BCS Superconductor Energy Gap Calculator
Superconductivity & condensed matter physics: Calculate Bardeen-Cooper-Schrieffer (BCS) energy gap ($\Delta(T)$), thermodynamic critical magnetic field ($H_c(T)$), condensation energy, and penetration depth.
Superconducting Material Properties
Superconducting State Performance
Energy Gap $\Delta(T)$
--
meV (Cooper binding)
Zero-Temp Gap $\Delta(0)$
--
meV ($1.764 \, k_B T_c$)
Critical Field $B_c(T)$
--
mT (thermodynamic)
London Penetration $\lambda_L(T)$
--
nm (screening depth)
Ginzburg-Landau $\kappa$
--
Type I vs Type II boundary
Condensation Energy $U_0$
--
$\text{kJ}/\text{m}^3$ ($B_c^2 / 2\mu_0$)
BCS Energy Gap & Critical Field vs Temperature
At $T = T_c$, both the energy gap and critical field collapse to zero via second-order phase transition.
Microscopic BCS Theory Formulation
The 1957 Bardeen-Cooper-Schrieffer (BCS) microscopic theory explains superconductivity through electron-phonon interactions forming Cooper pairs of opposite spin and momentum:
$$\Delta(0) = 1.764 \, k_B T_c$$
$$\Delta(T) \approx \Delta(0) \tanh\left( 1.74 \sqrt{\frac{T_c}{T} - 1} \right) \quad (T < T_c)$$
$$B_c(T) = B_c(0) \left[ 1 - \left(\frac{T}{T_c}\right)^2 \right]$$
The Ginzburg-Landau parameter $\kappa = \lambda_L / \xi_0$ distinguishes Type I ($\kappa < 1/\sqrt{2} \approx 0.707$, complete Meissner expulsion) from Type II ($\kappa > 0.707$, Abrikosov vortex state with mixed phase between $B_{c1}$ and $B_{c2}$).