Doping dependence of the low-energy excitations in superconducting Bi2Sr2CaCu2O8-d : Evidence for thermal phase fluctuations

By using a single-coil technique, we study the low-temperature dependence of the in-plane magnetic pen- etration depth l ab in several Bi 2 Sr 2 CaCu 2 O 8 1d ~ BSCCO ! single crystals prepared with various doping levels d ranging from the under- to the overdoped regime. Four samples exhibit a linear dependence of thermally activated low-energy excitations Dl ab [l ab (T) 2l ab (0) ; T. The thermal activation rate }] l ab / ] T is mini- mum, 8 Å/K, at optimum doping, in good quantitative agreement with the d-wave model of superconducting order parameter. The much larger rates, * 40 Å/K, observed in all under- and overdoped samples indicate that, in these samples, another type of low-energy excitations is relevant. These large rates are quantitatively consistent with a model of thermal phase fluctuations suitable for granular superconductors with short coher- ence length like cuprates. One underdoped sample exhibits a quadratic dependence suggesting an incipient crossover from thermal to quantum fluctuations.