High-precision measurement of the Ferrell-Glover-Tinkham sum rule in a cuprate high-temperature superconductor

The Ferrell-Glover-Tinkham (FGT) sum rule in superconductors defines the superfluid density, ${\rho }_s$, as the optical conductivity spectral weight (SW) that transfers into a $\delta$ function at $\omega =0$ due to the opening of the energy gap below $T_c$. In high-$T_c$ superconductors, strong electron-boson coupling, self-energy effects, and intertwining of energy scales can link ${\rho }_s$ to various high-energy processes, making the question of whether or not the FGT sum rule is valid in cuprates, and at what energy scale, central to a full understanding of the pairing mechanism. Here, we report high-precision measurements of the FGT sum rule in near-optimally doped $DyB{a}_{2}C{u}_3{O}_{7-\delta }$ thin films. We resolve the low-energy balance of SW by combining submillimeter-microwave interferometry, terahertz time-domain spectroscopy, and infrared ellipsometry to independently obtain the real and imaginary parts of the complex dielectric function between 0.8 meV and 1.1 eV (6–9000 cm${}^{-1}$). By applying a Kramers-Kronig consistency analysis to the measured spectra we find that the FGT sum rule is obeyed, and the total intraband SW is conserved to within $\pm 0.2$ below an energy scale $\sim 0.6$ eV. ... mehrWe attribute specific anomalies observed in the conductivity spectra below $\sim 0.6$ eV to coupling of charge carriers to the spectrum of collective antiferromagnetic spin fluctuations. The procedure presented here, applied to near-optimally doped $DyB{a}_{2}C{u}_3{O}_{7-\delta }$, lays out a protocol for how the FGT sum rule should be studied in other doping levels and compounds.