Doping-dependent critical Cooper-pair momentum in thin, underdoped cuprate films

We apply a recently-developed low-field technique to inductively measure the critical pair momentum $p_c$ in thin, underdoped films of Y$_{1-x}$Ca$_{x}$Ba$_{2}$Cu$_{3}$O$_{7-delta}$ and Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+delta}$ reflecting a wide range of hole doping. We observe that $p_c propto hbar/xi$ scales with $T_c$ and therefore superfluid density $n_s(Trightarrow0)$ in our two-dimensional cuprate films. This relationship was famously predicted by a universal model of the cuprates with a textit{doping-independent} superconducting gap, but has not been observed by high field measurements of the coherence length $xi$ due to field-induced phenomena not included in the theory.