Universal scaling of length, time, and energy for cuprate superconductors based on photoemission measurements of Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+\delta }$

A microscopic scaling relation linking the normal and superconducting states of the cuprates in the presence of a pseudogap is presented using angle-resolved photoemission spectroscopy. This scaling relation, complementary to the bulk universal scaling relation embodied by Homes’ law, explicitly connects the momentum-dependent amplitude of the $d$-wave superconducting order parameter at $T\sim 0$ to quasiparticle scattering mechanisms operative at $T\gtrsim T_c$. The form of the scaling is proposed to be a consequence of the marginal Fermi-liquid phenomenology and the inherently strong dissipation of the normal pseudogap state of the cuprates.

Figure 1

Raw ARPES spectra of OP91 (a) and UD70 (b) at the tip of the Fermi arc ${\phi }_c$ in the normal state just above $T_c$. Low-temperature spectra ($T=15$ K), after deconvolution, are shown for the OP91 sample in (c) and the UD70 sample in (d). (e) and (f) show the $T_c$ and $T=15$ K symmetrized EDCs for OP91 and UD70, respectively, as well as Lorentzian fits thereof. The energy scale of the low-temperature EDCs in (e) and (f) have been scaled up by a factor of $\pi$ to better illustrate Eq. (1) evaluated at $\phi ={\phi }_c$ and the intensity scaled to half that of the normal-state EDCs. The inset of (e) illustrates the Fermi-surface angle $\phi$ in relation to the normal-state Fermi surface and the cuts represented by the current experiment (solid lines).

Figure 2

(a) Superconducting gap vs Fermi-surface angle at $T=15$ K for UD70 (black squares) and OP91 (red circles). The gap is measured as the distance of the coherence peak in the superconducting state to $E_F$. (b) Inverse lifetime at $T\sim T_c$ vs Fermi-surface angle for UD70 (black squares) and OP91 (red circles). Fits described in the text are shown as dotted lines extrapolated toward the antinodal region (Ref. 17). All error bars are derived from the fits.

Figure 3

(a) Plot of $R\left(\phi \right)$ for OP91 (red circles) and UD70 Bi2212 (black squares). Error bars are formally propagated from those in Fig. 2. $R\left(\phi \right)$ obtained from the fits is superimposed on the data.

Figure 4

Summary of available ARPES data from the current study as well as Ref. 19 for $R\left(\psi \right)$. ${\phi }_c$ is taken in every case to simply be the last point to which a lifetime at the Fermi level could be reasonably ascertained. Error bars in the abscissa have been suppressed.