Page 146 - Principles and Applications of NanoMEMS Physics

134 Chapter 3

the superconducting state disappears (it becomes normal). Type II
superconductors, on the other hand, are characterized by two critical fields,
namely, a lower critical field H , below which the superconducting state
1 c
exists exclusively, and above which the superconductor is threaded by flux
lines that give rise to a lattice of vortices, and an upper critical field H ,
c 2
beyond which superconductivity disappears. The vortices are circulating
superconducting currents around normal regions, and are such that the onset
of a vortex occurs when the corresponding flux is that of a single fluxoid.
Quantitatively,

φ
H ≈ 0 , (180)
1 c πδ 2
L
and
φ
H ≈ 0 , (181)
c 2 πξ 2
ξ
where δ is the magnetic field penetration depth, and = v= 2 ∆ [28] is
L F
the coherence length, which captures the lattice constant of vortex lattice.

3.2.2 Photonic Band-Gap Crystals

Continuing with the topic of wave phenomena in periodic structures, we
now briefly take on the subject of electromagnetic wave propagation and
manipulation in periodic dielectric structures or photonic band-gap crystals
(PBCs) [51]. PBCs are 1-, 2-, or 3-dimensionally periodically patterned
materials whose dispersion relation, i.e., propagation constant versus
frequency response, exhibits ranges in which wave propagation is forbidden
(band gaps) and ranges in which it is allowed.

3.2.2.1 One-dimensional PBC Physics

The fundamental physics of a PBC are easily grasped from considerations
of a 1-D PBC, which is of finite extent and consists of alternating regions of
dielectric constant, ε and ε , respectively, see Fig. 3.24.
1 2