= Planar London screening
{c}
{title2=$B(z)=B_0e^{-z/\lambda_L}$}
Take a homogeneous material exhibiting <superconductivity> in the <half-space> $z\geq0$, bounded by vacuum at $z=0$. Fix the <temperature> below the <superconducting transition temperature>, take a constant positive <London penetration depth> $\lambda_L$, and assume a local, linear <London equation> description in a <Meissner state>. The applied <magnetic field> is weak enough that it neither nucleates a <quantum vortex> nor appreciably suppresses the material's <superconductivity>. Assume translation invariance in $x,y$, no imposed transport <electric current>, and a tangential boundary <magnetic field> $\mathbf B(0)=B_0\widehat{\mathbf x}$. Here $B_0$ is the local surface <magnetic flux density>, equal to the imposed vacuum <magnetic field> in this infinite planar geometry; it is not automatically the far-away applied value for a finite object.
For a bounded, decaying interior solution, the <London equations>, <magnetostatic Ampère law> and <Gauss's law for magnetism> give
$$
\nabla^2\mathbf B=\frac{\mathbf B}{\lambda_L^2},\qquad
\frac{d^2B_x}{dz^2}=\frac{B_x}{\lambda_L^2},\qquad
B_x(0)=B_0,\quad B_x(\infty)=0.
$$
The solution is
$$
\mathbf B(z)=B_0e^{-z/\lambda_L}\widehat{\mathbf x}.
$$
At one <London penetration depth> the <magnetic field> is $e^{-1}\simeq0.368$ of its surface value; at five it is $e^{-5}\simeq0.00674$. There is no finite depth where this exponential is exactly zero. The <differential equation> excludes the growing exponential through the deep-interior boundary condition. A constant normal component cannot decay in this geometry: <Gauss's law for magnetism> would require $\partial_zB_z=0$. A perpendicular applied <magnetic field> on a finite plate therefore requires a different spatial problem.<Lecture notes on the electrodynamics of metals and superconductors>[§2, p. 13, Eqs. (68)–(74): London screening equation; the planar boundary-value solution here follows directly]{ref}
\Image[/meissner-field-and-screening-current-profiles.png]
{title=<Magnetic field> and signed <Meissner screening current> profiles with finite <London penetration depth>}
{description=Original analytic profiles, not measured data. Left: the <Planar London screening> solution, with vacuum at $z<0$ and the material at $z\geq0$. Right: <Finite-thickness Meissner screening> in a slab of thickness $2\lambda_L$, with equal tangential surface <magnetic fields>. Blue is $B_x/B_0$; orange is the signed <current density> $\mu_0\lambda_LJ_y/B_0$. Different horizontal origins are labeled explicitly.}
Back to article page
