mathematical spine of superconductivity from London → GL → BCS.

Important Equations in Superconductivity

1. London Equations

Supercurrent density: \begin{equation} \mathbf{J}_s = n_s e \mathbf{v}_s \end{equation} First London equation: \begin{equation} \frac{\partial \mathbf{J}_s}{\partial t} = \frac{n_s e^2}{m} \mathbf{E} \end{equation} Second London equation: \begin{equation} \nabla \times \mathbf{J}_s = -\frac{n_s e^2}{m} \mathbf{B} \end{equation} Magnetic field decay: \begin{equation} \nabla^2 \mathbf{B} = \frac{1}{\lambda_L^2} \mathbf{B} \end{equation} London penetration depth: \begin{equation} \lambda_L = \sqrt{\frac{m}{\mu_0 n_s e^2}} \end{equation} Exponential decay of magnetic field: \begin{equation} B(x) = B_0 e^{-x/\lambda_L} \end{equation}2. Ginzburg--Landau Theory

Order parameter: \begin{equation} \psi(\mathbf{r}) = |\psi| e^{i\theta} \end{equation} Free energy density: \begin{align} F &= F_n + \alpha |\psi|^2 + \frac{\beta}{2} |\psi|^4 \\ &\quad + \frac{1}{2m^*} \left| \left(-i\hbar \nabla - 2e \mathbf{A}\right)\psi \right|^2 + \frac{|\mathbf{B}|^2}{2\mu_0} \end{align} First GL equation: \begin{equation} \alpha \psi + \beta |\psi|^2 \psi + \frac{1}{2m^*} (-i\hbar\nabla - 2e\mathbf{A})^2 \psi = 0 \end{equation} Supercurrent density: \begin{equation} \mathbf{J}_s = \frac{2e\hbar}{m^*} \mathrm{Im}(\psi^* \nabla \psi) - \frac{4e^2}{m^*} |\psi|^2 \mathbf{A} \end{equation} Coherence length: \begin{equation} \xi = \sqrt{\frac{\hbar^2}{2m^* |\alpha|}} \end{equation} Penetration depth: \begin{equation} \lambda = \sqrt{\frac{m^*}{4\mu_0 e^2 |\psi|^2}} \end{equation} Ginzburg--Landau parameter: \begin{equation} \kappa = \frac{\lambda}{\xi} \end{equation} Critical value: \begin{equation} \kappa_c = \frac{1}{\sqrt{2}} \end{equation}3. Flux Quantization

Phase quantization: \begin{equation} \oint \nabla \theta \cdot d\mathbf{l} = 2\pi n \end{equation} Magnetic flux quantization: \begin{equation} \Phi = n \frac{h}{2e} \end{equation} Flux quantum: \begin{equation} \Phi_0 = \frac{h}{2e} \end{equation}4. BCS Theory

BCS Hamiltonian: \begin{equation} H = \sum_{k,\sigma} \epsilon_k c_{k\sigma}^\dagger c_{k\sigma} - V \sum_{k,k'} c_{k\uparrow}^\dagger c_{-k\downarrow}^\dagger c_{-k'\downarrow} c_{k'\uparrow} \end{equation} Quasiparticle energy: \begin{equation} E_k = \sqrt{\xi_k^2 + \Delta^2} \end{equation} Gap equation: \begin{equation} \Delta = V \sum_k \frac{\Delta}{2E_k} \tanh\left(\frac{E_k}{2k_B T}\right) \end{equation} Zero-temperature gap: \begin{equation} \Delta(0) = 1.76\, k_B T_c \end{equation}5. Critical Fields

Thermodynamic critical field: \begin{equation} H_c^2 = \frac{2}{\mu_0}(F_n - F_s) \end{equation} Upper critical field (Type II): \begin{equation} H_{c2} \approx \frac{\Phi_0}{2\pi \xi^2} \end{equation}