in which the normal 16O was completely substituted by 18O (as verified by frequency shifts detected in Raman spectroscopy). Figure 5a provides the comparisons between the distributions of Ω(r) and Δ(r) in different samples containing complete substitutions of the two oxygen isotopes. For each sample, we take Δ(r) and Ω(r) and construct a two-dimensional histogram of the frequency of occurrence of spectra with a given pair of values (Δ,Ω). Each histogram is peaked along the vertical axis at the most common gap energy and along the horizontal axis at the most common boson energy . Comparison between 16O ΔΩ histogram (blue) and the 18O ΔΩ histogram (red) reveals immediately that for 18O shifts downwards by several meV compared to that of 16O. A quantitative analysis in Fig. 5b shows the distribution of boson energies Ω in two different samples containing complete substitutions of the two oxygen isotopes: 16O in blue and 18O in red. We find that the shift of upon substitution of 16O by 18O is -3.7 ± 0.8 meV. These results are found equally true for both filled E = -(Δ + Ω) and empty E = +(Δ + Ω) states, as expected for EBI in strong-coupling superconductivity theory2,3. Consequently, substitution of 18O for 16O reduces the mean boson energy scale of EBI by 6% —as expected for lattice vibrational modes involving the O atom.
These d2I/dV2-imaging studies alter several existing concepts of the EBI problem in Bi2Sr2CaCu2O8+δ. We demonstrate that it is modes involving lattice vibrations that generate the d2I/dV2 features in tunnelling. Further, the diminishing intensities of the d2I/dV2 peaks with coherence peak height (Fig. 1b), along with the necessity for gap referencing (Fig. 2), signify the primary involvement of the antinodal states (E ≈ Δ ,k ≈ (π,0)) in the interactions. The mean mode energy is ≈ 52 meV with a statistical spread of ± 8 meV. These modes exhibit intense atomic-scale disorder of interaction energies Ω(r), whose doping independence (Fig. 4b) points to a population native to the crystal. And perhaps most significantly, the Ω(r) are spatially anticorrelated with the superconducting energy gap disorder Δ(r) at all dopings (Fig. 3) and for both oxygen isotopes (Fig. 5a). Finally, in Fig. 5c we show the normalized correlations4 between the dopant atom locations O(r) and both Ω(r) and Δ(r). Whereas the zero-displacement correlations O(r):Δ(r) ≈ + 0.35 are as expected, and the Ω(r):Δ(r) ≈ - 0.30 correlations are consistent with Fig. 3, we find that Ω(r) and O(r) are uncorrelated. Therefore, correlations between Ω(r) and Δ(r) cannot be occurring trivially, through a similar effect of dopant disorder on both. A direct atomic-scale influence of Ω(r) on Δ(r) (or vice versa) is implied.
Taken together, these data present some intriguing new possibilities. The first is that superconducting energy gap disorder Δ(r) is a consequence of heterogeneity in the pairing potential caused by disorder in the frequencies and coupling constants of pairing-related vibrational modes31. But the strong dependence of superconducting electronic structure on hole density while the Ω(r) distributions remain unchanged (Fig. 4) appears to argue against this point of view. A second possibility is that the d2I/dV2 features are unconnected to pairing-related EBI—perhaps occurring because of inelastic stimulation of vibrational modes within the tunnel barrier itself32 or because of non-pairing-related electron lattice interactions. The primary difficulty here is that the ubiquitous anticorrelation between Ω(r) and Δ(r) cannot be explained trivially within such scenarios. A third possibility is that the d2I/dV2 features represent electron–lattice interactions related to a competing electronic ordered state (see, for example, ref. 13), and that the anticorrelation between Ω(r) and Δ(r) occurs because of this competition. To help to distinguish between these possibilities, an atomic-scale version of the McMillan–Rowell procedure1 may now become necessary.
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We thank the following people for discussions and communications: Ar. Abanov, P. W. Anderson, N. Ashcroft, T. P. Devereaux, T. Egami, M. Eshrig, C. Henley, P. J. Hirschfeld, A. Lanzara, D.-H. Lee, P. Littlewood, D. Morr, K. A. Müller, M. R. Norman, J. Orenstein, T. M. Rice, J. Rowell, D. J. Scalapino, Z.-X. Shen, C. M. Varma and A. Zettl. This work was supported by an LDRD from Los Alamos National Laboratory, a Grant-in-Aid for Scientific Research from the Ministry of Science and Education (Japan), the 21st-Century COE Program of JSPS, by Cornell University and by the Office of Naval Research; K.F. acknowledges Fellowship support from the ICAM International Materials Institute.
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