Approximate tight-binding sum rule for the superconductivity-related change of c-axis kinetic energy in multilayer cuprate superconductors

We present an extension of the c-axis tight-binding sum rule discussed by Chakravarty, Kee, and Abrahams [Phys. Rev. Lett. 82, 2366 (1999)] that applies to multilayer high-$T_c$ cuprate superconductors (HTCS) and use it to estimate—from available infrared data—the change below $T_c$ of the c-axis kinetic energy, $〈H_c〉,$ in ${\mathrm{YBa}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{7-\delta }$ $(\delta \approx 0.45,0.25,0.07),$ ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8,$ and ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{Ca}}_2{\mathrm{Cu}}_3{\mathrm{O}}_{10}.$ In all these compounds $〈H_c〉$ decreases below $T_c$ and except for ${\mathrm{Bi}}_2{\mathrm{Sr}}_2{\mathrm{CaCu}}_2{\mathrm{O}}_8$ the change of $〈H_c〉$ is of the same order of magnitude as the condensation energy. This observation supports the hypothesis that in multilayer HTCS superconductivity is considerably amplified by the interlayer tunneling mechanism.