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 $p_c\propto \mathrm{\hslash }\mathrm{/}\xi$ scales with $T_c$ and therefore superfluid density $n_s(T\to 0)$ in our two-dimensional cuprate films. This relationship was famously predicted by a universal model of the cuprates with a $\mathit{\text{doping-independent}}$ 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.