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We study the 2D full Ginzburg-Landau energy with a periodic rapidlyoscillating, discontinuous and [strongly] diluted pinning term using aperturbative argument. This energy models the state of an heterogeneous type IIsupercon-ductor submitted to a magnetic field. We calculate the value of thefirst critical field which links the presence of vorticity defects with theintensity of the applied magnetic field. Then we prove a standard dependance ofthe quantized vorticity defects with the intensity of the applied field. Ourstudy includes the case of a London solution having several minima. Themacroscopic location of the vor-ticity defects is understood with the famousBethuel-Brezis-H{\'e}lein renormalized energy. The mesoscopic location, i.e.,the arrangement of the vorticity defects around the minima of the Londonsolution, is the same than in the homogenous case. The microscopic location isexactly the same than in the heterogeneous case without magnetic field. We alsocompute the value of secondary critical fields that increment the quantizedvorticity. 1. Introdution This article studies the pinning phenomenon intype-II superconducting composites. Superconductivity is a property thatappears in certain materials cooled below a critical temperature. Thesematerials are called superconductors. Superconductivity is characterized by atotal absence of electrical resistance and a perfect diamagnetism.Unfortunately, when the imposed conditions are too intense, superconductivityis destroyed in certain areas of the material called vorticity defects. We areinterested in type II superconductors which are characterized by the fact thatthe vorticity defects first appear in small areas. Their number increases withthe intensity of the conditions imposed until filling the material. Forexample, when the intensity h ex of an applied magnetic field exceeds a firstthreshold, the first vorticity defects appear: the magnetic field begins topenetrate the superconductor. The penetration is done along thin wires and maymove resulting an energy dissipation. These motions may be limited by trappingthe vorticity defects in small areas. The behavior of a superconductor ismodeled by minimizers of a Ginzburg-Landau type energy. In order to study thepresence of traps for the vorticity defects we consider an energy including apinning term that models impurities in the supercon-ductor. These impuritieswould play the role of traps for the vorticity defects. We are thus lead to thesubject of this article: the type-II superconducting composites withimpurities.

Magnetic Ginzburg-Landau energy with a periodic rapidly oscillating and diluted pinning term