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}{2\pi\mu_0 d' j_c}},

where $\Phi_0$ is the magnetic flux quantum, $j_c$ is the critical current density $\mathrm{(A/m^2)}$ , and $d'$ characterizes the inductance of the superconducting electrodes

d'=d_I +\lambda_1 \coth\left(\frac{d_1}{\lambda_1}\right) +\lambda_2 \coth\left(\frac{d_2}{\lambda_2}\right),

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.

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