Concentration Behavior and Lattice Structure of 3D Surface Superconductivity in the Half Space

TY - JOUR

T1 - Concentration Behavior and Lattice Structure of 3D Surface Superconductivity in the Half Space

AU - Fournais, Søren

AU - Miqueu, Jean Philippe

AU - Pan, Xing Bin

PY - 2019

Y1 - 2019

N2 - We study a 3D Ginzburg-Landau model in a half-space which is expected to capture the key features of surface superconductivity for applied magnetic fields between the second critical field HC2 and the third critical field HC3. For the magnetic field in this regime, it is known from physics that superconductivity should be essentially restricted to a thin layer along the boundary of the sample. This leads to the introduction of a Ginzburg-Landau model on a half-space. We prove that the non-linear Ginzburg-Landau energy on the half-space with constant magnetic field is a decreasing function of the angle ν that the magnetic field makes with the boundary. In the case when the magnetic field is tangent to the boundary (ν = 0), we show that the energy is determined to leading order by the minimization of a simplified 1D functional in the direction perpendicular to the boundary. For non-parallel applied fields, we also construct a periodic problem with vortex lattice minimizers reproducing the effective energy, which suggests that the order parameter of the full Ginzburg-Landau model will exhibit 3 dimensional vortex structure near the surface of the sample.

AB - We study a 3D Ginzburg-Landau model in a half-space which is expected to capture the key features of surface superconductivity for applied magnetic fields between the second critical field HC2 and the third critical field HC3. For the magnetic field in this regime, it is known from physics that superconductivity should be essentially restricted to a thin layer along the boundary of the sample. This leads to the introduction of a Ginzburg-Landau model on a half-space. We prove that the non-linear Ginzburg-Landau energy on the half-space with constant magnetic field is a decreasing function of the angle ν that the magnetic field makes with the boundary. In the case when the magnetic field is tangent to the boundary (ν = 0), we show that the energy is determined to leading order by the minimization of a simplified 1D functional in the direction perpendicular to the boundary. For non-parallel applied fields, we also construct a periodic problem with vortex lattice minimizers reproducing the effective energy, which suggests that the order parameter of the full Ginzburg-Landau model will exhibit 3 dimensional vortex structure near the surface of the sample.

KW - 3D vortices

KW - Calculus of variation

KW - Estimate of eigenvalue

KW - Ginzburg-Landau equations

KW - Magnetic Schrodinger operator

KW - Partial differential equations

KW - Superconductivity

KW - Surface concentration

KW - Surface superconductivity

UR - http://www.scopus.com/inward/record.url?scp=85063161683&partnerID=8YFLogxK

U2 - 10.1007/s11040-019-9307-7

DO - 10.1007/s11040-019-9307-7

M3 - Journal article

AN - SCOPUS:85063161683

SN - 1385-0172

VL - 22

JO - Mathematical Physics Analysis and Geometry

JF - Mathematical Physics Analysis and Geometry

IS - 2

M1 - 12

ER -