Assembly of Photonic Band-Gap Crystals

Several materials including InGaAs, InAs, SiO2, and etc., have been used to fabricate PBG crystals [11]. Among those, SiO2 based PBG crystal is the most common one found whatever in nature or by man made.

There are several methods to establish PBG crystals, including nature sedimentation, electrophoresis method, template method, and so on [12]. Nature sedimentation is the most common and convenient method to fabricate PBG crystals. In order to set up a perfect PBG crystal, some points have to be paid attention to. Whatever a fabrication is applied, (1) slow sedimentation or assemble rate, (2) careful drying, (3) good dispersion, and (4) mono-size dispersibility of used particles are the most 4

important factors. If the speed of solvent evaporation too fast, particles will not be able to have enough time packing at proper sites, and thus various defects occur.

Small particles dispersed in water experienced energy turbidity in random direction which is called “Brownian motion”. It’s believed that van der waal force, electric potential force balanced with Brownian motion offer the driving energy for particles to move to the sites that have the lowest energy. Besides the Brownian motion, the solvent flow during sedimentation also should be concerned. In 2004, Norris [13] supposed that when particles depositing on an ordered packing layer, two kinds of the location were possible for sedimentation: clear and obstructed niches as pointed in Fig. 2- 5. Solvent flowed through the surface of particles. As the solvent evaporating from the system, more solvent traveled through the sites of clear niches than obstructed ones. In Fig. 2- 5(b), the distance for particles moving form clear to obstructed niches is four times than from clear to obstructed ones. This phenomenon may be a reason to explain why a planar PBG crystal often packs as a face-centered-cubic (FCC) structure.

There is still one more important key point to create a perfect PBG crystal during sedimentation: the dispersion. Mono-sized ceramic particles dispersing in water will have some interesting chemical reactions occur on the surface, and thus lead to the charging effects. The possible reasons of creation of surface charge on colloidal

(1) Hydration and base-acid reactions: colloidal surface adsorbed H⁺ or OH^-, and thus oxides compounds dispersed in solvent are affected by pH value, actually by electrical potential.

(2) Induced orientation of dipoles: colloid and solvent have different dielectric constant, so dipole-dipole moment may exist between these interfaces.

(3) Defects existing on colloidal surface: atoms or ions imperfectly packing may lead to the particle charged on the surface.

(4) Adsorption of foreign ions: foreign ions adsorbed by the colloid give rise to extra charge.

(5) Partial ionization in solvent: ions that dissolve in solution should satisfy its Ksp. therefore, the colloid and ion solubility are considered in balance.

So, if the particles have same sign of charging, such as strongly positively charged, then, these particles are kept in stable because the electrostatic repulsion force can keep colloids dispersed well in water without agglomeration. Otherwise, if the particles carry a low surface potential (usually less than 10 mV), they tend to coagulate together and

become unstable compared with the previous one. Equations. 2.1 and 2.2 describe the Coulomb force and ven der Waals force, where F is force, k is a constant (

4 0

1 πε ^{), d is}

distance between particles, q is charge, A is Hamaker constant, Rs is radius of particles.

2

12d2

F_a =− AR^s (2.2)

The competition between particles, i.e. attracting and repulsing forces, is mainly dependent upon eq. 2.1 and 2.2. A colloid with poor dispersion gives rise to a poor packing density [14, 15], and of course a perfect PBG crystal will not be obtained.

Measuring zeta- (ζ-) potential of colloid could give us the information to predict the behavior of ceramic particles in water. Table 2- 1 shows the IEP of some oxide materials [16]. Sometimes, IEP may vary with several factors: impurities, phase (single or polycrystalline), and etc. Al2O3 in Table 2- 1 shows a range. The reason is that bauxite contains significant impurities, including SiO2, Fe³⁺, Ti⁴⁺, Na⁺, and so on. These impurities may cause the particle surface reacted with water, and make the IEP shift from the pure oxide.

If we take SiO2 for an instance, the zeta potential of SiO2 approaches to zero as acidity of the aqueous solution is at pH = 2. The potential (absolute value) will rise up when pH value away from IEP. So it is concluded that dispersion SiO2 in a basic solution at pH = 10 leads to a stable suspension. As for the other cases [17], adding surfactant to modify the surface charge is also a good way to keep particles from agglomeration. Usually, surfactant that carries a specific function group can offer some mechanisms to achieve this goal, i.e. Coulombic repulsion and steric repulsion. As

group (-COOH, -NH2 or the others) can be ionized and become partial positive or negative. Therefore, the surface potential will be fully changed by polymer instead of the initial one carried by the particle surface. Base on this procedure, particles are dispersed well by the potential contributed from attached polymers. Another mechanism could be mentioned below. Some polymers attached on particles do not ionize. However, polymer branches will stretch or spread out in water, and this resulte in steric hindrance.

This phenomenon makes polymer molecules repulsing on to each other, and then the colloids are well dispersed.

Guoy and Chapman [18] explained this phenomenon by a model considering electric double layer. Particles that carry strong zeta-potential have a thick double layer, which offers the repulsion force to maintain the stability. When the double layer is compressed, the effect contributed by ven der Waals (attractive) force becomes dominated. Fig. 2- 6 illustrates the distance between two particles vs. net attractive or repulsive force. The maximum of the net force provides an energy barrier that allows particles without contact permanently. In the other hand, primary and second minima are the sites that the net force will become attractive, therefore, gives rise to particle agglomeration.

Im et al. [19] and Park et al. [20] created PBG crystals with PS (polystyrene) spherical particles by nature sedimentation. The particles were closed packed as FCC

ordered structure by self-assembly. Capillary force during solvent evaporation plays an important role because of causing the particles into a closed packing. Consolidation occurs at the interface of particles, solvent and air. The existence of meniscus between two particles drives them to stable position. After drying, these particles could be fixed on substrate. With the moving of liquid/air interface, consolidation occurs continuously until the whole suspension getting dried.

PBG crystals could also be assembled on template with patterns. Therefore, if there are periodic holes or grooves on a substrate, the particles will fill into the holes or grooves spontaneously. Base on this concept, a template with a designed pattern could be prepared for the assembly of particles. Xia et al. [20] prepared a Si substrate with coated photo resist, using lithography and RIE (reactive ion etching) to etch the required pattern. After that, 1 wt% silica colloidal suspension was spin-coated on the substrate.

The advantage for this method is that the template could be prepared with any designed 2 dimensional pattern, such as holes, rectangular, groove, or the holes arranged in FCC, HCP, etc. [23]

Fig. 2- 1 Images showing the natural PBG structures. The top one is an opal and the bottom ones are a butterfly and enlarged structure. [6]

Fig. 2- 2 A typical band-gap diagram representing a PBG crystal with diamond structure.

The components are in spherical shape and have a dielectric constant of 12. [11]

Fig. 2- 3 Schematic diagram that illustrates the silica glass tubes constructing a 2D-PBG crystal structure into fiber with the furnace operating at about 1800-2000^oC. [8]

Fig. 2- 4 SEM image illustrating the microstructure of a success waveguide consist of 12 stacked layers with a period of 4 μm. The designed waveguide itself is sandwiched by the upper and lower complete PBG crystals. [9]

Fig. 2- 5 A schematic diagram illustrating the behavior of particles by solvent flow along upward direction. Both clear and obstructed niches are possible packing sites for the moving particles. [13]

(a) (b)

Table 2- 1 A list showing the IEP point of some ceramic materials [16]

Materials IEP

WO3 0.43

SiO2 1-3

Soda lime silica glass 2-3

Potassium feldspar 3-5

SnO2 4-6

Zirconia 4-6 Apatite 4-6

TiO2 4-6

Kaolin 5-7 Mullite 6-8

Cr2O3 6-7

Fe2O3 8-9

Al2O3 6-10

ZnO 9-10

CaCO3 9-10

MgO 12

Fig. 2- 6 A potential diagram representing the net force resulting from summation of repulsive force (electric double layer) and attractive force (ven der Waals force). [16]