ass5: Introduce steps we'll take for 17

superconductivity_assignment5_kvkempen.tex

@@ -49,6 +49,21 @@
 \begin{document}
 
 \section{$T_c$ upper limit in BCS}
+In BCS theory, the formation of Cooper pairs is mediated by phonons.
+There is a phonon-electron interaction quantified by the dimensionless quantity 
+\[
+	\lambda := Vg(\epsilon_F)
+\]
+with $V$ Cooper's approximate potential and $g(\epsilon_F)$ the density of states near the Fermi surface for the electrons.
+A thorough discussion can be found in Annett's book \cite[chapter 6]{annett} and in the slides of week 6 of this course.
+The binding energy of the Cooper pairs (i.e. the energy gain of forming these pairs) is
+\[
+	-E = 2\hbar\omega_De^{-1/\lambda}.
+\]
+In the weak coupling limit of the BCS theory, the case we have considered so far, it is assumed that $\lambda << 1$.
+It is when this assumption breaks down, BCS does not work and we find an upper limit to the critical temperature $T_c$.
+We will look at a way to express the critical temperature in terms we can derive, and than look at the values that maximize this critical temperature whilst still following BCS theory.
+
 
 \section{Energy gap $\Delta$ et al.}