# What causes current to move to the surface in superconductors?

I’m really trying to understand superconductors in an intuitive way here, although many say you cannot, anyway Im unable to find anywhere that explains to me why the current is on or near the surface of a superconductor. I understand cooper pairing, and I understand the meissner effect which seems to be a result of this surface current, but there is no explanation as to why the current is at the surface.

## 1 Answer

The simplest picture is from energy. If we use the Landau-Ginzburg formalism we can define a current $$\rho {\bf v}= \frac{1}{2m^*}\left( \psi^*\nabla\psi-(\nabla \psi^*)\psi\right)\\ = \rho(\nabla\phi -2e{\bf A})/(2m_e)$$ where $$\rho= \psi^*\psi$$ and $$\psi=\sqrt{\rho} e^{i\phi}$$ and $$m^*=2m_e$$. Then the vorticity of the superconducting electron fluid
$$\omega\equiv \nabla\times {\bf v}$$ obeys the Meissner constriant $$m_e\omega+ e{\bf B}=0.$$ There is crucial difference here between a superconductor and a "perfect" conductor. In a perfect conductor a changing $${\bf B}$$ field gives rise to an electric field via $${\rm curl\,} {\bf E}= - \partial{\bf B}/{\partial t}$$ and this electric field gives rise to a $${\rm curl\,}{\bf v}$$ so that $$m_e\omega+ e{\bf B}$$ is constant, but this constant does not have to be zero. As a result, in a perfect conductor, the magnetic field is frozen into the fluid. This what happens in a highly conductive plasma, and is the source of Alfven waves.

In the superconductor, however, the constant must be exactly zero. In other words, in a superconductor, where there is a magnetic field the fluid is forced to have vorticity. Now vorticity costs kinetic energy, so having a magnetic field inside the superconductor is energetically costly. The lowest energy configuration puts a minimal $$\rho {\bf v}$$ screening current on surface that is just sufficient to keep the field out. The field has therefore been expelled from the interior of the superconductor. This is the Messner effect.

Now there is another way to reduce that the $$\rho |{\bf v}|^2/2$$ kinetic energy that has not been included yet. This is to reduce $$\rho$$ rather than $$|{\bf v}|$$. Such a reduction costs energy from the $$V(|\psi|)=$$ interaction term in the Landau-Ginzburg free energy. If the external field is large enough the system will reduce the $$\int d^3x \frac{1}{2\mu_0} |{\bf B}|^2$$ field energy at the expense of paying energy to reduce $$|\psi|$$. The field will then penetrate the superconductor either in the form of vortices or, in a type I superconductor, completely.

• I think I’m getting confused on the setup here.. is the magnetic field in question an externally applied magnetic field such as in the case of magnetic levitation? Or the INDUCED magnetic field from the current flowing through the superconductor? If it’s the latter then I think you are saying that the current induces a magnetic field in such a way that it sort of cancels the flow in the center and the flow increases radially outward reaching a maximum at the surface? – SuperYoughe Oct 29 '19 at 15:03
• I was thinking about an isolated SC in an external field. If we have an SC magnet as in a particle accelerator or MRI machine then yes, you are correct. In a normal conductor with a DC current, it can be uniform across the conductor. The SC current, however, runs mostly on the surface. – mike stone Oct 29 '19 at 15:26