Two-dimensional Quantum Critical Point in Underdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta }$

Underdoped Bi$_{2}$Sr$_{2}$CaCu$_{2}$O$_{8+\delta }$ films with T$_{c}$'s from 5 to 70 K are fabricated by sputtering and Pulsed Laser Deposition (PLD). Temperature dependences of superfluid densities are measured to study the superconductor-to-insulator quantum phase transition. Sputtered films, which tend have higher dopings, show superfluid densities that are weakly linear in T at low-T and drop dramatically where Kosterlitz-Thouless-Berezinski theory predicts, assuming that individual CuO$_{2 }$bilayers are uncoupled. However, our PLD films, which are more underdoped than the sputtered films, have superfluid densities that are roughly linear from low T to T$_{c}$. Also, There is no indication of thermal critical behavior near T$_{c}$. Underdoped YBCO crystals also lack critical behavior, even though critical behavior is strong in optimally doped and moderately underdoped samples. Near the superconductor-to-insulator phase transition, T$_{c}$ and n$_{s}$(0) have a linear relationship that mimics that of ultrathin, two-dimensional films of Ca-doped YBa$_{2}$Cu$_{3}$O$_{7-\delta }$, thereby indicating a 2-D quantum critical point at low doping.