Vertical temperature boundary of the pseudogap under the superconducting dome in the phase diagram of ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$

Combining electronic Raman scattering experiments with cellular dynamical mean field theory, we present evidence of the pseudogap in the superconducting state of various hole-doped cuprates. In ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_{8+\delta }$ we track the superconducting pseudogap hallmark, a peak-dip feature, as a function of temperature $T$ and doping $p$, well beyond the optimal one. We show that, at all temperatures under the superconducting dome, the pseudogap disappears at the doping $p_c$, between 0.222 and 0.226, where also the normal-state pseudogap collapses at a Lifshitz transition. This demonstrates that the superconducting pseudogap boundary forms a vertical line in the $T-p$ phase diagram.

Figure 1

Cartographies of the $B_{1g}$ Raman response difference $\chi (\omega ,T)-\chi (\omega ,T=250\text{K}$) versus temperature of (a) an overdoped (OD58, $p=0.226$) and (b) an underdoped (UD80, $p=0.123$) Bi-2212 single crystals. Cartographies were built from the $B_{1g}$ Raman responses plotted in panels (c) and (d) subtracted from the ones at 250 K. The energy is expressed in $2\mathrm{\Delta }$ units. $2\mathrm{\Delta }$ corresponds to $247{\mathrm{cm}}^{-1}$ and $558{\mathrm{cm}}^{-1}$ for OD58 and UD80, respectively. The blue (dark gray) and the red (bright gray) colors correspond respectively to a loss and gain of Raman spectral weight.

Figure 2

Superconducting and normal (just above $T_{\mathrm{c}}$) antinodal ($B_{1g}$) Raman responses ${\chi }_{B_{1g}}^{\prime \prime }(\omega ,T)$ of four distinct underdoped cuprates (a) Hg-1201 ($T_{\mathrm{c}}=88\mathrm{K}$), (b) Y-123 ($T_{\mathrm{c}}=90\mathrm{K}$), (c) Bi-2212 ($T_{\mathrm{c}}=80\mathrm{K}$), (d) Hg-1223 ($T_{\mathrm{c}}=133\mathrm{K}$). The Raman spectra in panels (b), (c), and (d) were obtained with the 532 nm laser line whereas the Hg-1201 one with the 647.1 nm line to reduce the Raman phonon activity. The PP-dip structure is also detected with the $532\mathrm{nm}$ laser line in Hg-1201 showing the PP-dip structure is not a Raman resonant effect. Sharp peaks correspond to phonon modes. In the insets, a closer view of the PP-dip structure is plotted by subtracting the normal Raman spectrum from the superconducting one.

Figure 3

Difference of the antinodal ($B_{1g}$) Raman response in the SC ($12\mathrm{K}$) and normal state (just above $T_c$) of Bi-2212 crystals for a set of doping levels from $p=0.096$ to 0.236. The Raman intensity has been normalized to the maximum intensity of the PP. The dip disappears above $p=0.222$ while the PP is still observable. The arrows indicate the PP and the dip bottom. Raw data are reported in the Appendix pp3.

Figure 4

Theoretical $B_{1g}$ Raman response obtained from Eq. (1) combined with the CDMFT. The PP and the dip-bottom energies are denoted by arrows and go down with doping but their difference in energy remains almost constant with doping, as observed experimentally in Fig. 3.

Figure 5

Temperature dependence of the difference between the SC $B_{1g}$ Raman response and the one just above $T_{\mathrm{c}}$, denoted $T_0$ for (a) an overdoped OD62 Bi-2212 compound ($T_0=70\mathrm{K}$), (b) an overdoped OD58 Bi-2212 compound ($T_0=60\mathrm{K}$). (c) and (d) Closer views of the dip energy range above the pair breaking peak.

Figure 6

Temperature-doping phase diagram of Bi-2212, showing the PG in the normal and SC phases. The normal state PG which develops between $T^{*}$ and $T_{\mathrm{c}}$ is obtained from the $B_{1g}$ spectral loss observed in Ref. [13]. The $T^{*}$ values are extracted from Ref. [13]. The dip is the PG-related feature in the SC state. The PG collapses abruptly (vertical line) between $p=0.222$ and $p=0.226$ in the SC state.

Figure 7

Two different energy locations of the pair breaking peaks in Raman responses (measured at 10 K) for two Bi-2212 single crystals with distinct doping levels: OD 58 (red/gray curve) and the OD 62 (black curve).

Figure 8

Raman responses of Bi-2212 crystals normalized to the maximum intensity of the pair-breaking peak. All the spectra have been corrected for the Bose factor and the optical constants as determined by ellipsometry measurements [66]. The $532\mathrm{nm}$ laser line was used for all the spectra.

Figure 9

Non-normalized Raman responses of Bi-2212 crystals corresponding to the normalized ones shown in Fig. 8.

Figure 10

Spectral function at $\mathbf{k}=(\pi ,0)$, calculated with the CDMFT for various doping levels.