Introduction to Superconductivity

The classic hallmarks of superconductivity: are:

Note that a superconductor is more than just a conductor with zero resistivity in the sense that a normal conductor in an external magnetic field is attracted by the magnetic field, however a superconductor is a diamagnet which means that it is repelled by an external magnetic field.
 * zero electrical resistance
 * a phenomenon known as the Meissner effect which is the result of perfect diamagnetism

= Conventional and Unconventional superconductivity = In conventional superconductors, the Cooper pairs originate from a small attractive electron-electron interaction mediated by phonons.

In the so-called unconventional superconductors, the pairing can originate even from purely repulsive interactions

= Experimental Electrical Resistivity of Metals = The electrical resistivity of most metals is dominated at room temperature (300K) by collisions of the conduction electrons with lattice phonons and at liquid helium temperature (4K) by collisions with impurity atoms and mechanical imperfections in the lattice.

To a good approximation the electron collision rates are independent from one another. This approximation is known as the Matthiessen's rule

$$\frac{1}{\tau} = \underbrace{\frac{1}{\tau_L}}_\text{Latice phonons} + \underbrace{\frac{1}{\tau_i}}_\text{imperfections}$$

Since we have $$\rho \propto 1/\tau$$, the resistivity due to each component (phonons and imperfections) can be summed up to get the total resistivity

$$\rho = \rho_L + \rho_i$$

And often, $$\rho_L$$ is independent of number of impurities and imperfections, whereas $$\rho_i$$ is independent of temperatures.

Debye Temperature
At temperatures above the Debye temperature the phonon concentration is proportional to the temperature, $$N_{ph} \propto T$$, therefore the resistivity becomes proportional to the temperature

$$\rho_L \propto 1/\tau \propto N_{ph} \propto T$$


 * In a metal the resistivity at low temperatures has a constant contribution from impurity scattering, a $$T^2$$ contribution from electron-electron scattering, and a $$T^5$$ contribution from phonon scattering.