London penetration depth (${\lambda }_L=\sqrt{m/({\mu }_0{n}_s{q}^2)}$)

For uniform vortex-free superconductivity, the London equation and stationary Maxwell equations give ${\nabla }^2\mathbf{B}=\mathbf{B}/{\lambda }_L^2$. A field tangential to a planar superconducting half-space decays as $\mathbf{B}(z)=\mathbf{B}(0)e^{-z/{\lambda }_L}$. A nonzero normal boundary field is incompatible with this translation-invariant decaying Meissner solution, because divergence-free magnetic flux would require that component to remain constant.

For scalar kinetic term $|D\phi {|}^2$ and condensate magnitude $|\phi |=v$ without a factor $1/\sqrt{2}$, the canonical vector mass is $m_A^2=2e^2{v}^2$. In a constant-phase gauge the current is $\mathbf{j}=-2e^2{v}^2\mathbf{A}$. A static magnetic field obeys ${\nabla }^2\mathbf{B}=m_A^2\mathbf{B}$, so its penetration length is $1/(\sqrt{2}|e|v)$. This normalization differs from defining the vacuum amplitude as $v/\sqrt{2}$.

For a slab of thickness $2a$ with parallel applied field, the local London equation gives $B(x)=B_a\cosh (x/\lambda )/\cosh (a/\lambda )$. Averaging gives $M=-B_a[1-\tanh (a/\lambda )/(a/\lambda )]/{\mu }_0$. Integrating $-Md{B}_a$ and comparing with the bulk condensation energy yields the displayed thin-slab enhancement. It assumes field-independent condensate density and can be limited by pair breaking or nonlocal electrodynamics.

For carrier number density $n$, charge $q$ and effective mass $m$, the plasma frequency is ${\omega }_p^2=n{q}^2/({ϵ}_{0}m)$. Combining the second London equation with the magnetostatic Ampère law and Gauss's law for magnetism gives ${\nabla }^2\mathbf{B}=({\omega }_p^2/c^2)\mathbf{B}$. Therefore the decaying planar solution has length ${\lambda }_L=c/{\omega }_p=\sqrt{m/({\mu }_{0}n{q}^2)}$.

In a local London equation description, the measured London penetration depth satisfies ${\lambda }_L^2=m/({\mu }_0{n}_s{e}^2)$ when $n_s$ counts superconducting electrons and $m$ is their effective mass. Thus screening measures $n_s/m$. If instead one counts Cooper pairs, use pair mass $2m$ and pair electric charge $-2e$, giving pair density $n_s/2$ for the same screening length. Identifying this superconducting carrier density with the total conduction-electron density requires the additional assumption that the relevant carriers all participate in screening at the measurement temperature.