Critical current density of superconducting-normal-superconducting Josephson junctions and polycrystalline superconductors in high magnetic fields

We investigate the in-field critical current density ${J}_{\text{c}}(B)$ of superconducting-normal-superconducting (SNS) Josephson junctions (JJs) and polycrystalline superconducting systems with grain boundaries modeled as Josephson-type planar defects, both analytically and through computational time-dependent Ginzburg-Landau (TDGL) simulations in two and three dimensions. For very narrow SNS JJs, we derive analytic expressions for ${J}_{\text{c}}(B)$ that are high-field solutions for ${J}_{\text{c}}(B)$ for JJs across the entire applied field range up to the effective upper critical field ${B}_{\text{c2}}^{*}$. They generalize the well-known (low-field) exponential junction thickness dependence for ${J}_{\text{c}}$ from de Gennes, often used in the Josephson relation. We then extend our analytic expressions to describe wider junctions using physical arguments, and we confirm their agreement with TDGL simulations. These results are then compared with the current densities found in superconductors optimized for high-field applications. They provide an explanation for the Kramer field dependence and inverse power-law grain size dependence widely found in many low-temperature superconductors, and the power-law field dependence ${J}_{\text{c}}(B)\ensuremath{\sim}{B}^{\ensuremath{-}0.6}$ found at intermediate fields in some high-temperature superconductors including powder-in-tube ${\text{Bi}}_{2}{\text{Sr}}_{2}{\text{Ca}}_{2}{\text{Cu}}_{3}{\text{O}}_{x}$ and $R{\text{Ba}}_{2}{\text{Cu}}_{3}{\text{O}}_{7}$ tapes ($R=\text{rare}\phantom{\rule{4.pt}{0ex}}\text{earth}$). By reanalyzing critical current density data using the mathematical framework derived here and confirmed using TDGL, we enable an analysis of ${J}_{\text{c}}$ data that provides the local properties of grain boundaries in high-field superconductors and hence a deeper understanding of how grain boundaries influence ${J}_{\text{c}}$ in high magnetic fields.