Extraction of normal electron self-energy and pairing self-energy in the superconducting state of the Bi${}_2$Sr${}_2$CaCu${}_2$O${}_8$ superconductor via laser-based angle-resolved photoemission

Super-high-resolution laser-based angle-resolved photoemission measurements have been performed on a high-temperature superconductor, Bi${}_2$Sr${}_2$CaCu${}_2$O${}_8$. The band back-bending characteristic of the Bogoliubov-like quasiparticle dispersion is clearly revealed at low temperature in the superconducting state, which gives rise to two peaks in the momentum distribution curves. This makes it possible to experimentally extract the normal electron self-energy and pairing self-energy in the superconducting state. The resultant normal and pairing self-energies exhibit features at $\sim$54 and $\sim$40 meV, in addition to the superconducting gap-induced structure at lower binding energy and a broad featureless structure at higher binding energy. This information can be used to further determine the Bosonic spectral function that will provide key insight and constraints on the origin of electron pairing in high-temperature superconductors.

Figure 1

Observation of two branches of Bogoliubov quasiparticle-like dispersions in the superconducting state of slightly underdoped Bi2212 ($T_c=89$ K). (a). Simulated single-particle spectral function $A(k,\omega )$ in the superconducting state, taking a superconducting gap of $\Delta =10$ meV and a linewidth broadening of $\Gamma =5$ meV. The simulated $A(k,\omega )$ multiplied by the Fermi distribution function $f\left(\omega \right)$ at a temperature of 70 K is shown in (b) and at 20 K in (c). (e1)–(e5) Photoemission images taken at different temperatures along the momentum cut marked as in (d). This cut nearly points toward ($\pi$, $\pi$) with an angle $\theta$ as defined. (f1)–(f5) Photoemission images in (e1)–(e5) divided by a Fermi distribution function at the corresponding temperatures.

Figure 2

Observation of band back-bending in the lower branch of the Bogoliubov quasiparticle dispersion in the superconducting state of Bi2212. (a) Photoemission image taken at 16 K along the cut shown in Fig. 1d (it corresponds to Cut $\#$4 in Fig. 3). This is the same data as in Fig. 1(e1) but with a different false-color scale. (b) Photoemission image taken at 16 K divided by the image taken at 107 K along the same cut. (c1)–(c5) Representative MDCs in the normal state at 107 K (red circles) and superconducting state at 16 K (blue circles) at five different binding energies. The solid lines represent fitted results. (d1)–(d5) MDCs at 16 K divided by MDCs at 107 K at different binding energies (black circles). The solid black lines represent the fitted MDCs at 16 K [blue lines in (c1)–(c5)] divided by the fitted MDCs at 107 K [red lines in (c1)–(c5)]; they are not independent fits of the divided data.

Figure 3

Momentum dependence of the lower branch of the Bogoliubov quasiparticle-like dispersion in Bi2212. (a1)–(a6) Photoemission images measured at 16 K for different momentum cuts with their location shown in the upper-right inset. (b1)–(b6) Photoemission images measured at 16 K divided by the corresponding images measured at 107 K. The white arrows in (b5) and (b6) mark an additional feature that is likely due to umklapp bands in Bi2212.

Figure 4

The normal and pairing electron self-energies obtained from the ARPES measurements of Bi2212 in the superconducting state. (a) and (b) show real part and imaginary part, respectively, of the normal electron self-energy $\Sigma \left(\omega \right)$ obtained for three different momentum cuts (Cuts $\#$3, $\#$4, and $\#$5 in upper-right inset of Fig. 3); (c) and (d) show the real part and imaginary part, respectively, of the pairing self-energy $\phi \left(\omega \right)$ for the three-momentum cuts. The superconducting gap size is 10, 12.5, and 15 meV on the Fermi surface crossing Cuts $\#$3, $\#$4, and $\#$5, respectively, while the maximum superconducting gap near the antinodal region is 33 meV.