Page 135 - Principles and Applications of NanoMEMS Physics

3. NANOMEMS PHYSICS: Quantum Wave Phenomena 123

§ p 2 ·
E η σ = ¨ ¨ − E F ¸ η σ + ∆E σ σ ζ , (128a)
σ
¸
© 2 m ¹
and

§ p 2 · *
- E ζ σ = ¨ ¨ − E F ¸ ζ ¸ σ + ∆ E σ σ η , (128b)
σ
© 2 m ¹

is obtained. Solving (128) for E one obtains E = ± ∆ 2 + v 2 (p − p ) .
2
F F
This is the dispersion relation for superconducting electrons. It represents a
parabola with a minimum at p = p , corresponding energy ∆ , and energy
F
gap ∆2 . Therefore, application of the Landau criterion for superfluidity, to
the present case of superconductivity, yields the critical velocity,
v = (∆ p ), below which electron transport experiences no electrical
c F
resistance, i.e., is superconductive. Next, we address the formation of Cooper
pairs.
In exploiting the superfluid physics analogy to describe
superconductivity, one must confront the issue of explaining how electrons,
which would ordinarily be precluded from binding, due to Coulomb’s
repulsion force, would bond/condense to form bosons. The clue to this
possibility was advanced by the discovery that [157], [158] in
superconducting elements, the product of the square root of their isotopic
mass and the critical temperature, M 2 / 1 T , is a constant. This experimental
c
fact, in turn, was interpreted by Fröhlich [154] to mean that the properties of
the zero-point or thermal lattice phonons, were involved in
superconductivity and, in particular, that electrons residing within the crystal
lattice were capable, via interactions mediated by these phonons, of
attracting one another. This phenomenon is demonstrated next.
To determine the nature of the phonon-mediated electron-electron
interaction, we assume the coexistence of phonons and electrons is described
by a Hamiltonian consisting of three terms, namely, the energy of the
electrons, the energy of the phonons, and the energy of interaction between
electrons and phonons, respectively. The first two terms are captured by the
“unperturbed” Hamiltonian:

H = ¦ E G c + c G + ¦ = ω a GG + a G
G
0 σ , k σ , k σ , k q q q . (129)
G G
σ , k q