Doping-dependent critical Cooper-pair momentum $p_{c}$ in thin underdoped cuprate films

We apply a low-field ($l100$ G) technique to measure the critical pair momentum ${p}_{c}$ in thin underdoped films of ${mathrm{Y}}_{0.7}{mathrm{Ca}}_{0.3}{mathrm{Ba}}_{2}{mathrm{Cu}}_{3}{mathrm{O}}_{7ensuremath{-}ensuremath{delta}}$ and ${mathrm{Bi}}_{2}{mathrm{Sr}}_{2}{mathrm{CaCu}}_{2}{mathrm{O}}_{8+ensuremath{delta}}$ reflecting a wide range of hole doping. We observe that ${p}_{c}ensuremath{propto}ensuremath{hbar}/ensuremath{xi}$ scales with ${T}_{c}$ and therefore superfluid density ${n}_{s}(Tensuremath{rightarrow}0)$ in our two-dimensional cuprate films. This relationship was famously predicted by a universal model of the cuprates with a doping-independent superconducting gap, but has not been observed by high field $(g{10}^{4}phantom{rule{0.28em}{0ex}}mathrm{G})$ measurements of the coherence length $ensuremath{xi}$ because high-field cuprate phenomena obscure the simple scaling we observe at much lower fields.