Surface superconductivity in presence of corners

The aim of this Thesis is to study the response of a type-II superconducting wire with non-smooth cross section to an external time-independent magnetic field hex parallel to it and with intensity varying in a certain regime. Superconductivity is a well known quantum critical phenomenon in which the electrons arrange in pairs, known as Cooper pairs, thanks to the relative attraction mediated by the crystal of ions. Superconductors can be divided into two types, according to how the breakdown of superconductivity occurs. For type-I, superconductivity is abruptly destroyed via a first order phase transition. In 1957 Abrikosov deduced the existence of a class of materials which exhibit a different behaviour, i.e., some of their superconducting properties are preserved when submitted to a suitably large magnetic field. Physically, these two classes can be identified by the value of a parameter κ, also known as the Ginzburg-Landau parameter, which is proportional to the inverse of the penetration depth. We consider in this work extreme type-II superconductors, i.e., we assume that the Ginzburg-Landau parameter satisfies κ ≫ 1. It is possible to describe the phase transitions in a type-II superconductor by identifying three increasing critical values of the magnetic field. When the first critical value Hc1 is reached, superconductivity is lost in the bulk of the sample at isolated points. Between the second and third critical fields, i.e., in the regime Hc2 ≤ hex ≤ Hc3, superconductivity survives only close to the boundary of the sample, as it was predicted by Saint-James and de Gennes in the 60’s and later observed in experiments. Above the third critical field Hc3, the sample comes back to its normal state. In this Thesis we focus on the surface superconductivity regime: in this regime superconductivity is confined near the boundary of the sample. We will perform our investigation in the framework of Ginzburg-Landau (GL) theory. It is indeed well known that close enough to the critical temperature Tc, GL theory provides an accurate description of the physics of superconductivity, despite having been formulated as a phenomenological theory and only later justified in terms of the microscopic BCS theory (BCS stands for Bardeen, Cooper, Schrieffer). Since in the regime of interest superconductivity survives only close to the boundary of the sample, a natural question is: how does the physics depend on the geometry of the boundary? The aim of this Thesis is to consider domains for which there are singularities along the boundary. We prove that for this domains, in the surface superconductivity regime, the leading order of the energy density is not affected by the presence of corners and that the density of Cooper pairs in the equilibrium state is approximately constant along the transversal direction. This implies that superconductivity is uniformly distributed near the boundary, at least to leading order. In addition, we introduce a new effective problem near the corner that allows us to prove a refined asymptotics and to isolate the contributions to the energy density due to the presence of corners. The explicit expression of the effective energy is yet to be found but we formulate a conjecture on it based on the behavior for almost flat angles. Indeed, for corners with angles close to π, we are able to explicitly compute the leading order of the corners effective problem and show that it sums up to the smooth boundary contribution to reconstruct the same asymptotics as in smooth domains.