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$H_{C_{2}}$ and the third critical field HC3$H_{C_{3}}$. 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.

中文翻译：

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半空间中 3D 表面超导的浓度行为和晶格结构

我们在半空间中研究了 3D Ginzburg-Landau 模型，该模型有望捕捉第二临界场 HC2$H_{C_{2}}$ 和第三临界场 HC3$ 之间外加磁场的表面超导性的关键特征H_{C_{3}}$。对于这种情况下的磁场，从物理学中可知，超导性应该基本上被限制在沿样品边界的薄层中。这导致在半空间上引入 Ginzburg-Landau 模型。我们证明了恒定磁场半空间上的非线性Ginzburg-Landau能量是磁场与边界所成角度ν的递减函数。在磁场与边界相切的情况下（ν = 0），我们表明，通过在垂直于边界的方向上最小化简化的一维泛函，能量被确定为领先。对于非平行外加场，我们还构建了一个周期性问题，涡晶格最小化器再现了有效能量，这表明完整的 Ginzburg-Landau 模型的阶参数将在样品表面附近表现出 3 维涡结构。