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3. NANOMEMS PHYSICS: Quantum Wave Phenomena 113

is exposed to a high critical magnetic field H [28]. We discuss the
c
principles of superconductivity here, mainly because of the importance of
superconductors (materials exhibiting superconductivity) as an alternative
means of implementing quantum bits (qubits). Our point of departure in
discussing superconductivity is the concept of superfluidity, from which it
may be understood in an intuitive fashion.

3.2.1.2.1 Superfluidity

Superfluidity refers to the property exhibited by a superfluid, i.e., a liquid
that flows without friction. A successful explanation of superfluidity was put
forth by Landau [153], [154]. Landau’s reasoning was as follows [131]. If
one assumes that the Bose quantum fluid of mass M is in its ground state at
absolute zero, and flowing within a capillary tube with velocity v, and
1 2
energy Mv , then, in a coordinate system anchored in the fluid, the fluid
2
would be at rest and the capillary would appear to be moving at a velocity –
v. If friction emerges between the capillary and the fluid, then the part of the
latter in contact with the tube would no longer be at rest, but would begin to
be carried along by the capillary wall. However, since this part of the fluid
would no longer be at rest, the act of it being carried along by the tube wall
must induce excitations from its ground state. These excitations, in turn,
would manifest as changes in its energy and momentum, E and p, so that the
G
p⋅
fluid’s total energy would now be E + G v + 1 Mv . Upon excitation, the
2
2
fluid itself would lose energy. Therefore, energy change must be negative,
i.e.,
G G
E + p⋅ v < 0. (88)

Since the fluid is a quantum system of Bose particles, its energy is quantized
and must change discretely. The smallest energy excitation, therefore, is that
G
G
G
G
for which E + p v ⋅ is a minimum, which occurs when p and v are
opposite. This means that one must have,
E − pv < 0 or v > E . (89)
p

This equation sets the minimum velocity at which excitations would begin,
as the critical velocity,