Large momentum-dependence of the main dispersion 'kink' in the high-T_c superconductor Bi₂Sr₂CaCu₂O_8+δ

The data presented here were obtained from Bi₂Sr₂CaCu₂O_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. Angle-resolved photoemission spectroscopy (ARPES) cuts were taken along the (π,π) direction of the Fermi surface (FS).

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 dependences (e.g. electron–electron 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

$$\begin{equation} \Sigma'_{\rm{eff}}(\omega)=\omega-v_{\rm{F}}^{\rm{eff}}[k_{m}(\omega)-k_{\rm{F}}], \end{equation} \tag{ 1 }$$

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 electron–hole 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 equation (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) FSs 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 FS angles θ = 0.9° and 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 to −200 meV that are constrained to pass through the MDC peak location at ω = −Δ(θ) (i.e. k_F).

Zoom In Zoom Out Reset image size

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 α²F(k,ν), where ν is the energy axis for bosons. In an ungapped system at T = 0, $\Sigma ''(\boldsymbol {k},\omega)=\pi \int _{0}^{|\omega |}\mathrm {d}\nu \alpha ^{2}F(\boldsymbol {k},\nu)$ [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. Firstly, Ω_kink evolves smoothly as a function of θ in the nodal region, shifting towards E_F by about 10 meV from θ = 0 to 15°. Secondly, 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 figures 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 parameterized by, e.g. the intensity and sharpness of the features in Σ_eff(ω,θ), as seen in figures 1(e) and (f).

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 + Δ) but more complicated for a d-wave superconductor in which Δ is strongly k-dependent. In the presence of an anisotropic gap Δ(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]

$$\begin{equation} |\Omega_{\rm{kink}}(\boldsymbol{k})|=\Omega_{\rm{boson}}(\boldsymbol{q})+\Delta(\boldsymbol{k'}) \end{equation} \tag{ 2 }$$

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, SF observed by inelastic neutron scattering (INS) are peaked at points on or near the (ξ,ξ,0) line [29–31]. 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(b) as a function of FS angle θ (red ▪). The blue circles (•) are the corresponding gap-referenced boson energies Ω*_boson. The asterisk (*) denotes that only the special cases of scattering vectors shown in that panel apply. Under these circumstances, Δ(k) = Δ(k'), leading to

$$\begin{equation} \Omega_{\rm{boson}}^{*}(\theta)=|\Omega_{\rm{kink}}(\theta)|+\Delta(\theta). \end{equation} \tag{ 3 }$$

In calculating Ω*_boson(θ), we used Δ(θ) based on our ARPES-measured values, which were found to have excellent agreement with the expected d-wave form, with maximum (antinodal) magnitude Δ₀ = 30 meV. The gap measurements shown here were performed using the newly developed tomographic density of states technique [35, 36], and we have checked that the analysis and results that follow are essentially unchanged if the symmetrized energy distribution curve method is employed [37].

Zoom In Zoom Out Reset image size

The extracted q-space dispersions of Ω_kink and Ω*_boson under these various scattering scenarios are shown in figures 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 FS 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 figures 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 FS (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 figures 2(c) and (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 FS. Meanwhile, the diagonal intra- and inter-hole-pocket vectors ('Intra' and 'Inter' in figures 2(e) and (f) respectively) are also treated independently, since they are distinct in how they couple the topology of the FS.

In figures 2(c)–(f), the extracted Ω*_boson(q) curves (blue •) are compared to various Cu–O phonon dispersions in YBa₂Cu₃O_6+x (YBCO) with x = 0 (▿) and x = 1 (▵) [27], as well as two phonon branches observed in Bi₂Sr_1.6La_0.4Cu₂O_6+δ (Bi2201, ◊) [23]. Additionally, a sketch of the dispersion of a high-energy branch of incommensurate SF is shown in figure 2(f) (green shaded line).

Overall, the extracted Ω*_boson(q) curves do not provide clear support that the kink primarily originates from electron–phonon interactions, although the data may not be wholly inconsistent with this possibility. For instance, in figure 2(d), there is a limited q-space region (q ≈ (0.3–0.4,0,0)) where the extracted boson dispersion roughly overlaps with the Cu–O bond stretching phonon branch, as pointed out in [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 SF 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 [28–31, 39]. 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 [40–43]. 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 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 FS [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 SF-related, 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.

Zoom In Zoom Out Reset image size