Evidence of two-dimensional (2D) quantum critical fluctuations is observed in the superfluid density $n_s$($T$) ∝ λ${}^{-2}$($T$) of extremely underdoped Bi${}_2$Sr${}_2$CaCu${}_2$O${}_{8+x}$ (Bi-2212) films, indicating that, in general, quantum fluctuations play a dominant role in underdoped cuprates. Two-dimensional (2D) fluctuations are expressed by the linear scaling $T_c$ ∝ $n_s$(0). 2D scaling in Bi-2212 contrasts with three-dimensional scaling that is seen in the much less anisotropic YBa${}_2$Cu${}_3$O${}_{7-\delta }$. Quantum critical fluctuations could also account for the absence of thermal critical behavior in λ${}^{-2}$($T$) of strongly underdoped Bi-2212 samples, $T_c<48$ K.

• Received 17 October 2011

DOI:https://doi.org/10.1103/PhysRevB.85.180507