Evidence of two-dimensional quantum critical behavior in the superfluid density of extremely underdoped Bi ₂Sr ₂CaCu ₂O _8+x

Evidence of two-dimensional (2D) quantum critical fluctuations is observed in the superfluid density n _s(T) λ ^-2(T) of extremely underdoped Bi ₂Sr ₂CaCu ₂O _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 ₂Cu ₃O _7-δ. 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.