19.1 Photonic Band Gap Materials
493
Table 19.1 Various photonic band gap structures and some of their properties. The band gap is between the n-th and
(n + 1)th band, ˜
ω is given for air/silicon ( ≈ 12)
Name
Crystal type
n
˜
ω (%)
Refs.
Diamond
diamond
2
29
[1598]
Yablonovite
fcc
2
19
[1602]
Woodpile
fc tetragonal
2
20
[1603]
Spirals
sc
4
17
[1604]
Square-spirals
tetragonal
4
24
[1599]
Layered 3D
bc orthorhombic
4
23
[1605]
Inverted scaffold
sc
5
7
[1606]
Inverse opal
fcc
8
4.25
[1607]
Inverse hcp
hcp
16
2.8
[1608]
band gap must not arise above the first band, relaxing problems due to asymmetry of the Brillouin zone
(cf. Table 19.1).
Maxwell’s equations (zero charge density) for monochromatic waves ∝ exp(iωt) (and isotropic
dielectric function)
∇ · D = 0
(19.31)
∇ × E = i
μ ω
c
H
(19.32)
∇ × H = i
ω
c
D
(19.33)
∇(μ H) = 0 ,
(19.34)
together with D(r) = (r) E(r) and μ = 1 they are combined into the wave equation
∇ ×
−1
(ω, r) ∇ × H(r)
+
ω
2
c 2 H(r) = 0 .
(19.35)
This equation is numerically solved for planar waves with wavevector k.
In the following, results are shown for various structures. In a fcc lattice of air spheres in a dielectric
medium with n = 3.6 (a typical semiconductor), no band gap can be achieved (Fig. 19.9a), only a
pseudogap (Fig. 19.9b) appears.
In a diamond lattice (imagine as two fcc lattices shifted by 1/4 111), a complete photonic band
gap is possible [1598] (Fig. 19.10). Recently, a periodic array of spirals (Fig. 19.11) has been predicted
to exhibit a large photonic band gap [1599]. Glancing-angle deposition [1600] (GLAD) is a way
to realize such structures. Another method to fabricate structures with arbitrary geometry within a
material is two-photon lithography or two-photon holography. Another path to PBG structures are
so-called inverted opals. First, a close-packed structure of spheres, e.g. monodisperse silica spheres,
is fabricated by sedimentation or self-assembly. The gaps are filled with a high-index medium and the
template is subsequently removed, e.g. by etching or dissolving. The resulting structure is shown in
Fig. 19.12a. Such a structure has a photonic band gap (Fig. 19.12b) if the refractive index is sufficiently
high (> 2.85) [1601]. The band gap in this case is between the 4th and 5th band. Table 19.1 offers a
compilation of various PBG structures and their properties.
493
Table 19.1 Various photonic band gap structures and some of their properties. The band gap is between the n-th and
(n + 1)th band, ˜
ω is given for air/silicon ( ≈ 12)
Name
Crystal type
n
˜
ω (%)
Refs.
Diamond
diamond
2
29
[1598]
Yablonovite
fcc
2
19
[1602]
Woodpile
fc tetragonal
2
20
[1603]
Spirals
sc
4
17
[1604]
Square-spirals
tetragonal
4
24
[1599]
Layered 3D
bc orthorhombic
4
23
[1605]
Inverted scaffold
sc
5
7
[1606]
Inverse opal
fcc
8
4.25
[1607]
Inverse hcp
hcp
16
2.8
[1608]
band gap must not arise above the first band, relaxing problems due to asymmetry of the Brillouin zone
(cf. Table 19.1).
Maxwell’s equations (zero charge density) for monochromatic waves ∝ exp(iωt) (and isotropic
dielectric function)
∇ · D = 0
(19.31)
∇ × E = i
μ ω
c
H
(19.32)
∇ × H = i
ω
c
D
(19.33)
∇(μ H) = 0 ,
(19.34)
together with D(r) = (r) E(r) and μ = 1 they are combined into the wave equation
∇ ×
−1
(ω, r) ∇ × H(r)
+
ω
2
c 2 H(r) = 0 .
(19.35)
This equation is numerically solved for planar waves with wavevector k.
In the following, results are shown for various structures. In a fcc lattice of air spheres in a dielectric
medium with n = 3.6 (a typical semiconductor), no band gap can be achieved (Fig. 19.9a), only a
pseudogap (Fig. 19.9b) appears.
In a diamond lattice (imagine as two fcc lattices shifted by 1/4 111), a complete photonic band
gap is possible [1598] (Fig. 19.10). Recently, a periodic array of spirals (Fig. 19.11) has been predicted
to exhibit a large photonic band gap [1599]. Glancing-angle deposition [1600] (GLAD) is a way
to realize such structures. Another method to fabricate structures with arbitrary geometry within a
material is two-photon lithography or two-photon holography. Another path to PBG structures are
so-called inverted opals. First, a close-packed structure of spheres, e.g. monodisperse silica spheres,
is fabricated by sedimentation or self-assembly. The gaps are filled with a high-index medium and the
template is subsequently removed, e.g. by etching or dissolving. The resulting structure is shown in
Fig. 19.12a. Such a structure has a photonic band gap (Fig. 19.12b) if the refractive index is sufficiently
high (> 2.85) [1601]. The band gap in this case is between the 4th and 5th band. Table 19.1 offers a
compilation of various PBG structures and their properties.