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Opal-like photonic crystal with diamond lattice

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One of the most important goals in photonic band gaps (PBG) technology is the search for methods to produce 3D systems with reasonably large full gaps in the visible and infrared regions of the spectrum. The approach many researchers have followed is the search for appropriate assemblies that present the most spherical Brillouin zone e.g. face centered cubic (fcc) or diamond structures as a particular case thereof. Structures with a diamond lattice can show large gaps providing a refractive index contrast above 2 is achieved. So far there are, mainly, two approaches that allow to obtain PBG materials with reasonably broad full gaps. One involves lithographic techniques to build up woodpile arrangements or layered structures; these methods are time consuming, technologically sophisticated and allow stacking reduced numbers of layers. Alternatively, the opal templating route is very attractive because it is an easy and cheap method. Inverted opals can develop full gaps when refractive index contrast is above 2.8 as has recently been reported for inverted silicon and germanium. However, the full gap, that appears between the 8th and the 9th bands, is very fragile as it is strongly influenced by the presence of defects. This problem would be solved if routes leading to the packing of micro-spheres in a diamond lattice by self-assembly techniques could be found. Unfortunately, unlike fcc arrangements, which are obtained simply by gravity deposition or template directed sedimentation, this structure is far from minimum free energy requirements to self-assemble.

Here we present a method based on robot-aided micromanipulation that opens the possibility of building up 3D photonic crystals with diamond structure (20 microns in size typically). Robot-aided manipulation of single microspheres on a template substrate allows assembling a body centered cubic (bcc) lattice, which in fact, is the combination of two diamond structures. The crucial point at this moment is designing a way in which the right atoms from the bcc structure can be removed in order that the remaining ones are arranged in the sought for diamond structure. This can be achieved if the sacrificial atoms are of a class, in some sense, chemically different from the other atoms thus providing a means for the dissolution. This can be done with inorganic vs. organic microspheres arranged in the points of a bcc lattice. So our method starts with the construction, of a heterogeneous structure of mixed inorganic and organic (e.g. silica or silicon and latex) spheres in a bcc lattice. This structure, composed of two kinds of spheres, we will call, from now on, mixed body centered cubic (mbcc). Subsequently, upon selective removal of the organic particles a diamond structure of the inorganic ones is obtained. As adjacent spheres are in contact, the remaining diamond opal made of inorganic particles should be stable.

FIG. 1. Computer design showing the layout of the first four layers in a mixed body centered cubic lattice where dark and light gray spheres symbolize organic and inorganic particles respectively. (a) Stacking along the (001) direction; the fifth layer would be exactly as the first one (b) staking along (111) orientation. The insets represent a top view.

The construction of an mbcc lattice as an intermediate step presents some advantages over direct diamond lattice assembly. The diamond lattice has a very low filling fraction for touching spheres (~34%) which makes it too unstable to be constructed directly. On the other hand, mbcc lattice allows stacking spheres along, at least, two orientations: (001) and (111). In both cases sphere positions are minimum energy locations once the first layer is in place. In either case the initial layer should be ordered on a template substrate.

Growth in the (001) orientation consists in stacking layers of mixed i-o spheres with a square lattice configuration where nearest neighbors are of opposite character leading to diagonals of like character reminding a checkerboard. The cyclic arrangement of subsequent layers is explained in Fig. 1(a). Each layer is shifted with respect to the layer underneath by half the pitch (both along the x and y directions). The pitch is 1.15·d (where d stands for the sphere diameter) and the distance between successive layers results in f100=0.58·d.

The stacking procedure along (111) direction is shown in Fig. 1(b). Here each layer presents a triangular lattice configuration and, as opposed to the (001) case, layers are homogeneous in composition (completely organic or inorganic). The layering sequence reminds that of an FCC lattice (…ABC…) but, in this case, two layers of each material are laid successively. For example, AiBoCo-AiBiCo-AoBiCi…, where sub-indices indicate inorganic (i) or organic (o) particles. The in-plane distance between neighbor spheres is 1.63·d and interlayer distance is f111=d/3.

Once the mbcc has been constructed along either (001) or (111) orientation, organic particles are selectively removed to leave a diamond structure of inorganic particles (see step (b) in Fig. 2). At this point two different routes can be pursued depending on the refractive index contrast of the structure. If the spheres used have a refractive index above 2.0 an opal with an omnidirectional gap results.3 Then, gap width may be fine tuned through a sintering process similar to that reported for templating silica opals (step (c) in Fig. 2). It has been recently proposed a method to produce monodisperse spheres from several oxides with high refractive indices that could serve this purpose. Photonic band gap calculations predict full gaps width as large as 13% for a diamond lattice of high refractive index spheres e.g. silicon (e=12) in air background when filling fraction is 43 %.

FIG. 2. Computer simulation showing, in five steps, the fabrication of an inverse diamond structure with a full photonic band gap. First a mixed body centred cubic lattice is assembled (a) after which latex sublattice is removed (b); then the structure is sintered to a filling fraction of ca. 50 % (c); after that silicon or germanium infiltration (d) takes place and finally silica elimination (e).

When the refractive index of inorganic particles is not above the threshold for full gap opening one should proceed with the inverse diamond structure depicted in steps (d) and (e) of Fig. 2. This possibility holds when silica is used as inorganic material, for instance, in the case of silica-latex particles the following procedure would be performed: First, the mbcc structure should be arranged on a patterned substrate with the aid of micro-robot technique. Next, latex spheres are selectively removed by a calcinations process. The remaining diamond crystal is then infiltrated with a high dielectric constant material like silicon or germanium, and finally, the removal of the silica spheres produces the inverse opal. Since our ultimate goal is to obtain an inverse structure with a complete PGB, one should take into account that touching air cavities (filling ratio ~34%) in a dielectric medium provide a very narrow gap unless air filling ratio is increased. A sintering process, as depicted in Fig. 2(c), performed prior to the infiltration, provides the necessary tool to have an inverse structure with a tailored full gap.

FIG. 3. Photonic Band diagrams of a silicon/silica composite diamond opal (upper panel) and that made of air spheres in silicon resulting from the removal of the silica spheres from the former (lower panel). The filling fraction for silicon being 50%. The inset shows the real space corresponding structures.

Fig. 3 shows the photonic band structure calculations for a silicon infiltrated diamond opal of silica spheres along with that of the inverse structure resulting from silica etching. Filling fraction of the silica diamond opal is taken to be 50% (a reasonable value for sintering process) which determines the air filling fraction of the inverse silicon structure. Maximum values of the gap width are obtained for air spheres filling fractions of 78%, however, such a high value can be unrealistic from a practical point of view.

Other possibilities can be considered when device design is envisaged. Imperfections or uncontrolled defects in the structure are more critical in photonic crystals with larger refractive index contrast. Therefore, one should also cons der diamond opals with smaller dielectric contrast (as silicon/silica composites) providing they have a full gap. Table 1 shows the parameters involved in the fabrication of several silicon diamond lattices with different dielectric contrasts.

It can be seen that even for small dielectric contrasts (as silica diamond lattice infilled with silicon) one can obtain reasonable gaps of 4%. It corresponds to a band width value of about 7 THz in the 1.5 microns region, and it corresponds to a diamond lattice parameter of 0.6 microns. This figure is quite reasonable for many applications as Dense Wavelength Division Multiplexing (DWDM) devices.

Table 1. Values of full gap width (Dw/w) and midgap position (a/l,
where a is the lattice parameter) for different configurations in which materials and filling fraction (ff) percentages have been varied.