Doping-dependent Critical Cooper-pair Momentum in Thin Underdoped Cuprate Films

We apply a low-field ($<100$ G) technique to measure the critical pair momentum ${p}_{c}$ in thin underdoped films of ${\mathrm{Y}}_{0.7}{\mathrm{Ca}}_{0.3}{\mathrm{Ba}}_{2}{\mathrm{Cu}}_{3}{\mathrm{O}}_{7\ensuremath{-}\ensuremath{\delta}}$ 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 ${n}_{s}(T\ensuremath{\rightarrow}0)$ 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 $(>{10}^{4}\phantom{\rule{0.28em}{0ex}}\mathrm{G})$ measurements of the coherence length $\ensuremath{\xi}$ because high-field cuprate phenomena obscure the simple scaling we observe at much lower fields.