Page 141 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 129

The zero-temperature binding energy (gap) is given by,
2 ω
=
∆
2 = D = E . (157)
2 b
e VN ) 0 ( − 1
The binding energy (157) determines how far apart the electrons forming a
Cooper pair may separate while still acting as bound. In this context, the
radius R of a Cooper pair has been estimated as [160],

= 2 k
R ~ F , (158)
mE
b
which, numerically, is of the order of µ 1 m . The implication of the binding
energy is as follows. At absolute zero, an energy greater than the binding
energy is required to separate Cooper pairs and, thus, create excited electrons
which are generated in pairs. At energies close to this threshold, E , the
b
current will consist of both Cooper pairs and single (normal) electrons
resulting from the breaking of the pairs, giving rise to a two-fluid model
transport. Abrikosov has shown that as the temperature increases E
b
decreases until it reaches zero a the critical temperature, T . This is
c
temperature dependence is given by,
E = . 3 06 T (T − ) T . (159)
b c c

Next, we consider the phenomenon of magnetic field exclusion from a
superconductor. We examine the supercurrent in a superconductor containing
a density of n electrons moving with velocity v and, thus, given by
s s G
J = en v , in the presence of a vector potential field A . In general, the
'
s s s
particle velocity in a vector potential is given by,
G 1 § G q G ' ·
v = ¨ − A ¸ . (160)
p
M © c ¹
In the case of the superconductor, M = 2m , and q = 2e . If we let Ψ be
e
the wavefunction of the electron pair (boson), then we can express (160) as,