  We report on measurements of the low temperature dependence of the magnetic
penetration depth lambda_ab in several single crystals of La_2-xSr_xCuO_4,
Bi_2Sr_2CaCu_2O_8 and YBa_2Cu_3O_6+x at various doping levels ranging from the
under- to the overdoped regimes by using a novel single-coil technique
achieving 10 pm resolution. We have found a linear dependence of lambda_ab in
all samples with a rapidly increasing slope dlambda_ab/dT as doping decreases.
Our analysis of the data indicates that a superconducting gap with d-wave
symmetry is sufficient to quantitatively account for the above slope values in
the optimally or over- doped samples. In the underdoped samples, the d-wave
model predicts much smaller values than those measured by assuming realistic
values for the zero-temperature lambda_ab and gap delta. The experimental data
are compatible with a model of thermodynamic phase fluctuations of the order
parameter. Therefore, we put forward the hypothesis that the gapless properties
observed in cuprates may have qualitatively different physical origins
depending on the doping level.
