Re: 3D doubts

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