  We study the 2D full Ginzburg-Landau energy with a periodic rapidly
oscillating, discontinuous and [strongly] diluted pinning term using a
perturbative argument. This energy models the state of an heterogeneous type II
supercon-ductor submitted to a magnetic field. We calculate the value of the
first critical field which links the presence of vorticity defects with the
intensity of the applied magnetic field. Then we prove a standard dependance of
the quantized vorticity defects with the intensity of the applied field. Our
study includes the case of a London solution having several minima. The
macroscopic location of the vor-ticity defects is understood with the famous
Bethuel-Brezis-H{\'e}lein renormalized energy. The mesoscopic location, i.e.,
the arrangement of the vorticity defects around the minima of the London
solution, is the same than in the homogenous case. The microscopic location is
exactly the same than in the heterogeneous case without magnetic field. We also
compute the value of secondary critical fields that increment the quantized
vorticity. 1. Introdution This article studies the pinning phenomenon in
type-II superconducting composites. Superconductivity is a property that
appears in certain materials cooled below a critical temperature. These
materials are called superconductors. Superconductivity is characterized by a
total absence of electrical resistance and a perfect diamagnetism.
Unfortunately, when the imposed conditions are too intense, superconductivity
is destroyed in certain areas of the material called vorticity defects. We are
interested in type II superconductors which are characterized by the fact that
the vorticity defects first appear in small areas. Their number increases with
the intensity of the conditions imposed until filling the material. For
example, when the intensity h ex of an applied magnetic field exceeds a first
threshold, the first vorticity defects appear: the magnetic field begins to
penetrate the superconductor. The penetration is done along thin wires and may
move resulting an energy dissipation. These motions may be limited by trapping
the vorticity defects in small areas. The behavior of a superconductor is
modeled by minimizers of a Ginzburg-Landau type energy. In order to study the
presence of traps for the vorticity defects we consider an energy including a
pinning term that models impurities in the supercon-ductor. These impurities
would play the role of traps for the vorticity defects. We are thus lead to the
subject of this article: the type-II superconducting composites with
impurities.
