Superconductivity

Superconductivity is a macroscopic quantum phenomenon in which a material's electrical resistance drops to exactly zero below a critical temperature T_c, and magnetic flux is expelled from its interior (Meissner effect).

BCS Energy Gap

The BCS theory (Bardeen–Cooper–Schrieffer, 1957) explains superconductivity via Cooper pairs — electron pairs bound by phonon-mediated attraction. At T = 0, the energy gap is:

$\Delta(0) = 1.764 \, k_B T_c$

This gap must be broken to create quasiparticle excitations. Converting to eV: Δ(eV) = Δ(J) / e.

London Penetration Depth

Magnetic fields are screened from the superconductor interior over the London penetration depth:

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

where n_s is the density of Cooper-pair electrons (superfluid density). Typical values: 20–500 nm.

Coherence Length and GL Parameter

The coherence length ξ is the spatial scale over which the superconducting order parameter varies. The Ginzburg-Landau parameter distinguishes two types:

$\kappa = \frac{\lambda_L}{\xi}$

• Type I: κ < 1/√2 ≈ 0.707 (e.g., aluminum, tin) — sharp transition, full Meissner effect
• Type II: κ > 1/√2 (e.g., niobium, YBCO) — allows partial flux penetration as vortices

Your Task

import math

def bcs_gap_J(T_c_K):
    # Delta = 1.764 * k_B * T_c, k_B = 1.381e-23
    pass

def bcs_gap_eV(T_c_K):
    # Convert bcs_gap_J to eV using e = 1.602e-19
    pass

def london_penetration_depth_m(n_s_m3):
    # lambda_L = sqrt(m_e / (mu0 * n_s * e^2))
    # m_e=9.109e-31, mu0=4*pi*1e-7, e=1.602e-19
    pass

def gl_parameter(lambda_L_m, xi_m):
    # kappa = lambda_L / xi
    pass