Coexistence of Two Sharp-Mode Couplings and their Unusual Momentum Dependence in the Superconducting State of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathbf{O}}_{8\mathbf{+}\delta }$ Revealed by Laser-Based Angle-Resolved Photoemission

High-resolution laser-based angle-resolved photoemission measurements have been carried out on ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ (Bi2212) superconductors to investigate momentum dependence of electron coupling with collective excitations (modes). Two coexisting energy scales are clearly revealed over a large momentum space for the first time in the superconducting state of the overdoped Bi2212 superconductor. These two energy scales exhibit distinct momentum dependence: one keeps its energy near 78 meV over a large momentum space while the other changes its energy from $\sim 40\mathrm{meV}$ near the antinodal region to $\sim 70\mathrm{meV}$ near the nodal region. These observations provide a new picture on momentum evolution of electron-boson coupling in Bi2212 that electrons are coupled with two sharp modes simultaneously over a large momentum space in the superconducting states. Their unusual momentum dependence poses a challenge to our current understanding of electron-mode-coupling and its role for high-temperature superconductivity in cuprate superconductors.

Figure 1

Identification of two coexisting energy features. (a) Original photoemission data measured at 17 K for OD82K sample. Second-derivative image with respect to energy for OD82K (b) and OD73K (c). (d) Second-derivative image of the simulated single-particle spectral function which considers electron coupling with two Einstein bosonic modes at 50 meV and 78 meV, as indicated by two arrows [20]. (e) Dispersion by MDC fitting for OD82K sample. (f) Effective real part of the electron self-energy for OD82K (black circles) and OD73K (blue circles). (g) MDC width for OD82K (black circle) and the corresponding difference (red square) by subtracting a straight line (red dashed line). (h) and (i) Photoemission spectra (EDCs) showing multiple peak-dip-hump structures for OD82K and OD73K. Typical EDCs at the Fermi momentum ($k_F$) (red) and near $k_F$ are expended to show the structures more clearly and the arrows indicate the dip positions.

Figure 2

Momentum dependence of the two energy features in Bi2212 OD82K sample. (a1)–(a5) Original images measured along five momentum cuts at 17 K. (b1)–(b5) Corresponding EDC second-derivative images. The location of the five cuts is shown in the inset. (c) EDCs at the Fermi momenta $k_F$ of these five cuts. Black and red arrows point to the two energy features. (d) Effective real part of electron self-energy of cut 1 to cut 4 [20]. For clarity, they are offset vertically with the offset values shown on the right side. (e) Momentum dependence of the two energy features, extracted from the effective real part of electron self-energy (solid circles and squares) and dip position in EDCs (empty circles and squares). The $\Phi$ angle is defined in the inset of (b5) with $\Phi =45°$ representing the nodal direction.

Figure 3

Temperature dependence of the two types of energy features for the Bi2212 OD82K sample. (a),(b) EDC second-derivative images of cut 1 measured below (at 17 K) and above (at 100 K) $T_c$. (c),(d) and (e),(f) are the same measurements for cut 2 and cut 3 respectively. (g)–(i) Corresponding effective real part of electron self-energy, measured at different temperatures, for cuts 1, 2, and 3, respectively. For a given cut, a straight line connecting a high binding energy point on the 100 K dispersion and its $k_F$ at the Fermi level is taken as a common empirical bare band for all the temperatures [20]. Peaks of two energy features are marked by arrows, with the red one representing the LW energy feature and the black one representing the HI energy feature.

Figure 4

Doping dependence of the two types of energy features in Bi2212 samples. Effective real part of electron self-energy measured in superconducting state for OD82K sample (a), Opt91K sample (b), and UD89 sample (c) [20]. The location of the momentum cuts are given by $\Phi$ angles as defined in Fig. 2(b5). For clarity, the effective real part of electron self-energy is offset vertically with the offset values marked on the right side of the curves. (d) EDCs at the Fermi momentum $k_F$ along the cut $\Phi =18°$ for the three samples with different doping levels. Part of the EDCs are expanded (blue curves) in order to highlight the possible multiple peak-dip-hump structures. Arrows mark the position of dips. (e) Summary of the momentum dependence of the two types of energy features in Bi2212 with different doping levels. Both EDC and electron self-energy are used to obtain the energy positions.