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3. NANOMEMS PHYSICS: Quantum Wave Phenomena 139

Specifically, the properties of these structures are given by arguments
advanced by Bragg [64], whose essence (for a 2-D periodic lattice) is that
the path difference (phase shift) between incoming and scattered rays,
G
G
G
∆ = ⋅ a (k − k′ ), see Fig. 3-27(b), determines whether the transmission of the
structure exhibits a maximum or a minimum; a maximum when ∆ is an
G
G
integer multiple of π2 , and a minimum when it is an odd multiple of π. For
G
a 3-D PBC, on the other hand, ∆ = R ⋅ (k − k′ ) must be valid simultaneously
G
for all vectors R that are Bravais lattice vectors [64].
A large number of computational techniques to obtain the properties of
general PBCs have been developed, most of which derive from the solid
state physics literature on computing band structures [162]-[166]. Obviously,
it would be impossible to engage in detailing these techniques here, thus we
instead provide a number of analytical results derived by Joannopoulos et al.
[162] that capture some general properties of PBCs and facilitate one’s
intuition when thinking about them.
3.2.2.2.1 General Properties of PBCs

Initially, techniques for computing the properties of dielectric PBCs
exploited previously introduced methods for computing the band structures
of semiconductors. Indeed, a comparison between the equations of quantum
mechanics (QM), used to describe semiconductors, and electromagnetics
(EM), used to describe dielectric PBCs, shows many similarities, Table 3-1.

Table 3-1. Comparison between quantum mechanics and electromagnetics
formulations. [159]. G G
G
G
Field Ψ ( ) t,r G = Ψ ( )er iω t H ( ) Ht,r G = ( )er iω t
G
G
2
Eigenvalue problem H Ψ = E Ψ Ξ H = ( cω ) H
2
Hermitian operator − (= ∇ 2 ) G § 1 ·
H = + V () r Ξ = ∇ × ¨ ¨ G ∇ × ¸ ¸
2 m © ε () r ¹
A key difference, however, which restricts the general applicability of the
QM formulation to solve PBC problems is the scalar nature of the QM
problem compared to the vector nature of the EM problem. Fortunately,
however, unlike the QM semiconductor band structure problem, in which the
Bohr radius introduces a fundamental length scale and, as a result, similar
lattices with differing dimensions give rise to different behaviors, the EM
problem possesses no fundamental length scale constant. This means that the
properties of PBCs which differ only via a length expansion or contraction
of all distances, are related by simple expressions. In particular, given an EM
eigenmode obeying the equation,