## Historic Overview

In the beginning of the 20th century the question of resistance behaviour of metals at low temperatures in a range of single Kelvins arose. Three different cases where highly discussed:

- J. Dewar 1904: extrapolation of high temperature data,

$\rho(T) \rightarrow 0$ for $T\rightarrow 0$K. (red curve) - H. F. L. Matthiesen 1864: residual resistivity due to impurities and defects,

$\rho(T) = \rho_\text{R} + \rho_\text{ph}$ remains constant. (blue curve) - W. Lord Kelvin 1902: electrons are bound to the atoms at low temperatures and become localized,

$\rho(T)\rightarrow\infty$. (green curve)

In 1908 Heike Kamerlingh Onnes was able to produce liquid helium and thus, enabling a new range of temperatures, new experiments became available.

After his first experiments with gold, he concluded that the Matthiesen rule seems to fit very well and the residual resistivity can be lowered by purifying the material. Thus, he switched to mercury, as it was the purest metal available. During his measurements he observed a phase transition to a new state — the superconducting state.

After that ground breaking discovery the search for a fundamental theory began.

Heike Kamerlingh Onnes

Investigations into the properties of substances at low temperatures, which have led, amongst otherthings, to the preparation of liquid helium Nobel Lecture, December 11, 1913

## London Theory

The two German brothers Fritz and Heinz London developed a first, phenomenological theory of superconductivity in 1935. Their description involve the interaction between electromagnetic fields and the current inside the superconductor. Therefore, they derived the two Londen-Equations

\begin{align}\frac{\partial \vec{j}_\text{S}}{\partial t} &= \frac{n_\text{S}\cdot e^2}{m}\vec{E},\label{eq:London1}\\

\nabla\times\vec{j}_\text{S} &= -\frac{n_\text{S}\cdot e^2}{m}\vec{B}.\label{eq:London2}\end{align}

From the second equation the *London penetration depth*

\begin{align}

\lambda_\text{L} &= \sqrt{\frac{m}{\mu_0\cdot n_\text{S}\cdot e^2}}

\label{eq:LondonPenDep}

\end{align}

can be derived. This theory was an major step for the understanding of superconductivity as it was able to describe the behaviour of a superconductor regarding external fields like the Meissner-Ochsenfeld-Effect.

## Ginzburg-Landau-Theory

In 1950 Vitaly Ginzburg and Lev Landau derived a theory from thermodynamics and second-order phase transitions called *Ginzburg-Landau-Theory*. In this theory two lengths play an important role:

With the Ginzburg-Landau parameter $\kappa = \frac{\lambda}{\xi}$ one can distinguish Type I $\left(\kappa<\frac{1}{\sqrt{2}}\right)$ and Type II $\left(\kappa>\frac{1}{\sqrt{2}}\right)$ superconductors.

This theory was another important step towards a full understanding of superconductivity. Alexei Abrikosov was able to explain the penetration of thin films in 1957. Flux tunnels, so called Abrikosov vertices, form a Abrikosov lattice, while each vortex contain a flux equal to exactly a single flux quantum $\Phi_0$.

## BCS Theory

The first microscopic quantum theory was published by John Bardeen, Leon Cooper and John Schrieffer in 1957. This theory describes an interaction between two electrons and a phonon, leading the electrons to condensate into a bosonic *Cooper pair*. For this theory they recieved the Nobel Prize in Physics in 1972. With this theory superconductivity was fully understood until the discovery of high temperature superconductors by Bednorz and Müller in 1986.

Note: A more detailed explaination of the BCS theory will follow in an extra post.