Large momentum-dependence of the main dispersion"kink"in the high-Tc superconductor Bi2Sr2CaCu2O8+{\delta}

Ultrahigh resolution angle-resolved photoemission spectroscopy with low-energy photons is used to study the detailed momentum dependence of the well-known nodal"kink"dispersion anomaly of Bi2Sr2CaCu2O8+{\delta}. We find that the kink's location transitions smoothly from a maximum binding energy of about 65 meV at the node of the d-wave superconducting gap to 55 meV roughly one-third of the way to the antinode. Meanwhile, the self-energy spectrum corresponding to the kink dramatically sharpens and intensifies beyond a critical point in momentum space. We discuss the possible bosonic spectrum in energy and momentum space that can couple to the k-space dispersion of the electronic kinks.


Introduction
One of the defining characteristics of the electronic structure of the high-T c cuprates is the presence of an especially prominent anomaly, or "kink", in the electronic dispersion, which corresponds to a strong feature in the complex electronic self-energy spectrum Σ(k, ω) = Σ (k, ω)+iΣ (k, ω). The origin of the kink -whether it is due to interactions of the electrons with bosons (particularly phonons [1,2] or magnetic excitations [3,4,5]) or some other phenomenon [6] -is still heavily debated. Likewise the kink's connection to superconductivity, and whether the interactions it signifies may either form or break Cooper pairs, or be altogether irrelevant, remains unknown.
At the nodes of the d-wave superconducting gap, this kink appears at a binding energy of roughly 60-70 meV [7,8,9,10]. Meanwhile near the antinode, a seemingly stronger kink is located at about 20-40 meV, depending on doping [11,12,13]. While a possible connection between the nodal and antinodal kinks remains a mystery, the new data here fills in details of the evolving physics between these points. Such information is crucial for obtaining a complete understanding of the behaviour and origin of the kink and hence the electron-boson coupling in the high-T c superconductors.

Experimental
The data presented here were obtained from Bi 2 Sr 2 CaCu 2 O 8+δ (Bi2212) near optimal doping with T c ≈ 89 K. Rotational alignment of the sample better than 1 • was performed by Laue diffraction. The data were collected in the superconducting state at 10 K using a photon energy of 7 eV. Compared to conventional photon energies, the low photon energy greatly improves the photoelectron escape depth, momentum resolution, and overall spectral sharpness [14]. Total combined energy resolution of the light source and analyser was about 7 meV. ARPES cuts were taken along the (π, π) direction of the Fermi surface (FS).

Momentum-dependent self-energy
In the present work, we are especially concerned with the self-energy contribution due to electrons coupling to a collective mode over a sharp energy range, and we wish to isolate this from other interactions with smooth energy dependencies (e.g., electronelectron scattering [15]). This is accomplished by assuming a smooth (in this case linear) effective bare band eff (k) for each ARPES cut that connects points on the dispersion far from the main kink. The real part of the effective self-energy is then simply where k m (ω) is the measured dispersion, v eff F is the slope of eff (k), and k F is the Fermi momentum. Σ eff (ω) is then the Kramers-Kronig transformation of Σ eff (ω). Unlike Σ (ω), Σ eff (ω) is well-behaved at its endpoints (by construction), and its Kramers-Kronig transformation is easily computed. Our routine sets the in-gap points of Σ eff (ω) to zero and computes the transformation by Fast Fourier transform assuming electronhole symmetry. We have verified by simulations that a possible violation of electron-hole symmetry [16] should not significantly alter the findings here. We note that Eq. 1 is a conventional definition of Σ eff . Recently it was shown that this definition undervalues the "true" bosonic part of the self-energy by an overall scaling factor related to the coupling strength of electron-electron interactions, λ el-el [17]. As this factor influences the magnitude of the self-energy, not its distribution along ω or k , neglecting it will not affect the conclusions of the present study. A systematic assessment of λ el-el in Bi2212, and hence the correct scaling factor to be applied to Σ eff , is currently underway. Figure 1(a) shows ARPES data collected along the FS in the first quadrant of the Brillouin zone. The colour scale represents the measured intensity 10 meV below E F . The thick solid lines in Figure 1(a) are sketches of the antibonding (AB) and bonding band (BB) Fermi surfaces based on a tight-binding model [18]. For 7-eV photons, only the AB is detected [19], which greatly simplifies the analysis. Two representative raw data cuts, corresponding to Fermi surface angles θ = 0.9 • and θ = 16.3 • , are indicated by the red curves labelled i and ii, respectively. The spectra from these cuts are shown in Figure 1(b). The dispersions from momentum distribution curve (MDC) fits are overlaid on the spectra (solid black curves). The dashed red lines are assumed effective bare bands used to calculate corresponding effective self-energy spectra Σ eff (ω) at each θ. These effective bare bands are determined by linear fits from -230 meV to -200 meV that are constrained to pass through the MDC peak location at ω = −∆(θ) (i.e., k F ). Figure 1(c) shows the Lorentzian MDC widths for each cut from i to ii, while Figure 1(d) depicts Σ eff (ω) (black, right axis) and Σ eff (ω) (red, left axis) for cuts i and ii. The full spectrum of Σ eff (θ, ω) is plotted as a colour scale in Figure 1(e). We define the kink energy Ω kink as the location of the peak in Σ eff (ω) at each θ. These values are determined by quadratic fits over a range ±20 meV about the maximum of each spectrum. The error bars show the standard deviations (±σ) returned from the fits. Figure 1(f) depicts −∂Σ eff /∂ω as a function of ω and θ. To reduce noise in the derivative, some light smoothing was applied to the Σ eff spectrum. Together panels (e)-(f) highlight the evolution of the self-energy over the nodal region, which exhibits both dispersive behaviour and sharpening. The results are fully consistent with the behaviour of the MDC widths in panel (c) and the electronic dispersion anomalies in (b), providing an important verification of the self-consistency of the data and analysis methods. It is worth noting that the quantity −∂Σ eff /∂ω in panel (f) is somewhat related to a useful parameter of strong coupling theory -the Eliashberg boson coupling spectrum α 2 F (k, ν), where ν is the energy axis for bosons. In an ungapped system at T = 0, Σ (k, ω) = π´| ω| 0 dνα 2 F (k, ν) [20], although the anisotropic gapping in cuprates can significantly alter this relationship. Addressing this issue via suitable "gap referencing" is a key objective of the present work. Two key points are evident from Figure 1. First, Ω kink evolves smoothly as a function of θ in the nodal region, shifting toward E F by about 10 meV from θ = 0 to θ = 15 • . Second, the nature of Σ eff appears to change abruptly past a critical point in k-space. While by eye the kink perhaps becomes more dramatic going from node to antinode [21], this fact alone does not necessarily mean that the self-energy strengthens, since Σ is related to the bare band velocity, which decreases away from the node. Indeed, the results in Figure 1(c)-(f) show that over much of the near-nodal region Σ eff (ω) is relatively unchanged, despite the visual appearance that the kink is "getting stronger". However, for θ 10 • Figure 1(e) shows a rapid increase in the peak of Σ eff (ω). This corresponds with sharpening of the step in Σ eff (ω) seen in Figure 1(f). The findings in Figure 1 contrast with a previous study of overdoped Pb-Bi2212 where it was argued that the scattering rate near E F is independent of k [22]. We also point out that these results contradict recent claims that the energy Ω kink is constant near the node and then suddenly jumps at a "crossover" point on the FS ∼ 15 • away from the node [23,24], though there still may be a crossover parameterised by, e.g., the intensity and sharpness of the features in Σ eff (ω, θ), as seen in Figure 1(e)-(f).

Scattering q-space analysis of the kink momentum dependence
The large nodal ARPES kink seen in cuprates is generally explained as the result of the coupling of the electrons to a bosonic mode of energy Ω boson . In particular, Ω kink may be able to tell us which electrons interact with which bosons, and in principle this can yield information about which (or even whether) bosons act as the "glue" responsible for the formation of the Cooper pairs. The new finding of the large, smooth dispersion of Ω kink (k) is therefore an important result that may connect directly to the coupling mechanism of the electrons within a pair. Here we consider how to best connect the k-dispersion of the kink to known data of the q-space dependence of various bosonic modes.
In the simplest picture, the kink energies Ω kink will be exactly those of the coupling boson [8], though this ignores the "gap referencing" which is simple for an s-wave superconductor (|Ω kink | = Ω boson +∆) but more complicated for a d-wave superconductor in which ∆ is strongly k-dependent. In the presence of an anisotropic gap ∆(k), a bosonic mode with energy Ω boson (q) scattering an electron purely from k to k is expected to produce an ARPES dispersion kink below E F at [25] |Ω kink (k)| = Ω boson (q) + ∆(k ) (2) which can be deduced by considering the set of photoholes at k that can be annihilated via electrons decaying from k and emitting bosons Ω boson (q). An argument along these lines (but for an isotropic gap) is presented in section 7.3 of [20]. We will make use of this gap referencing relationship throughout the present work in order to identify the boson dispersions appropriate to particular scattering scenarios. Additional corrections for relating Ω kink to Ω boson are believed to be too small to account for the dispersive behaviour of Ω kink (k) [26] and therefore should not qualitatively alter the present work. There is reason to believe that the predominant electron-boson scattering relevant to the nodal kink falls along some symmetry direction, thus simplifying the connection between k-and q-space. For instance, spin fluctuations observed by inelastic neutron scattering (INS) are peaked at points on or near the (ξ, ξ, 0) line [29,30,31]. Likewise, both experiment [32] and theory [1,33] suggest the Cu-O "half-breathing" phonon mode scatters electrons primarily along (ξ, 0, 0) [equivalently (0, ξ, 0)], and there is evidence that this mode couples strongly to electrons [34] and, in particular, may contribute to the nodal kink [19].
In the case of phonons, numerical calculations find that, on the whole, the scattering matrix elements are fairly complicated [1,33]. Nevertheless, they are expected to evolve smoothly over the FS and exhibit some preference for particular directionalities. Thus, despite the complexity of the full scattering problem, one can reasonably expect to find qualitative agreement between the actual phonon dispersion and the inference from ARPES -at least over a limited portion of the FS.
To proceed with our analysis, assumed scattering q's along symmetry directions are illustrated in Figure 2(a). We consider cases where electrons may scatter horizontally, vertically, or diagonally via inter-or intra-hole-pocket vectors. The kink energies obtained in Figure 1(e) are plotted in Figure 2 In calculating Ω * boson (θ), we used ∆(θ) based on our ARPES-measured values, which were found to have excellent agreement with the expected d-wave form, with maximum (antinodal) magnitude ∆ 0 = 30 meV. The gap measurements shown here were performed using the newly-developed tomographic density of states (TDoS) technique [35,36], and we have checked that the analysis and results that follow are essentially unchanged if the symmetrised energy distribution curve (EDC) method is employed [37].
The extracted q-space dispersions of Ω kink and Ω * boson under these various scattering scenarios are shown in Figure 2(c)-(f). The insets in each of these panels illustrate how the q values on each horizontal axis were determined. For simplicity and generalizability of the analysis, we consider each scattering channel independently. For an assumed bosonic mode that would scatter electrons in the (ξ, 0, 0)/(0, ξ, 0) directions, a given k point on the Fermi surface could couple via two orthogonal vectors -one shorter than the node-node distance in q-space (ξ ∼ 0.35) and the other longer [labelled "H/V short" and "H/V long" in Figure 2(c) and (d), respectively]. However, in general these short and long q channels would not be expected to contribute equally to the appearance of the kink, but rather their relative scattering intensities would evolve around the Fermi surface (only matching at the node-node distance, where they have the same length). To disentangle these paired interactions, the Ω kink (q) and Ω * boson (q) curves were extracted by treating the short and long scattering vectors separately, as plotted in Figure 2(c), (d). This is a logical choice, since one or the other (either the short or long vector) would probably be more influential at any given point on the Fermi surface. Meanwhile, the diagonal intra-and inter-hole-pocket vectors ["Intra" and "Inter" in Figure 2(e) and (f) respectively] are also treated independently, since they are distinct in how they couple the topology of the Fermi surface.
In Figure 2 [23]. However, for larger values of q extending toward (0.5, 0, 0), Ω * boson (q) diverges from the Cu-O bond stretching phonon branch with a different slope. In this regard, our data show greater overall similarity between Ω * boson (q) and the SF dispersion in Figure 2(f), which differ by merely a simple offset in ω and/or q, perhaps reflecting systematic differences between the techniques and/or samples used in the studies. The rough correspondence between the kink and the SF dispersion compares favorably with spectral analysis of the spin response function extracted from ARPES data by Chatterjee et al [38], though their formalism only considered spin fluctuations and did not provide a comparison to phonon dispersions. Moreover, that approach treated the spectral function holistically, rather than isolating the kink feature and considering its explicit connection to the bosonic spectral function.
The SF dispersion in Figure 2(f) is the high-energy branch of incommensurate spin excitations so far observed in many cuprates [29,30,39,31,28]. It converges with a low-energy branch near ∼ 40 meV, where there is a well-known q = (0.5, 0.5)-centred "resonance" peak in the spin susceptibility at low T [40,41,42,43]. The effective self-energy obtained from our analysis intensifies significantly at Ω * boson somewhat near the resonance energy. This is depicted in Figure 3, where the peak height of Σ eff (red triangles) is plotted versus Ω * boson . INS data from optimally-doped Bi2212 (open circles) show the difference in scattered neutron intensity from 100 K to 10 K, illustrating the location of the resonance [43]. Notably, within the context of an orbital-overlap model, coupling to the Cu-O bond-stretching phonon suggested by Figure 2(d) is not expected to intensify in this manner at θ corresponding to Ω * boson close to the resonance [44]. With that said, the data is again only in rough agreement with the SF picture, and other studies imply that the strength of Σ eff is monotonic around the Fermi surface [45], meaning that it would not obey the peak-shaped trend of the INS data reproduced in Figure 3. However, this does not fully undermine the possibility that the kink is SFrelated, as it could instead signal a contribution to Σ eff at low energies (i.e., near the antinode) from additional types of electron-boson interactions, as we will discuss.

Discussion
The above analysis has relied on the key assumption of a dominant scattering mode and directionality, which, while consistent with interpretations of some INS and inelastic x-ray scattering (IXS) data and calculations, is not exhaustive. A natural potential counter example is the case where all points on the FS couple primarily to the van Hove singularities at the antinodes -a situation in which the results would not be directly comparable with conventional INS/IXS data, since the q's would no longer fall along a straight line. Perhaps the best that can be said is that coupling strictly to the antinodes would shift the Ω kink (q) dispersion down universally by ∆ 0 ≈ 30 meV such that Ω boson (q) would span an energy range of roughly 25-35 meV. These energies are home to many phonons in the cuprates [46], which in principle could combine their effects in some intricate way to produce the observed kink behaviour. A strong antinodal coupling appears unlikely, however, because this scenario would imply a huge shift of the nodal kink energy between the normal and superconducting states and/or as a function of doping. Multiple ARPES studies find no evidence of such a shift [10,12,21,47,25], although one case where the kink was interpreted to be composed of multiple phonon mode couplings arguably shows evidence of node-antinode scattering [48].
Reviewing the results, the analysis of Ω * boson (q) for "long" (ξ, 0, 0)/(0, ξ, 0) scattering vectors found some region of agreement with the dispersion of a Cu-O bond stretching phonon [ Figure 2(d)], although the trends diverge as ξ extends out to 0.5. On the other hand, Ω * boson (q) extracted for diagonal inter-hole-pocket scattering is similar to the dispersion of spin fluctuations [ Figure 2(f)], merely differing by a simple offset in energy and/or q. Additionally, Figure 3 illustrates that the strength of the self-energy associated with the kink, plotted with respect to Ω * boson , has a qualitative resemblance to spin fluctuations, and this is contrary to the behaviour predicted for electron-phonon coupling [44]. However, to the extent the data might be viewed as favoring the spin fluctuation picture, it poses an intriguing apparent paradox; A very detailed low-hν ARPES study found an isotope shift in the energy of the nodal kink [19], giving strong merit to the phonon scenario. One possible explanation for this conflict is that electronphonon interactions might constitute a finite but relatively small contribution to the total self-energy [49]. Alternatively, the results may signal a role for coupling between spin and lattice degrees of freedom [50,51,52].
The discovery of the large momentum dependence of the main nodal kink adds to the richness of strong electron-boson coupling phenomena in cuprates. It was recently shown that a newly-discovered ultra-low-energy kink ∼ 10 meV below E F [53,25,54,55] has its own distinct momentum dependence that runs counter to the behaviour of the deeper-energy kink studied here [56]. Specifically, unlike the larger main kink 65-55 meV below E F , which evolves toward lower binding energy while moving from node to antinode, the ultra-low-energy kink closely follows the contour of the superconducting gap in the nodal region, moving to higher binding energy approaching the antinode. A natural, but spectroscopically demanding, next course of study will be to investigate the possible convergence of these two energy scales near the antinodal point and to see whether either or both this these connect with the antinodal feature observed near 20-40 meV, which historically has been regarded as a separate kink. Hence the data here, in concert with [56], open the possibility that interactions with distinct physical origins could combine within a narrow energy range in the antinodal region -perhaps with major implications for high-T c superconductivity.
In conclusion, using low photon energy ARPES, we have mapped the detailed momentum dependence of the primary kink in the nodal k-space region of near-optimal Bi2212. From a simplifying treatment of the data that takes into account effects of the d-wave superconducting gap, the kink's dispersion seems inconsistent with most phonons, though over a limited range of momentum transfer [q ≈ (0.3-0.4, 0, 0)] it bears some semblance to scattering due to a Cu-O bond-stretching mode. However, in terms of the momentum dependence of the location and sharpness/intensity of the selfenergy feature, the greatest similarity is found with the dispersion of the upper branch of incommensurate spin fluctuations.
Note added -During review of this manuscript, a related article was published [57].

Acknowledgments
Funding was provided by the DOE under project number DE-FG02-03ER46066. Experiments were conducted at BL-9A of the Hiroshima Synchrotron Radiation Center and BL5-4 of the Stanford Synchrotron Radiation Lightsource (SSRL). SSRL is operated