Photonic band structure, or photonic bandgaps, of a diamond lattice of spheres * diamond sphere structures * pyrochlore structures

Photonic Band Structure of a Diamond Lattice of Spheres

   Indisputably, one of the hallmarks in the study of photonic band gap structures is an article by Ho, Chan, and Soukoulis [1]. It opened the way for a fabrication of the first photonic structure with a complete photonic band gap (CPBG) [2] and advanced the field considerably. Its main conclusions were that, regarding a CPBG, the diamond structure (for geometrical arrangement of spheres in the diamond lattice see here) fares much better than a simple face-centered-cubic (fcc) one. Namely: However, the bulk photonic KKR method [4] yielded results quantitatively different from the earlier plane-wave calculations of Ho, Chan, and Soukoulis [1]. These results have been first discussed in my comment. Additionally, you can also download a few data ASCII files, as obtained by the bulk photonic KKR method [4], and two PDF figures (courtesy of Mischa Megens), which show a comparison of the results obtained by the bulk photonic KKR method against those obtained by the MIT Photonic-Bands (MPB) package. Full account of this story you will find in Ref. [4], which also includes metallo-dielectric spheres and zinc-blende structures. Here a brief summary is provided.

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

2*N*(lmax+1)**2 x 2*N*(lmax+1)**2,

where 2 stands for two independent polarizations, N is the number of atoms in the lattice unit cell, and lmax is the cutoff on the orbital angular momentum used in the expansions into spherical waves. Convergence is tested with increasing the cutoff value of lmax. Computational times with and without dispersion of the dielectric contrast are the same. I have run calculations till lmax=9. You can download here the (lmax=9) data for the two lowest photonic bands, calculated with lmax=9 (in units [c/A], where c is the speed of light in vacuum and A is the lattice constant of the conventional cubic unit cell of the diamond lattice). In the first column, there are labels of (between any pair of special points, equidistantly spaced) points on the characteristic path used in Fig. 1 of the comment. The special points carry a label 1+7*n, where n is an integer. The second column of the (lmax=9) data contains eigenvalues (zeros of the photonic KKR determinant). I have used such a high value of lmax to maintain precision of eigenvalues within 0.001 in the whole frequency range. In order to have an idea of the convergence properties of the photonic KKR method, you can download here the respective (lmax=2) data and (lmax=8) data for the two lowest photonic bands, calculated with the respective lmax=2 and lmax=8. You will find that, already for lmax=2 (i.e., with the secular matrix of the size 36 x 36), the lowest two bands are determined within 6% of the converged values (see figure). The (lmax=8) data are within 0.001 of the (lmax=9) data (see figure). For a comparison of the low frequency dispersion with the Garnett formula [5,6] see comment and Ref. [4].

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.


  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)

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