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Both types are mediated by electron-phonon coupling, but there are quite some differences. +Using figure \ref{fig:typevs}, we will go through their differences. \begin{figure} - \label{fig:type-i-vs-ii} - \includesvg[width=\textwidth]{Lecture-2-slides-for-printing-type-i-vs-ii.svg} + \label{fig:typevs} + \includegraphics[width=\textwidth]{Lecture-2-slides-for-printing-type-i-vs-ii.pdf} \caption{Figure is borrowed from the presentation of week two of the Superconductivity course by Alix McCollam.} \end{figure} +Type-I superconductors (sc) are described by BCS theory. +An example of a type-I sc is lead (Pb). +Below a critical temperature $T_c$, these materials exhibit perfect diamagnetism. +Inside the sc, a magnetic field is generated to expell the externally applied field, +such that the net field is zero. +This is called the Meissner effect. +The perfect diamagnetism is quantized as having susceptibility $\chi = -1$. +There is, however, a limit to how large a field can be completely expelled. +This is called the critical field $B_c$. +If $B_c$ is exceeded, the superconductivity breaks down, thus the material will cease to expell the field, +the diagmagnetism drops to $\chi = 0$. +This effect is seen in the bottom-left plot in figure \ref{fig:typevs}. +The relation between these two critical values is given as +\[ + B_c(T) = B_c(0) \left[ 1 - \frac{T}{T_c}^2 \right]. +\] +An example of this curve is plotted in the top-left of figure \ref{fig:typevs}. +Two phases can be distinguished: +the Meissner (or superconducting) state under the graph, +and the normal state outside it. +The transition between these states is a first-order phase transition due to the discontinuity in the magnetization $M = \frac{dF}{dB}$, the first derivative of the free energy to the applied field. + +Then there are type-II sc, for example, niobium (Nb). +Instead of only having a Meissner or sc phase, they have another phase: +the vortex state. +Below $T_c$ and lower critical field $B_{c1}(T)$, the type-II sc is in the Meissner state, +and the material thus completely cancels the externally applied field. +Above the upper critical field $B_{c2}(T)$, the material is in the normal state. +When the field is between these two critical fields, $B_{c1}(T) < B < B_{c2}(T)$, +the material is in the vortex state. +In the vortex state, the externally applied magnetic field is not completely expelled. +Instead, the material consists of normal and sc regions, the former called vortices. +These vortices are normal conducting regions and allow magnetic flux to pass. + + Type-I superconductors are thought to be described by BCS theory. They are phonon-mediated. 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