On Thu, 9 Nov 2006, Vincent Paeder wrote:
I want to simulate a 3D slab of rods (see attached file rods-3d.ctl). I started with the 2D case and supposed that, if the slab fills the cell entirely in the z direction, I'd find the same band diagram as in 2D.However, I don't, and I don't either understand why. With increasing z, one can see flat bands appearing here and there. I don't see for what kind of physical reason that may happen. Therefore, I thought of groping around the following things: 1) influence of the resolution for fixed z 2) influence of the mesh size for fixed z 3) influence of the cell size (in z) 4) influence of the number of bands Nothing seems to help. Changing z is the only thing that really makes a difference and it's the only thing that shouldn't. Now I'm running out of ideas. Does anybody have a clue?
Your assumptions are incorrect. The 3d case *should* have additional bands compared to the 2d case.
In two dimensions, you are only looking at light propagating in the plane (for kz = 0).
When you include a finite cell size S in the z direction, then you are including out-of-plane wavevectors kz + 2*pi*n/S for all integers n, by the periodic boundary conditions. These out-of-plane wavevectors are what are giving you your additional states, which depend on S. This is not a numerical effect, it is a physical consequence of the question you are asking MPB to solve.
You can think of this as the "folding" into the finite Brillouin zone of the vertical periodicity S (recall that MPB uses periodic boundary conditions). (In the 2d case where S=0 the Brillouin zone in the vertical direction is infinite and there is no folding.)
Or, if you like, you can think of this in terms of higher-order "standing-wave" modes in the vertical direction. That is, the vertical mode profile need not be constant, it can be cos or sin of 2*pi*n/S for any n.
See e.g. my paper in Phys. Rev B vol. 60, p. 5751 (1999) for further discussion of the consequences of slab thickness. Or some of my tutorial presentations at ab-initio.mit.edu/photons/tutorial
Cordially, Steven G. Johnson _______________________________________________ mpb-discuss mailing list mpb-discuss@ab-initio.mit.edu http://ab-initio.mit.edu/cgi-bin/mailman/listinfo/mpb-discuss
