Photonic Crystals: Computational Studies

Abstract

Abstract People Publications Cooperations Alumni

R. Hillebrand, S. Matthias, and F. Müller

In this page a photonic crystal (PC) approach to an inverted diamond-like structure with a large complete band gap is introduced. We theoretically study the dispersion behavior of PCs being fabricated by modulating macroporous silicon. The widening of single narrow pores to air bulges of spherical or ellipsoidal shape follows the lattice sites in the diamond cell. This could be accomplished by an appropriate pre-structuring of the etch pits. The inverted diamond-like structure could be achieved starting from a square basis or from a hexagonal arrangement of etch pits on the Si wafer. For the 2D square lattice the pore growth direction is parallel (001).

Fig. 1 reveals the 3D arrangement of the sphere/pore - system in the Si block with one cell marked by dotted lines. It has to be noted that for graphical reasons the air volume is shaded, whereas the Si volume fraction is shown in white.

The distance between two successive air bulges in a pore is one lattice parameter a for all pores assuming that the neighboring pores have offsets of a/4.

We compile photonic band structure calculations as a function of i) the radius of the air spheres, ii) the radius of the connecting pores, and iii) of the shape (sphere/ellipsoid) of the air bulges (for details see: [1]). The band structure computations presented here were done with the mit-package (http://ab-initio.mit.edu/photons/index.html).

Fig. 2 shows the band structures of a Si-based photonic crystal of inverted diamond type. For an air sphere radius of r_spere= 0.33a the band gap size is 28.2%, related to the midgap frequency of 0.609 [a/λ]. The Brillouin zone is given as inset.

Now it is assumed that there are connecting air pores parallel to the (001) direction, as shown in Fig. 1. The related band structure of Si-based photonic crystals of inverted diamond type with connecting cylinders is displayed in Fig. 3.

The parameters are: Air sphere radius r_spere= 0.30a, cylinder radius r_pore=0.10a, gap: 5.7%, central frequency f=0.61 [a/λ]. It is obvious that connecting air pores cause a shrinkage of the complete band gap. The perfect symmetry of the diamond cell is distorted.

Fig. 4 studies the variations of the complete band gap size vs. radius changes of the connecting pores.

At the left, the air sphere radius r_sphere in the diamond cell varies in the interval [0.24a, 0.34a], the undistorted diamond cell provides the black curve. For the air cylinders radii of 0.05a, 0.10a are assumed, which clearly reduces the gap size attainable (Notice that the vertical dotted lines refer to the right part of Fig. 4). The right part shows the decrease of the complete band gap size for an increasing radius of the connecting air pores for two selected sphere radii (0.28a, 0.33a).

Up to now we assumed that the air bulges in the diamond cell are perfectly spherical. The more general case is to experimentally fabricate and to numerically simulate ellipsoidal air volumes. Possible effects on the complete band gap size are studied in detail in Fig. 5.

For an air volume corresponding to r_eff=0.30a and r_pore=0.1a the size of the complete band gap increases from 5.7% for air spheres to 7.88% for an ellipsoid of the eccentricity of d_{_|_}/d_|| =1.6. The ratio of 1.6 means that the air bulge is lens shaped. Lens-shaped ellipsoids can considerably increase the gap size (r_pore=0.09, gap: 10.2%) and partially compensate the distortions caused by the connecting pores.