Photonic Band Structure of a Diamond Lattice of Spheres

• i) the threshold value of the dielectric contrast eps to open a CPBG is 4 (8.2 for an fcc structure [3]),
• ii) a CPBG opens between the 2nd and 3rd bands (the 8th-9th bands for an fcc structure), and, consequently, is much more stable against disorder, and
• iii) a CPBG is significantly larger (15% and 5%, for the respective diamond and fcc closed packed lattice of spheres with a dielectric contrast eps=12.96).

The KKR method is based on the first-principle multiple-scattering theory and turns out to be the most efficient in dealing with both highly dispersive materials and highly symmetric scatterers. The size of a secular matrix in the bulk photonic KKR method [3,4] is

For those who would like to reproduce these results, the cutoff on the number of direct lattice vectors used in the Ewald summation was set to NRMAX=343, whereas the cutoff on the number of reciprocal lattice vectors used in the Ewald summation was set to NKMAX=1400. The value of the Ewald parameter in the photonic KKR method [3] was taken to be Q=1.6 (in the notation of Ref. [7]). This choice of the Ewald parameter ensured that the two cutoffs NRMAX and NKMAX have never been exceeded in my calculations and yet ensured that the structure constants have been determined within 5 digits of accuracy in the whole frequency range.

In the comment, it has been discussed that results computed in early days of the plane-wave expansion method have often turned out imprecise, an example being Ref. [8]. Indeed, for a two-dimensional square lattice of air circular rods with a filling fraction f=67% in a dielectric material of refractive index n=4.25, Villeneuve and Piche predicted a 2D complete band gap of 11% (see Figs. 1b and 2 of Ref. [8]). However, the 2D bulk photonic KKR method [9] predicts that the complete band gap actually disappears (see figure). That the results of Villeneuve and Piche [8] are imprecise, has been, privately, confirmed by the authors. This can also be verified independently, using a transfer-matrix of Pendry and McKinnon [10]. Seemingly, when using the plane-wave expansion method, one has to exercise extreme care when handling systems with a stepwise changes in the dielectric contrast, which is a generic case of photonic crystals. The reason is an unaviodable presence of the Gibbs instability, resulting in a poor convergence of the Fourier transform of the hard-sphere dielectric function, as has been nicely discussed in Ref. [11] (see, for instance, Fig. 2 there). Even up-to-date variants of the plane-wave expansion method, in particular the MPB package of Steven G. Johnson, still needs an extrapolation to infinitely many plane waves to yield converged results. Recently, using the MPB package, Mischa Megens performed such an extrapolation of the plane wave method results obtained with N**3 plane waves, where N=16 and 32. A linear extrapolation of the N=16 and N=32 frequencies as a function of 1/N to 1/N --> 0 yielded result almost identical to that of the photonic KKR method but for the two lowest bands (see Fig. here). The band edges of the two lowest bands are cca 1.5% lower than those calculated by the photonic KKR method. A convergence of the 2nd-3rd gap width as a function of N is presented here. The last two figures are courtesy of Mischa Megens who also kindly provided his control file, which you can download here.

Prof. C. Soukoulis has agreed on the validity of my calculations.

A follow-up [12] has been recently published, which deals with photonic band structure calculations for diamond and pyrochlore crystals and which compares the KKR results against those obtained by the MIT-package. Please, see for details here.

Some additional information you can find in the PhD thesis of Esther C. M. Vermolen on Manipulation of colloidal crystallization (cca 9.4 MB) from 2008.

References

1. K. M. Ho, C. T. Chan, and C. M. Soukoulis, Existence of a photonic gap in periodic dielectric structures, Phys. Rev. Lett. 65, 3152-3155 (1990).
2. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, Photonic band structure: The face-centered-cubic case employing nonspherical atoms, Phys. Rev. Lett. 67, 2295-2298 (1991).
3. A. Moroz and C. Sommers, Photonic band gaps of three-dimensional face-centered cubic lattices,
   J. Phys.: Condens. Matter 11, 997-1008 (1999). [physics/9807057] [story behind this article]
4. A. Moroz, Metallo-dielectric diamond and zinc-blende photonic structures,
   Phys. Rev. B 66, 115109 (2002). [cond-mat/0209188] [pdf]
5. J. C. Maxwell Garnett, Colours in metal glasses and in metallic films, Phil. Trans. R. Soc. London 203, 385-420 (1904).
6. A. Moroz, A simple formula for the L-gap width of a face-centered-cubic photonic crystal,
   J. Opt. A: Pure Appl. Opt. 1, 471-475 (1999). [physics/9903022]
7. A. R. Williams, J. F. Janak, and V. L. Moruzzi, Exact Korringa-Kohn-Rostoker energy-band method with the speed of empirical pseudopotential methods, Phys. Rev. B 6, 4509-4517 (1972).
8. P. R. Villeneuve and M. Piche, Photonic band gaps in two-dimensional square lattices: Square and circular rods, Phys. Rev. B 46, 4973-4975 (1992).
9. H. van der Lem and A. Moroz, Towards two-dimensional complete photonic-bandgap structures below infrared wavelengths,
   J. Opt. A: Pure Appl. Opt. 2, 395-399 (2000).
10. J. B. Pendry and A. MacKinnon, Calculation of photon dispersion relations, Phys. Rev. Lett. 69, 2772-2775 (1993).
11. H. S. Sozuer, J. W. Haus, and R. Inguva, Photonic bands: Convergence problems with the plane-wave method, Phys. Rev. B 45, 13962-13972 (1992).
12. E. C. M. Vermolen, J. H. J. Thijssen, A. Moroz, M. Megens, and A. van Blaaderen, Photonic band structure calculations for diamond and pyrochlore crystals,
    Optics Express 17(9), 6952-6961 (2009).

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Alexander Moroz, August 1, 2001 (last updated on April 1, 2011)