  Vortices in a thin-film superconductor interact logarithmically out to a
distance on the order of the two-dimensional (2D) magnetic penetration depth
$\lambda_\perp$, at which point the interaction approaches a constant. Thus,
because of the finite $\lambda_\perp$, the system exhibits what amounts to an
{\it intrinsic} finite size effect. It is not described by the 2D Coulomb gas
but rather by the 2D Yukawa gas (2DYG). To study the critical behavior of the
2DYG, we map the 2DYG to the massive sine-Gordon model and then perform a
renormalization group study to derive the recursion relations and to verify
that $\lambda_\perp$ is a relevant parameter. We solve the recursion relations
to study important physical quantities for this system including the
renormalized stiffness constant and the correlation length. We also address the
effect of current on this system to explain why finite size effects are not
more prevalent in experiments given that the 2D magnetic penetration depth is a
relevant parameter.
