  Mechanisms of vortex dynamics and pinning and self-field effects which could
account for the thickness dependence of the critical current density J_c of
superconducting films are addressed. It is shown that at low magnetic fields B,
the 2D single-vortex pinning and thermal fluctuations yield J_c proportional to
(1/sqrt{d})exp[-(d_m/d)^{1/2}], if the film thickness d is smaller than the
pinning correlation length along B. The model describes the dependence of J_c
on d observed on YBa_2Cu_3O_7 coated conductors for d < 2(mu)m. Measurements of
J_c(d) in strong magnetic fields are proposed to probe whether the dependence
of J_c on d is mostly determined by the 2D pinning or changes in the material
microstructure as the film gets thicker.
