Superconductivity

Abstract

In 1911 H. Kamerlingh Onnes measured the electrical resistivity of mercury and found that it dropped to zero below 4.15 K. He could do this experiment because he was the first to liquefy helium and thus he could work with the low temperatures required for superconductivity. It took 46 years before Bardeen, Cooper, and Schrieffer (BCS) presented a theory that correctly accounted for a large number of experiments on superconductors. Even today, the theory of superconductivity is rather intricate and so perhaps it is best to start with a qualitative discussion of the experimental properties of superconductors.

Problems

• 8.1 Show that the flux in a superconducting ring is quantized in units of h/q, where q = |2e|.

• 8.2 Derive an expression for the single-particle tunneling current between two superconductors separated by an insulator at absolute zero. If E^T is measured from the Fermi energy, you can calculate a density of states as below.

Note:

$$ \begin{aligned} E^{T} & = \sqrt {\varepsilon^{2} + \varDelta^{2} } \left( {{\text{compare }}\left( {8.93} \right)} \right), \\ D_{S} \left( {E^{T} } \right) & = \frac{{{\text{d}}n\left( {E^{T} } \right)}}{{{\text{d}}E^{T} }} = \frac{{\frac{{{\text{d}}n\left( {E^{T} } \right)}}{{{\text{d}}\varepsilon }}}}{{\frac{{{\text{d}}E^{T} }}{{{\text{d}}\varepsilon }}}} \cong D\left( 0 \right)\frac{{E^{T} }}{{\sqrt {\left( {E^{T} } \right)^{2} - \varDelta^{2} } }}, \\ \end{aligned} $$

where D(0) is the number of states per unit energy without pairing.