Supercurrent density equations with derivation

Derivation of Supercurrent Density

The supercurrent density \mathbf{J}_s in a superconductor can be derived from quantum mechanics.

1. London Equation

The first London equation is:

\begin{equation} \mathbf{J}_s = -\frac{n_s e^2}{m}\mathbf{A} \end{equation}

\begin{aligned} \text{where } \mathbf{J}_s &\text{ is the supercurrent density,} \\ n_s &\text{ is the density of superconducting carriers,} \\ e &\text{ is the electron charge,} \\ m &\text{ is the effective mass, and} \\ \mathbf{A} &\text{ is the vector potential.} \end{aligned}

2. Relation to the Wave Function

The macroscopic wave function is:

\begin{equation} \psi = |\psi| e^{i\theta} \end{equation}

The supercurrent density can be written as:

\begin{equation} \mathbf{J}_s = \frac{n_s e}{m}\mathbf{v}_s \end{equation}

The superfluid velocity is:

\begin{equation} \mathbf{v}_s = \frac{\hbar}{m}\nabla \theta \end{equation}

3. Substitution

Substituting:

\begin{equation} \mathbf{J}_s = \frac{n_s e}{m}\left(\frac{\hbar}{m}\nabla \theta\right) \end{equation}

which simplifies to:

\begin{equation} \mathbf{J}_s = \frac{n_s e \hbar}{m^2}\nabla \theta \end{equation}