Page 133 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 121

3.2.1.2.2 Superconductivity
Our understanding of superfluidity, gained in the previous section,
facilitates that of superconductivity. Superconductivity, the absence of
electrical resistance to electron transport, may be conceptually visualized as
the “superfluidity of electrons”. A qualitative analogy between these two
phenomena may be summarized as follows. Whereas a superfluid embodies
a boson condensate of, e.g., helium atoms, a superconductor, on the other
hand, embodies boson condensates of, e.g., bound electron pairs. Electrons,
as is known, due to the Coulomb force of repulsion between them, do not,
strictly speaking, condense. However, under certain circumstances, an
effective binding force may be present that overcomes the force of repulsion
between electron pairs and turns these pairs, effectively, into bosons. These
electron pairs, which behave as bosons, are called Cooper pairs and have
zero spin (just as the helium atoms). Thus, while a boson condensate of
helium atoms may behave as a superfluid, under appropriate circumstances,
and when it does so it exhibits transport without friction, so too a condensate
of an aggregate of Cooper pairs, behaves as a superconductor. Continuing
with the analogy, while superfluid transport exists for velocities less than a
critical velocity, v ~ min (E ) p , so too superconductive transport exists
c
below a critical velocity v ~ (∆ p ), where ∆2 in this case is the binding
c 0
energy of a Cooper pair. Finally, while dissipation and fluid vortices (rotons)
appear above v in the superfluid, so too ohmic dissipation and so-called
c
vortex states, i.e., circulation of superconducting currents in vortices
throughout the system, appear beyond v in the superconductor. With these
c
qualitative preliminaries, we next address the salient aspects of
superconductivity, namely, the criterion for superconductivity in light of its
conceptual relationship to superfluidity, the binding energy of Cooper pairs,
the inhibition of a magnetic field inside superconducting materials, the
conditions for the extinction of superconductivity.
In analogy with (105), the equation for a single electron moving in a
superconductor may be written as,

∂ ψ ( tx, ) = 2
i= σ = − ∇ 2 ψ σ + g ψ * σ ψ σ ψ , (122)
σ
∂t 2m

where g represents charge, σ =↑ or ↓ represents the spin state, and
ψ * σ ψ is a 2-index summation that embodies the density from all spins. In
σ
this context, the wave function of a pair of electrons is a product given by,