斜磁场下Lawrence-Doniach泛函的极小值
Minimizers of the Lawrence-Doniach Functional with Oblique Magnetic Fields
摘要：
我们研究了层状结构超导体的Lawrence-Doniach能量的极小化，假定Floquet-周期边界条件。具体地说，我们考虑了在超导平面上倾斜施加恒定磁场的影响，在假设$s=\eps^α$与$0<α<1$的条件下，将其作为层间距$s\to 0$和Ginzburg-Landau参数$κ=\eps^{-1}\to\infty$。通过导出极小元能量的精确匹配上下界，我们确定了下临界场和通量格的方向，使之在参数$\eps$中达到最优阶.对于首阶，诱导场用$\RR^3$中的凸极小化问题表征。我们观察到一个“磁通锁定过渡”，其中磁通线被钉扎到水平方向的小倾角的应用领域，这是不存在于各向异性的金兹伯格-朗道模型的极小化。我们获得的能量分布表明存在“阶梯涡”，这在物理学文献中已定性描述。
英文摘要：
We study minimizers of the Lawrence--Doniach energy, which describes equilibrium states of superconductors with layered structure, assuming Floquet-periodic boundary conditions. Specifically, we consider the effect of a constant magnetic field applied obliquely to the superconducting planes in the limit as both the layer spacing $s\to 0$ and the Ginzburg--Landau parameter $κ=\eps^{-1}\to\infty$, under the hypotheses that $s=\eps^α$ with $0<α<1$. By deriving sharp matching upper and lower bounds on the energy of minimizers, we determine the lower critical field and the orientation of the flux lattice, to leading order in the parameter $\eps$. To leading order, the induced field is characterized by a convex minimization problem in $\RR^3$. We observe a ``flux lock-in transition'', in which flux lines are pinned to the horizontal direction for applied fields of small inclination, and which is not present in minimizers of the anisotropic Ginzburg--Landau model. The energy profile we obtain suggests the presence of ``staircase vortices'', which have been described qualitatively in the physics literature.
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