Energy scale directly related to superconductivity in high-$T_c$ cuprates: Universality from the temperature-dependent angle-resolved photoemission of Bi${}_2$Sr${}_2$Ca${}_2$Cu${}_3$O${}_{10+\delta }$

We have performed a temperature-dependent angle-resolved photoemission spectroscopy (ARPES) study of the trilayer high-$T_c$ cuprate superconductor (HTSC) Bi${}_2$Sr${}_2$Ca${}_2$Cu${}_3$O${}_{10+\delta }$ (Bi2223), and have shown that the $T_c$ (=110 K) and “effective” superconducting gap ${\Delta }_{\mathrm{sc}}$ defined at the end point of the Fermi arc follow the relationship 2${\Delta }_{\mathrm{sc}}\approx 4k_{\mathrm{B}}{T}_c$. Combining this result with previous ARPES results on single- and double-layer cuprates, we show that the relationship 2${\Delta }_{\mathrm{sc}}\approx 4k_{\mathrm{B}}{T}_c$ holds for various HTSCs.

Figure 1

ARPES spectra of optimally doped Bi2223 ($T_c=110$ K) in the SC ($T=10$ K) and normal ($T=115$ K) states. (a), (b) ARPES intensity plots divided by the Fermi-Dirac function convoluted with the resolution function along the cuts shown by red lines in inset across the FS for the OP and IP bands, where OP and IP stand for the outer and inner CuO${}_2$ planes, respectively.

Figure 2

Energy gaps for OP and IP of optimally doped Bi2223 below and above $T_c$. (a)–(d) Symmetrized energy distribution curves (EDCs) at Fermi momenta $k_{\mathrm{F}}$'s for the OP and IP bands in the SC $(T=10$ K) and normal ($T=115$ K) states. $k_{\mathrm{F}}$'s have been determined by the minimum gap locus (Ref. 3). (e), (f) The values of the energy gaps shown in (a)–(d) are plotted as functions of the $d$-wave order parameter $|\cos \left(k_{x}a\right)-\cos \left(k_{y}a\right)|$/2. Slightly above $T_c$ (= 115 K), the gap closes in the nodal region (curves 1–10 for OP and 1–5 for IP), while it remains open in the anti-nodal region (curves 11–18 for the OP and 6–11 for the IP) (Ref. 29). The effective SC gap ${\Delta }_{\mathrm{sc}}$ is defined by a gap at the end point of the Fermi arc. (g), (h) Schematic figures of the energy gaps for OP and IP. The energy gap near the nodal gap ${\Delta }_0$, anti-nodal gap ${\Delta }^{*}$, Fermi arc length $K_a$ and effective SC gap ${\Delta }_{\mathrm{sc}}$ are shown.

Figure 3

Relationship between the nodal gap ${\Delta }_0$, the Fermi arc length $K_a$, the effective SC gap ${\Delta }_{\mathrm{sc}}$, and $T_c$ for various HTSCs. (a) ${\Delta }_0$ plotted against $T_c$. The ${\Delta }_0$ data have been taken from the ARPES results for LSCO,[1, 21] Bi2201,[4] Bi2212,[8, 12, 22, 23] and Ba${}_2$Ca${}_3$Cu${}_4$O${}_8$F${}_2$ (F0234).[24] Arrows indicate the decrease of hole concentration for each of $n=1$, 2, and 3. (b) $K_a$ relative to that of the full Fermi surface 2$\pi k_{\mathrm{F}}$ as a function of hole concentrations. Dashed line is a guide to the eye. (c) Effective SC gap ${\Delta }_{\mathrm{sc}}$ plotted against $T_c$. The dashed line indicates $2\Delta =4k_{\mathrm{B}}{T}_c$.

Figure 4

Quasi-particle (QP) width $\Gamma$ (HWHM) at various temperatures. QP scattering rate $\Gamma$ on the FS of OP and IP is plotted as a function of $d$-wave order parameter [(a) OP, (b) IP]. Blue dashed lines in (a) and (b) indicate the nodal SC gap ${\Delta }_0$. The Fermi arc appears in the momentum region where the gap magnitude $\Delta \left(k\right)\le \Gamma$ is safisfied just above $T_c$, 115 K. The crossing point of the SC gap curve and the $\Gamma$(115 K) curve yields the end point of the Fermi arc and ${\Delta }_{\mathrm{sc}}$. Inset shows the temperature dependence of $\Gamma$ in the nodal region. The experimental energy resolution $\Delta E$ is indicated. (c) Schematic figure showing the relationship between $K_a$, $\Gamma$ and $T_c$ for HTSCs in the SC and normal states. Blue dashed line shows the energy gap in the SC state, and red solid and dotted curves indicate the Fermi arc formation just above $T_c$. Gray region shows the energy at which the width $\Gamma$($T_c$) of QP at $T_c$ falls. The energy gap within the gray area collapses just above $T_c$ in the normal state.