Meissner Effect

The Meissner effect is the expulsion of magnetic flux from the interior of a superconductor when it transitions below its critical temperature T_c. It distinguishes a superconductor from a perfect conductor.

London Penetration Depth

The magnetic field inside a superconductor decays exponentially from the surface:

$B(x) = B_0 \, e^{-x/\lambda_L}$

The London penetration depth λ_L is:

$\lambda_L = \sqrt{\frac{m_e}{\mu_0 n_s e^2}}$

where n_s is the superfluid density (superconducting electron density).

Critical Field (Type I)

A type-I superconductor loses superconductivity above the critical field:

$H_c(T) = H_{c0}\left[1 - \left(\frac{T}{T_c}\right)^2\right]$

Type II Superconductors

Type-II superconductors have two critical fields:

• Lower critical field H_c1 — flux vortices begin to penetrate
• Upper critical field H_c2 — superconductivity is destroyed:

$H_{c2} = \frac{\Phi_0}{2\pi \mu_0 \xi^2}$

where Φ₀ = h/(2e) = 2.068×10⁻¹⁵ Wb is the flux quantum and ξ is the coherence length.

Implement

def london_depth_m(n_s_m3):
    # Returns London penetration depth in metres
    ...

def field_inside_superconductor_T(B0_T, x_m, lambda_L_m):
    # Returns B(x) in Tesla
    ...

def critical_field_H_c_A_m(H_c0_A_m, T_K, T_c_K):
    # Returns H_c(T) in A/m
    ...

def upper_critical_field_H_c2_A_m(xi_m):
    # Returns H_c2 in A/m
    ...