S. Matthias, R. Hillebrand, and F. Müller
Large area three-dimensional (3D) photonic crystals were fabricated by macroporous silicon etching. This is possible by controlled strong diameter modulations. It is theoretically proven that simple cubic three-dimensional photonic crystals have a complete photonic band gap with a width of 4.99% in the IR. This page is focused on numerical simulations concerning the tolerable disorder allowing still for a complete photonic band gap. Fabrication tolerances for the simple cubic arrangement of intersecting air-spheres in silicon are deduced.
Fig. 1 shows a scanning electron micrograph (SEM) of a simple cubic 3D network of intersecting air-spheres obtained by isotropic etching of a strongly modulated macroporous silicon sample.
The following band structure calculations systematically analyze the influence of i) radius variations of the spheres, and ii) that of ellipsoidal deformations of the spheres. In  you can find details on the effect of tetragonal lattice distortions. Band structures of a simple cubic lattice of intersecting air spheres in Si (ε=11.6) have been compiled using the well-established MIT package http://ab-initio.mit.edu/photons/index.html.
In Fig. 2 the photonic bands are plotted along the k-directions of the irreducible Brillouin zone. The maximum band gap appears for a sphere diameter dSphere=1.20a around a central frequency of f0 = 0.482 [a/λ]. The size of the band gap is 4.99% related to the central gap frequency f0. Fig. 2(b) indicates the important geometrical parameters of the structure model.
In Fig. 3(a) the diameter of the air spheres dsphere is altered around the optimum value of dsphere=1.20a. The complete photonic band gap between the 5th and 6th band (cf. Fig. 2) remains open for remarkable size variations of the spheres. In Fig. 3(b) we continue to assume perfect cubic lattice geometry. The ellipsoidal air volumes arranged in the cell are characterized by their eccentricity dz/dx,y = [0.90, 1.10], assuming that z is the growth direction. The effective air volumes correspond to sphere diameters deff of 1.15a, 1.21a, 1.30a. The band structure calculations clearly show that perfect spheres with appropriate air filling ratio provide the maximum complete band gap.
Let us assume an optical device, which works with a fixed frequency near the center of the gap map, utilizing the effect of inhibited photon transfer. The computations, providing Fig. 4(a), show that a variation of the air sphere diameter of 0.032 a, which is 2.65% relative to the total sphere diameter, is sufficient to push the working frequency outside the gap. Expressed in terms of the air filling fraction, the value of 80.5% is optimum. A change of the air volume of ±4% will prevent an overall gap for one fixed frequency. Nevertheless, there are possible applications that might profit from a very narrow, precise complete band gap.
If it can be managed to keep the air filling fraction of the photonic crystals constant on a large scale, solely the shape effects of the air volumes can influence the photonic properties. Fig. 4(b) shows a specific gap map, where the air filling is assumed to be constant. The graph shows that pure elliptical deformations are uncritical if the porosity is near optimum. The figure has little slope, which makes the photonic band gap relatively robust with respect to ellipsoidal deformations of up to ±10%.