Raman-Scattering Measurements and Theory of the Energy-Momentum Spectrum for Underdoped ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCuO}}_{8+\delta }$ Superconductors: Evidence of an $s$-Wave Structure for the Pseudogap

We reveal the full energy-momentum structure of the pseudogap of underdoped high-$T_c$ cuprate superconductors. Our combined theoretical and experimental analysis explains the spectral-weight suppression observed in the $B_{2g}$ Raman response at finite energies in terms of a pseudogap appearing in the single-electron excitation spectra above the Fermi level in the nodal direction of momentum space. This result suggests an $s$-wave pseudogap (which never closes in the energy-momentum space), distinct from the $d$-wave superconducting gap. Recent tunneling and photoemission experiments on underdoped cuprates also find a natural explanation within the $s$-wave pseudogap scenario.

Figure 1

Theoretical spectral intensity at $T=0.06t$ and $p_{\mathrm{th}}=0.05$ (a) in the energy-momentum space and (b) in the first quadrant of the Brillouin zone at low energy ($\omega \sim 0$). Black circles and white squares denote, respectively, the peak positions of quasiparticle and in-gap bands, which are separated by an $s$-wave pseudogap. The dotted blue curve plots the position of the maximal-scattering rate (i.e., the imaginary part of self-energy) at each momentum. The green (dark gray) arrow in (b) denotes the momentum cut used in (a). (c) Local DOS in a wide energy range displaying the Hubbard bands ($\omega <0$ and $\omega \sim 6t$). The shaded area denotes the low-energy region plotted in (a). (d) Temperature dependence of the scattering rate within the pseudogap at $\omega =0.36t$ (node) and $0.28t$ (antinode). (e)–(g) Spectral function along the cuts depicted in (b), comparable to available ARPES data [6]. The dashed line indicates the upper energy limit reached in Figs. 3(e)–3(g) of Ref. 6. (h) Tunneling conductance at $T=0.06t$, comparable to, e.g., Fig. 5(b) (inset) of Ref. 38.

Figure 2

$B_{2g}$ Raman spectra obtained by experiments (a) on an underdoped Bi2212 $T_c\sim 74\mathrm{K}$, $p\sim 0.11$) and by the CDMFT (b) in the underdoped regime ($p_{\mathrm{th}}=0.05$). (c),(d) The same for $B_{1g}$ spectra. The blue arrows mark the starting energy points of pseudogap depression. The insets in (a) and (c) plot the integrated Raman weight normalized at $T=300\mathrm{K}$ as a function of temperature. (e) Experimental $B_{2g}$ Raman response in the normal $T=150$ and 80 K) and superconducting $T=10\mathrm{K}$) states. In (a), (c), and (e), the red (light gray), green (gray), black, and blue (dark gray) curves correspond respectively to 80 K, 150 K, 250 K, and 10 K. (f) Theoretical spectral intensity calculated with $\mathrm{CDMFT}+\mathrm{\text{exact}}$ diagonalization method for a $2\times 2$ cluster in the superconducting state. The white arrows denote excitations beyond the pseudo- and superconducting (SC) gaps in the nodal region, which contribute to the $B_{2g}$ Raman intensity.

Figure 3

Energy distribution curves of the single-particle spectra along the $(\pi ,0)-(\pi /2,\pi /2)$ line (a) at $T=0.06t$ and (b) at $T=0.08t$ and $0.12t$. The black circles (white squares) denote the quasiparticle (in-gap) peak plotted in Fig. 1a. The arrow at the top curve denotes the pseudogap. For clarity, the curves are offset by 0.3.