Dear Steven,thanks for your advices, and I have make an input file
accordingly to make the supercell, as follows
; supercell scheme
; * * * o o o o * * * --------layer one
;* * * o o o o o * * * --------layer two
(define-param supercell-x 9) ; the (odd) number of lateral supercell
periods,layer two
(set! geometry-lattice (make lattice
(basis1 (/ (sqrt 2) 2) (/ (sqrt 2) 2))
(basis2 (/ (sqrt 2) -2) (/ (sqrt 2) 2))
(size supercell-x 2 no-size)))
(define-param eps 12) ; the dielectric constant of background
(define-param barhigh (* (sqrt 2) 0.2)) ; the heighty of the air bars in the
bulk crystal
(define-param barwidth (* (sqrt 2) 0.5)) ; the Width of the air bars in the
bulk crystal
(define-param slabhigh (+ (* (sqrt 2) 4) 1)) ; the heighty of the air area
in the Left side
(define-param slabwidth (+ (* (sqrt 2) 2) 1)) ; the Width of the air area in
the Left side
(define-param airwidth (* (sqrt 2) 3)) ; the Width of the air area in the
right side
(define-param airhigh (+ (* (sqrt 2) 2) 1)) ; the heighty of the air area in
the right side
(set! default-material (make dielectric (epsilon eps)))
;make periodic crossing holes
(set! geometry
(list (make block (center 0 0 0) (material air) (size barwidth
barhigh infinity)))
(append
(list (make block (center 0 0 0) (material air) (size barhigh
barwidth infinity)))))
(set! geometry
(append
; duplicate the crossings over the supercell:
(geometric-objects-lattice-duplicates geometry)))
;make Left air area
(set! geometry
(append (list (make block (center (- (* (sqrt 2) -3.5) 0.25) (+
(sqrt 2) 0.25) 0)
(material air) (size airwidth airhigh infinity)))))
;make Right air area
(set! geometry
(append (list (make block (center (+ (* (sqrt 2) 3.5) 0.25) (+
(sqrt 2) 0.25) 0)
(material air) (size airwidth airhigh infinity)))))
The problem is that the crossing is not duplicated. It seems that the
"Apend" command doesnot work, why? Thanks a lot.
First of all, that paper was looking at two-dimensional photonic crystals,
*not* photonic-crystal slabs (although of course one could do an analogous
calculation for a slab).
Although I don't have the input file that was used for that figure in the
paper, I don't think you can infer the supercell from figure 3(a), for two
reasons.
First of all, for a surface-state calculation, you want to use only a
single unit cell in the periodic direction(s) parallel to the surface. In
figure 3(a), several unit cells are shown for illustration purposes.
Second, in the direction perpendicular to the surface, you want a
supercell, with enough air (on one side of the surface) and photonic
crystal (on the other side of the surface) that the exponentially
localized surface mode(s) don't "see" the boundary of the supercell. How
big your supercell has to be therefore depends on the structure and on
what wavevector you are looking at, which determines how localized your
states are.
I'm not sure exactly what you mean by "one-dimensional photonic crystal".
It is true that a surface in a 2d photonic crystal is periodic only along
one dimension, as I mentioned above. But the dielectric function is not
univariate, so the system is not "one-dimensional" per se.
Cordially,
Steven G. Johnson
PS. Because of the periodic boundary conditions of the supercell, your
surface-state calculation will have two surfaces of the photonic crystal
(there is no way to make the crystal semi-infinite). There are two
approaches to dealing with this. One approach is to make the surfaces
symmetric, as in figure 3 of the PRB, and look at say the even modes (or
the odd modes, it doesn't matter), which will have identical surface
states on both surfaces, with the same dispersion relation if the
supercell is large enough. Another approach is to terminate *one* of the
surfaces with a termination that doesn't support surface states; this
requires some experimentation if you don't know what termination you need,
however.
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