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

We apply a low-field (\textless 100 G) technique to measure the critical Cooper pair momentum $p_{c}$ in thin, underdoped films of Y$_{\mathrm{0.7}}$Ca$_{\mathrm{0.3}}$Ba$_{\mathrm{2}}$Cu$_{\mathrm{3}}$ O$_{\mathrm{7-\delta }}$ and Bi$_{\mathrm{2}}$Sr$_{\mathrm{2}}$CaCu$_{\mathrm{2}}$O$_{\mathrm{8+\delta }}$, where doping is effected by adjusting the oxygen stoichiometry through post-deposition annealing. The technique is based on applying a perpendicular magnetic field to the center of a superconducting film and measuring the field at which screening of the field catastrophically fails. Theory together with measurements on thin films of conventional superconductors Nb and MoGe argue for the validity of the technique. In underdoped cuprates, spectroscopy identifies multiple characteristic energy scales, e.g., the pseudogap and the ``nodal'' gap, neither of which is proportional to $T_{c}$. On general grounds, we expect to find that $p_{c} \propto 1 \mathord{\left/ {\vphantom {1 \xi }} \right. \kern-\nulldelimiterspace} \xi $ is proportional to the characteristic superconducting energy scale. We observe that $p_{c} \propto T_{c} $ as $T_{c}$ decreases with underdoping, identifying $k_{B} T_{c} $ as the characteristic energy. While this result is trivial in conventional superconductors whose spectroscopic gaps are proportional to $T_{c}$, it is significant in cuprates.