For some metals, the resistance drops to
zero when the temperature falls below a critical value
Naively, one may be tempted to associate zero resistance with an infinite conductivity. However, the superconducting state cannot be so described, as will be shown below.
Substituting the Ohm's law for a metal,
into the Faraday's law
we get
Hence,
i.e., B
is time
However, the superconducting state was
found experimentally to be a thermodynamic state with perfect diamagnetism (
According to the BCS theory, transition to the superconducting phase is a Bose- Einstein condensation, in momentum space, of Cooper pairs. Here, a Cooper pair is a boson representing a bounded state of 2 electrons of opposite spins interacting under an attractive effective potential caused by electron- phonon interactions.
Thus, the superconducting state is a
macroscopic quantum state. The onset of
superconductivity is an order- disorder phase transition with an effective wave
function Ψ as order parameter. Here,
Superconductivity can be destroyed by an
applied field
where
with
Using
we have, along the coexistence curve,
Þ
where we've used
Eq(3.55) is the Clausius- Clapeyron equation for superconductivity.
Using (3.53), eq(3.55) becomes
Þ
so that
For sufficiently low T, the cubic term will be smaller than the linear one so that
Note also that
in agreement with the 3^{rd} law.
Integrating (3.57) at fixed T gives
Using
Finally, with the help that, on the coexistence curve,
we can combine (3.59-60) into
Thus, the condensation energy is
with