Loss of antinodal coherence with a single $d$-wave superconducting gap leads to two energy scales for underdoped cuprate superconductors

We address the question of the two energy scales in the superconducting state of hole-doped copper-oxide superconductors below the optimal doping level. We show by performing temperature-dependent Raman scattering measurements on ${\text{Bi}}_2{\text{Sr}}_2{\text{CaCu}}_2{\text{O}}_{8+\delta }$ that the two energy scales often assigned to two distinct gaps, are both associated with coherent Bogoliubov quasiparticles and can be simply explained by a one single $d$-wave gap and the loss of antinodal coherence. We then establish that the critical temperature $T_c$ is controlled by the fraction of coherent Fermi surface $f_c$ around the nodes such as: $k_B{T}_c\propto f_c\cdot {\Delta }_{max}$, where ${\Delta }_{max}$ is the maximum amplitude of the superconducting gap.

Figure 1

(Color online) Temperature dependence of the Raman spectra. [(a) and (b)] Raman spectra, ${\chi }^{″}\left(\omega ,T\right)$ of Bi-2212 single crystals for several doping levels in $B_{1g}$ (antinodal) and $B_{2g}$ (nodal) geometries. [(c)and (d)] Raman spectra subtracted from the one measured at 10 K above $T_c$ for each sample in each geometry. A direct visual comparison of the subtracted spectra to a reference energy (chosen as the peak position for the most overdoped sample, drawn as a guide to the eyes) clearly reveals the distinct doping dependence of these two energy scales. (e) Temperature dependence of the normalized areas of the $B_{1g}$ and $B_{2g}$ peaks with respect to the area measured at $T=10\text{K}$. The inset displays these areas for Hg-1201 crystals. (f) Energy scales associated to the $B_{1g}$ and $B_{2g}$ peaks for both Bi-2212 and Hg-1201 crystals. The dashed lines are guides to the eyes and track the locations of the superconducting peak maxima. The doping value $p$ is inferred from $T_c$ using equation of Presland et al. (Ref. 30): $1-T_c/T_c^{max}=82.6{\left(p-0.16\right)}^2$. $T_c$ has been determined from magnetic susceptibility measurements for each doping level.

Figure 2

(Color online) Three scenarios for underdoped cuprates. [(AI)–(CI)] Doping evolution of the superconducting gap in the three scenarios A–C. [(AII)–(CII)] Angular dependences of the quasiparticle spectral weights $Z\Lambda \left(\varphi \right)$ as a function of doping level for each scenario. The angular dependence of the $B_{2g}$ Raman vertex is shown in dotted line (see AII). [(AIII)–(CIII) and (AIV)–(CIV)] Calculated Raman spectra for each scenario in $B_{1g}$ and $B_{2g}$ geometries. The locations of the $B_{1g}$ and $B_{2g}$ peaks are respectively controlled by the gap energy at the antinodes $\varphi =0$ and at the angle ${\varphi }_N$. Note that the low energy slope of the $B_{2g}$ Raman response is constant with doping as observed experimentally (Ref. 21). [(AV)–(CV)] Tunneling spectra for the three scenarios.

Figure 3

(Color online) Fermi surface coherent fraction and a single $d$-wave superconducting gap. (a) Scenario A where coherent Bogoliubov quasiparticles are partially suppressed on restricted parts of the Fermi surface and the superconducting gap has a single d-wave shape. (b) Doping evolution of the coherent fraction of the Fermi surface $\left(f_c\right)$ in the three scenarios A–C. The $f_c^{\text{ARPES}}$ curve is deduced from $L_{arc}/L_{full}\left(T_c\right)=1-0.70\frac{T_c}{T^{\ast }}$ (see Ref. 41), expressed as a function of the doping level. $f_c^{\text{HC}}$ is extracted from the zero temperature specific heat coefficient $\gamma {\left(0\right)}_n$ (Refs. 24, 25).