Superheating fields of semi-infinite superconductors and layered superconductors in the diffusive limit: structural optimization based on the microscopic theory
Takayuki Kubo
Abstract
Abstract The superheating field <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">h</mml:mi> </mml:mrow> </mml:msub> </mml:math> of the Meissner state is thought to determine the theoretical field-limit of superconducting accelerator cavities. We investigate <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">h</mml:mi> </mml:mrow> </mml:msub> </mml:math> of semi-infinite superconductors and layered structures in the diffusive limit using the well-established quasiclassical Green’s function formalism of the BCS theory. The coupled Maxwell–Usadel equations are self-consistently solved to obtain the spatial distributions of the magnetic field, screening current density, penetration depth, pair potential, and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">h</mml:mi> </mml:mrow> </mml:msub> </mml:math> . For a semi-infinite superconductor in the diffusive limit, we obtain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">h</mml:mi> </mml:mrow> </mml:msub> <mml:mo>=</mml:mo> <mml:mn>0.795</mml:mn> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> at the temperature <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>T</mml:mi> <mml:mo stretchy="false">→</mml:mo> <mml:mn>0</mml:mn> </mml:math> . Here, <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi>c</mml:mi> <mml:mn>0</mml:mn> </mml:mrow> </mml:msub> </mml:math> is the thermodynamic critical-field at the zero temperature. By laminating a superconducting film (S) with the thickness <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>d</mml:mi> </mml:math> on a semi-infinite superconductor (Σ), we can engineer <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">h</mml:mi> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:mi>d</mml:mi> <mml:mo stretchy="false">)</mml:mo> </mml:math> of the layered structure. When <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mi>d</mml:mi> </mml:math> is the optimum thickness <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>m</mml:mi> </mml:msub> </mml:math> , <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mi mathvariant="normal">s</mml:mi> <mml:mi mathvariant="normal">h</mml:mi> </mml:mrow> </mml:msub> </mml:math> can be larger than that of the simple semi-infinite superconductors made from the S and Σ materials: <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mrow> <mml:msub> <mml:mi>H</mml:mi> <mml:mrow> <mml:mtext>sh</mml:mtext> </mml:mrow> </mml:msub> <mml:mo stretchy="false">(</mml:mo> <mml:msub> <mml:mi>d</mml:mi> <mml:mi>m</mml:mi> </mml:msub> <mml:mo stretchy="false">)</mml:mo> <mml:mtext> ></mml:mtext> <mml:mi>max</mml:mi> <mml:mo>{</mml:mo> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mrow> <mml:mtext>sh</mml:mtext> </mml:mrow> <mml:mrow> <mml:mo stretchy="false">(</mml:mo> <mml:mtext>S</mml:mtext> <mml:mo stretchy="false">)</mml:mo> </mml:mrow> </mml:msubsup> <mml:mo>,</mml:mo> <mml:msubsup> <mml:mi>H</mml:mi> <mml:mrow> <mml:mtex