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Doping dependence of the low-energy excitations in superconductingBi2Sr2CaCu2O8+δ:Evidence for thermal phase fluctuations

By using a single-coil technique, we study the low-temperature dependence of the in-plane magnetic penetration depth λ a b in several Bi 2 Sr 2 CaCu 2 O 8 + δ (BSCCO) single crystals prepared with various doping levels δ ranging from the under- to the overdoped regime. Four samples exhibit a linear dependence of thermally activated low-energy excitations Δλ a b ≡ λ a b (T) - λ a b (0) ∼ T. The thermal activation rate ∂λ a b /∂T is minimum, 8 A/K, at optimum doping, in good quantitative agreement with the d-wave model of superconducting order parameter. The much larger rates,? 40 A/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 coherence length like cuprates. One underdoped sample exhibits a quadratic dependence suggesting an incipient crossover from thermal to quantum fluctuations.