1. Superconductivity

The discussions of the quantum Hall effect and superconductivity both started with the same Hamiltonian,

\begin{equation*} H = \frac{1}{2m}\left(\vec{p}-q\vec{A}(\vec{r},t)\right)^2 + qV(\vec{r},t) \end{equation*}

1. The concept of field momentum was introduced to describe a charged particle in a magnetic field. What is field momentum? Does field momentum exist in superconductors?
2. What is the essential difference between the quantum Hall effect and superconductivity? (Hint: it is related to the entropy in the superconducting state).
3. How would you detect a superconducting transition with a magnetometer?
4. What happens to the thermal conductivity of a metal when it becomes superconducting?

Solution

1. The concept of field momentum was introduced to describe a charged particle in a magnetic field. What is field momentum? Does field momentum exist in superconductors?

   For a particle feeling an external field with scalar potential $V$ and vector potential $\vec{A}$ the Lagrangian is

   \begin{equation*} \mathcal{L}(\vec{r},\vec{v},t) = \frac12 mv^2 - qV(\vec{r},t) + q\vec{v}\cdot\vec{A}(\vec{r},t), \end{equation*}

   which can be verified by inserting into the Euler-Lagrange equation. The conjugate momentum

   \begin{equation*} \vec{p} = \nabla_{\vec{v}}\mathcal{L} = m\vec{v} + q\vec{A}(\vec{r},t) \end{equation*}

   consists of a kinetic momentum and a field momentum $q\vec{A}$. This field momentum describes the additional work that must be done when accelerating the particle in order to establish the magnetic field around it (a moving charge is a current and therefore creates a magnetic field according to Ampere's law).

   In principle field momentum does exist in superconductors. A phenomenological quantum mechanical calculation (see http://lampz.tugraz.at/~hadley/ss2/superconductivity/macroscopic.php) shows that the current in a superconductor is given in terms of the phase of the wave function $\theta$ and the vector potential $\vec{A}$ as

   \begin{equation*} \vec{j} = -\frac{ehn_{cp}}{m_e}\left(\nabla\theta + \frac{2e}{\hbar}\vec{A}\right), \end{equation*}

   where $n_{cp}$ is the density of cooper pairs in the superconducting state.

   However, both the magnetic field and the current decay exponentially in superconductors (Meissner effect) and thus inside the material we have

   \begin{equation*} \vec{j} = -\frac{ehn_{cp}}{m_e}\left(\nabla\theta + \frac{2e}{\hbar}\vec{A}\right) = 0 \quad\implies\quad \nabla\theta + \frac{2e}{\hbar}\vec{A} = 0. \end{equation*}

   Due to gauge invariance the gradient of any scalar function can be added to the vector potential without changing observable quantities. Choosing the London gauge

   \begin{equation*} \vec{A} \to \vec{A} - \nabla\left(\frac{\hbar}{2e}\theta\right) \end{equation*}

   leads to $\vec{A} = 0$ and therefore also the field momentum $-e\vec{A}$ vanishes inside the superconductor.

2. What is the essential difference between the quantum Hall effect and superconductivity? (Hint: it is related to the entropy in the superconducting state).

   In the quantum Hall effect the resistance-free transport is only possible in edge channels, because inside the sample the Fermi energy is between two Landau levels and therefore no current can flow. This leads to a finite current for very small voltages (which makes the resistivity go to zero), but the edge states can only carry a limited current (a few μA).

   In the superconducting state the entropy approaches zero since the formed cooper-pairs forbid scattering (which would increase entropy). Unlike in the quantum Hall effect, where the resistance at finite temperatures is just extremely small, the resistance in superconductors is really zero.

3. How would you detect a superconducting transition with a magnetometer?

   Superconductors are perfect diamagnets ($\chi = -1$). When the magnetic H-field rises and reaches a critical point, magnetisation drops to zero. A magnetometer can be used to measure this effect.

4. What happens to the thermal conductivity of a metal when it becomes superconducting?

   At low temperatures thermal conductivity is generally dominated by electron scattering. In superconductors electrons form cooper pairs. For those small excitations such as scattering are forbidden. Therefore the thermal conductivity approaches zero.