Re: Re: Point defects in slab structure - resonant frequencies

On Fri, 6 Apr 2007, Rob wrote:
> 1. I added statements for compute-energy-in-objects, but found most of 
> the percentages were low (the highest was 15%).  When I looked at the Hz 
> field patterns (I am using a hexagaonal lattice of air holes in Si, r/a 
> = 0.3, h/a = 0.6) at z=0 output using h5topng expanded for a couple of 
> periods, I do not see field localization in the center defect, but do 
> see field localization in the expanded periodic copies of the defect. 
> Please see: http://img235.imageshack.us/img235/8948/b75ap1.jpg

Are you sure that this isn't just a phase issue?  i.e. if you are looking 
at the real part of the Hz field, try looking at the imaginary part or 
vice versa.

> Is there a physical reason why the periodic copies of the supercell and its
> defect seem to have field localization, but not the center one?  Does this
> explain why my compute-energy-in-objects percentages are low for a cylinder
> object placed at the center of the simulation area?

The compute-energy-in-objects percentages are most likely low because your 
structure has a low Q, and hence the fraction of the energy in the slab is 
small.  Alternatively, your cylinder may be too small.

> 2. For a 2D point defect, the resonant frequency should be equal at all k
> points.

Only in the limit as the supercell size becomes large (the bandwidth 
should decrease exponentially fast with the supercell size).

> Should this hold for slab structures?  When I plot all of the bands in 
> the gap at once for a slab structure, one can visually see flat bands, 
> but when I plot individual bands, the frequencies seem to vary with k.

Many of the bands "in the gap" for a slab will correspond to radiative 
modes from the light cone, not resonances at all, and their frequencies 
will certainly vary with k (with the bandwidth decreasing only linearly 
with supercell size).

Even for the resonant modes, the bandwidth should decrease relatively 
slowly (quadratically, I think?) with supercell size, because they are not 
exponentially localized except in the plane of the slab.  However, the 
constant factor should be much smaller.

Leaky modes in a supercell calculation are somewhat of a tricky business.

Steven