Quantum critical behaviour in the superfluid density of strongly underdoped ultrathin cuprate films

A central issue in the physics of high temperature superconductors is to understand superconductivity within a single copper-oxide layer or bilayer, the fundamental structural unit in the cuprates, and how it is lost with underdoping. As mobile holes are removed from the CuO_2 planes, the transition temperature T_C and superfluid density n_S decrease in a surprisingly correlated fashion in crystals and thick films. We seek to elucidate the intrinsic physics of bilayers in the strongly underdoped regime, near the critical doping level where superconductivity disappears. We report measurements of n_S(T) in films of Y_{1-x}Ca_xBa_2Cu_3O_{7-\delta} as thin as two copper-oxide bilayers with T_C's as low as 3 K. In addition to seeing the two-dimensional (2D) Kosterlitz-Thouless-Berezinski transition at T_C, we observe a remarkable scaling of T_C with n_S(0) that demonstrates that the disappearance of superconductivity with underdoping is due to quantum fluctuations near a T = 0 2D quantum critical point.