Josephson penetration depth

In superconductivity, Josephson penetration depth characterizes the typical length on which an externally-applied magnetic field penetrates into the long Josephson junction. Josephson penetration depth is usually denoted as $lambda\_J$ and is given by the following expression (in SI):

$lambda\_J=sqrt\{frac\{Phi\_0\}\{2pimu\_0\; d\text{'}\; j\_c,$

where $Phi\_0$ is the magnetic flux quantum, $j\_c$ is the critical current density $mathrm\{(A/cm^2)\}$, and $d\text{'}$ characterizes the inductance of the superconducting electrodes

$d\text{'}=d\_I\; +lambda\_1\; cothleft(frac\{d\_1\}\{lambda\_1\}\; ight)\; +lambda\_2\; cothleft(frac\{d\_2\}\{lambda\_2\}\; ight),$

where $d\_I$ is the thickness of the Josephson barrier (usually insulator), $d\_\{1,2\}$ are the thicknesses of superconducting electrodes, and $lambda\_\{1,2\}$ are their London penetration depths.