Energy and vorticity of the Ginzburg–Landau model with variable magnetic field

Abstract

We consider the Ginzburg-Landau functional with a variable applied magnetic field in a bounded and smooth two dimensional domain. The applied magnetic field varies smoothly and is allowed to vanish non-degenerately along a curve. Assuming that the strength of the applied magnetic field varies between two characteristic scales, and the Ginzburg-Landau parameter tends to +∞, we determine an accurate asymptotic formula for the minimizing energy and show that the energy minimizers have vortices. The new aspect in the presence of a variable magnetic field is that the density of vortices in the sample is not uniform.

The Ginzburg–Landau Functional with Vanishing Magnetic Field

“…As such, the assumption on the magnetic field in Theorem 1.1 is significant when b(κ)κ ≤ H ≤ M κ 2 and M ∈ (0, c 0 ] is a constant. Note also that our theorem gives a bridge between the situations studied by Attar in [1,2] and Pan-Kwek in [20]. Remark 1.4.…”

“…Earlier results corresponding to vanishing magnetic fields have been obtained recently in [1,2]. The assumption on the strength of the magnetic field was H ≤ Cκ, where C is a constant.…”

The density of superconductivity in the bulk regime

“…Again, if b ∈ [ǫ, 1) for some ǫ > 0 , the constants κ 0 and C can be selected independently from b, but they will depend on ǫ . More details can be found in [5,6], where it is allowed for ǫ to depend on κ, ǫ = ǫ(κ), and approach 0 as κ → +∞ .…”

On the Ginzburg–Landau energy with a magnetic field vanishing along a curve

“…• If H C 1 < H < H C 2 , the superconductor is in the mixed phase, where both the superconducting and normal states co-exist in the bulk of the sample; the most interesting aspect of the mixed phase is that the region with the normal state appears in the form of a lattice of point defects, covering the whole bulk of the sample [25] ; • If H C 2 < H < H C 3 , superconductivity disappears in the bulk but survives on the surface of the superconductor ; • If H > H C 3 , superconductivity is destroyed and the superconductor returns to the normal state . The case of a non-uniform sign changing magnetic field has been addressed first in [23] then recently in [4,5,6,17,19]. In the presence of such magnetic fields, the behavior of the superconductor (and the associated critical magnetic fields) differ significantly from the case of a uniform 1 applied magnetic field.…”

“…The asymptotic analysis of E gs (κ, H) has been carried for other regimes of the magnetic field strength, down to H ≈ κ 1/3 , in [4,5,19]. The case where the function B 0 is only Hölder continuous or a step function has been discussed in [17,3].…”