  The energy needed to create a vortex core is the basic ingredient to address
the physics of thermal vortex fluctuations in underdoped cuprates. Here we
theoretically investigate its role on the occurrence of the
Beresinskii-Kosterlitz-Thouless transition in a bilayer film with
inhomogeneity. From the comparison with recent measurements of the penetration
depth in two-unit cell thin films of
Y$_{1-x}$Ca$_{x}$Ba$_{2}$Cu$_{3}$O$_{7-\d}$ (YBCO) by Hetel et al. [Nat. Phys.
3, 700 (2007)] we can extract the value of the vortex-core energy $\mu$, and
show that $\mu$ scales linearly with $T_c$ at low doping.
