JOURNAL OF APPLIED PHYSICS
VOLUME 94, NUMBER 7
1 OCTOBER 2003
Multiple photonic band gaps in the structures composed
of core-shell particles
V. Babin and P. Garstecki
Institute of Physical Chemistry PAS Dept. III, Kasprzaka 44/52, 01224 Warsaw, Poland
R. Hołyst
Institute of Physical Chemistry PAS Dept. III, Kasprzaka 44/52, 01224 Warsaw, Poland
and WMP-SNS, Cardinal Stefan Wyszyński University Dewajtis 5, Warsaw, Poland
共Received 10 February 2003; accepted 27 June 2003兲
We present a detailed study of the full photonic band gaps for the spherical multilayered core-shell
particles arranged in bcc, fcc, and diamond fcc lattices. We find that layered structure of the particles
does not lead to an increase of the gap width in case of the fcc arrangement. On the other hand full
photonic band gap opens up for the bcc lattice composed of such particles. The most promising is
the diamond fcc arrangement. For such ordering of core particles covered by a thin shell we find
three reasonably wide gaps. © 2003 American Institute of Physics. 关DOI: 10.1063/1.1604932兴
I. INTRODUCTION
three shells arranged in the bcc, fcc, and dfcc lattices and
point out the best candidates for the applications. Here we
can change the number of shells in a particle and the type of
lattice 共bcc, fcc, or dfcc兲 and check which structure gives the
widest band gaps or the biggest number of wide band gaps.
We show that the core-shell spherical particles, with one or
two shells, arranged in the dfcc lattice have a very rich photonic band gap structure with up to three reasonably wide
band gaps.
Since the discovery of the photonic crystal,1 i.e., a material with periodic distribution of the refractive index, which
has a photonic band gap, a lot of theoretical work has been
devoted to the search for the structures which exhibit wide
band gaps.2– 4 Such studies are extremely important for technology because of a large number of possible applications of
photonic crystals ranging from telecommunication 共transmission of data at high density兲 to optical computers.
Theoretical5 and experimental6 studies show that the colloidal fcc and diamond fcc 共dfcc兲 crystals are very promising
for the optical applications. In the bcc crystal composed of
homogeneous particles, only pseudogaps have been observed
and therefore such structures have been ruled out as possible
candidates for photonic applications. The fcc and bcc colloidal crystals can be produced by various methods. For example, colloidal particles in suspensions self-assemble and
form such a crystal.7 Only the dfcc structures have been hard
to obtain. However, the recent studies of surfactant solutions8
show that there are interactions favoring diamond fcc ordering. Additionally, the recent successful efforts9 to fabricate a
colloidal crystal characterized by the density smaller than the
close packed density open up possible ways to obtain dfcc in
colloidal suspensions. Another possibility to obtain such ordering is the usage of anisotropic solvents.10 Finally, a different approach to the formation of such crystals has been
presented in Ref. 11—the authors used a nanorobot to assemble the dfcc lattice of spherical particles.
It has been pointed out recently12 that coating homogeneous spherical particles of high refractive index by a thin
shell of small refractive index leads to the widening of the
pseudogaps found in the fcc colloidal crystals. The particles
are called core-shell particles.
The emerging ability to produce and assemble more and
more complicated periodic structures calls for a systematic
study which could guide the experimental efforts. The purpose of this article is to present a thorough exploration of the
photonic band spectra of the core shell particles with up to
0021-8979/2003/94(7)/4244/4/$20.00
II. DETAILS OF THE COMPUTATIONS
The photonic crystal is a material composed of regions
with different dielectric constants ␧ which are periodically
arranged in space. The laws governing the propagation of
electromagnetic waves in such a medium are given by Maxwell’s equations. Ignoring the nonlinear effects, possible solutions of Maxwell’s equations can be expressed as a superposition of the time-harmonic modes. The frequencies of
these modes are eigenvalues of the linear Hermitian
operator13 forming discrete sequence of bands 共‘‘dispersion
relations’’兲 ␻ n (k) as a functions of the ‘‘wave vector’’ k. It is
sufficient to specify all the bands only for those vectors k
which belong to the first Brillouin zone. Values of k leading
to different ␻ n can be restricted further to the irreducible part
of the first Brillouin zone using point symmetries of the periodic structure.
There are few common approaches to the eigen decomposition of Maxwell’s equations. Two most frequently used
are the frequency-domain and time time-domain techniques.
In the frequency-domain approach one expands the fields in
some basis subjected to a finite truncation. Most often the
planewave basis is used. Then a resulting linear eigenproblem is solved. The time-domain techniques involve direct
simulations of Maxwell’s equations in time on a discrete grid
using finite-difference time-domain algorithms. The frequencies 共band structure兲 are then extracted via a Fourier trans4244
© 2003 American Institute of Physics
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J. Appl. Phys., Vol. 94, No. 7, 1 October 2003
Babin, Garstecki, and Hołyst
4245
ternatively, the matrix has to be filled with a material with
the same dielectric constant as the dielectric constant of the
outer shell of the particles. For the sake of maximizing the
relevancy of our work to the experiment we have investigated structures composed of two kinds of material with distinct dielectric constants only—␧ low and ␧ high , respectively.
The structures have been tested in the experimentally
accessible19 range of ␧ with ␧ low⫽1 and ␧ high⭐13.
III. RESULTS
FIG. 1. Schematic representation of a core-shell particle. The dielectric
constant ␧ changes from one spherical subshell to another. ␧ 1 , ␧ 2 can be
either ␧ high or ␧ low . The particles are embedded in the matrix and arranged
in the fcc, bcc, or diamond fcc lattice. r max denotes the radius of a sphere in
a close packed lattice which is different for different lattices: r max,bcc
⬇0.433a, r max,fcc⬇0.354a, and r max,dfcc⫽0.216a (a is the lattice spacing of
the conventional cubic cell兲.
form of the fields. Since we are interested in the photonic
band structure of the colloidal crystals, the frequencydomain approach is the most natural choice. We have used a
freely available MPB14,15 package to find fully vectorial
eigenmodes of Maxwell’s equations and the corresponding
dispersion relations. The MPB can be applied even to the
sharply discontinuous dielectric structures since it uses a
smoothed effective dielectric tensor.16 We have modeled the
unit cells of the studied structures with resolutions 32, 40,
48, 64, and 96 points in each of three spatial dimensions. The
accuracy of the determination of ␻ n (k) for n⬍40 has been
checked to be better than 0.5%. We have paid additional
attention to sample the Brillouin zone with a fine enough k
resolution to obtain smooth ␻ n (k) dispersion relations. This
is crucial for a proper estimation of the gap’s width. For
testing purposes we have reproduced some recently published dispersion relations originally computed in different
computational schemes. We have found perfect agreement
for the triply periodic bicontinuous morphologies3,17 and periodically arranged colloidal particles.4,18
The multilayered core-shell particles 共see Fig. 1兲 are
composed of a core and a number of shells. The volume
occupied by the core is bounded by a sphere of radius r 1 .
The first shell is between spheres r 1 and r 2 . Consequently
the nth shell is bounded by the spheres of radii r n and r n⫹1 .
In our study we have taken into account homogeneous
spherical particles 共no shells兲 of radius 兵 r 1 其 , core-shell particles characterized by two radii 兵 r 1 ,r 2 其 , double core-shell
particles with three radii 兵 r 1 ,r 2 ,r 3 其 and triple core-shell particles with four radii 兵 r 1 ,r 2 ,r 3 ,r 4 其 . In all cases the particles
are immersed in a matrix. Each subvolume has been associated with one of the two dielectric constants: ␧ low or ␧ high in
such a manner that they interchange with every subvolume
as one goes from the center of the particle to the matrix. The
centers of particles are arranged in the fcc, bcc, or dfcc lattice. It is important to note here that the outer shell radius has
not been kept to be equal to the radius of a particle in a close
packed lattice. Experimentally this means that in order to
fabricate such a material by conventional aggregation into a
close packed lattice, the particle has to be covered by an
additional layer which can be then selectively etched or, al-
We have performed computations of the photonic band
structure for colloidal crystals composed of multilayered
spherical particles consisting of a core coated by up to three
spherical shells with alternating dielectric constant. The systems considered can be conveniently distinguished by the
total number of subvolumes with different ␧. For instance,
homogeneous particles in a matrix constitute two subvolume
共2V兲 system. We have considered two 共2V兲, three 共3V兲, four
共4V兲, and five 共5V兲 subvolume systems. Apart from the number of shells the structures can be divided into two distinct
classes—the ones with a dielectric constant of a matrix
␧ matrix equal to the ␧ high and those with ␧ matrix⫽␧ low .
For all of these morphologies the following values of r
where sampled r i 苸 兵 1r 0 ,2r 0 , . . . ,nr 0 其 , where r 0 ⫽r max /n
with the obvious r i ⬍r i⫹1 constraint imposed. First of all we
have checked all the structures for n⫽6. After identification
of interesting structures several additional runs with a finer r
grid have been performed.
A. fcc lattice
The radius of a sphere in a close packed fcc lattice is
r max⬇0.354a, where a is the lattice spacing of the conventional cubic cell. In agreement with the previous study5 we
find that the systems with ␧ matrix⫽␧ low offer only a small
number of narrow gaps. For ␧ matrix⫽␧ high 2V structures exhibit a full band gap between the 8th and 9th bands. For
dielectric contrast ␩ ⫽␧ high /␧ low⫽13 the relative width f
⫽100 ⌬ ␻ / ␻ 0 ( ␻ 0 is a midgap frequency兲 of this band gap
reaches f ⬇6 for slightly overlapping particles (r 1 ⫽0.36a).
The band gap closes down monotonically with decreasing
dielectric contrast and disappears at ␩ ⬇8.25. No other significant band gaps have been observed in more complicated
structures 共3V, 4V, and 5V兲.
B. bcc lattice
Again in the first step all of the possible structures for
n⫽6 (r max⬇0.433a) have been checked. In the spectra of
2V samples no gaps have been found. Two 3V structures
exhibiting pseudogaps have been identified: one with ␧ matrix
⫽␧ low and r 1 ⫽r 0 , r 2 ⫽3r 0 共between bands 11 and 12兲, and
one with ␧ matrix⫽␧ high : r 1 ⫽2r 0 , r 2 ⫽4r 0 共between bands 8
and 9兲. Additionally the 5V structure with ␧ matrix⫽␧ low displaying a full photonic band gap between bands 12 and 13
has been found. The relative gap width reaches maximum
( f ⬇7) for slightly overlapping particles for (r 1 ⫽0.062a,
r 2 ⫽0.165a, r 3 ⫽0.309a, r 4 ⫽0.371) and is very sensitive to
the variation of r 2 –gap exist for 0.13a⭐r 2 ⭐0.195a.
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4246
J. Appl. Phys., Vol. 94, No. 7, 1 October 2003
FIG. 2. The relative gap width f ⫽100⌬ ␻ / ␻ 0 共⌬␻ is the gap width,
␻ 0 —midgap frequency兲 vs the radius of a homogeneous particle in the dfcc
lattice with ␧ matrix⫽␧ low for dielectric contrast ␩ ⫽␧ high /␧ low⫽13. The arrow marks r max⫽0.216a. The inset presents the 2–3 共for r 1 ⫽0.235) and
8 –9 共for r 1 ⫽0.17) relative gap widths f as a function of ␩.
C. dfcc lattice
For the dfcc lattice r max⬇0.216a. As before the first
search has been carried out with n⫽6. None of the tested
geometries with ␧ matrix⫽␧ high presented significant band
gaps. The spectra of structures with ␧ matrix⫽␧ low do have
gaps. Within the 2V structures, for the dielectric contrast ␩
⫽13 and not overlapping particles the widest gap is between
the bands 8 and 9 (r 1 ⫽0.17a, f ⬇11). If overlapping is allowed the 共2–3兲 gap is wider (r 1 ⫽0.235a, f ⬇13.5). The
dependence of f on r 1 and dielectric contrast ␩ is presented
in Fig. 2. Figure 2 also shows which geometries offer two or
three independent gaps.
The computations for 3V structures with ␧ matrix⫽␧ low
共core with ␧⫽␧ low , spherical shell with ␧⫽␧ high and matrix
with ␧⫽␧ low) structures have revealed several multi gap
spectra 共an example is presented in Fig. 3兲. Figure 4 shows
values of parameters corresponding to the materials possessing two or three full photonic band gaps. 3V structures provide the widest single gap also. For dielectric contrast ␩
⫽13 it is the gap between the bands 8 and 9 (r 1 ⫽0.072a,
r 2 ⫽0.18a, f ⬇14.4). The 共8 –9兲 gap is wider than the one in
case of the homogeneous particles. It disappears at ␩ ⬇7.5.
The 共2–3兲 gap ( f ⬇13.3 for r 1 ⫽0.03a, r 2 ⫽0.235a and ␩
⫽13) persist down to ␩ ⬇4.2.
4V dfcc with ␧ matrix⫽␧ low : the widest gaps have been
observed for r 3 ⫽r max and r 1 苸 兵 0.02,0.05其 a. The 4V structure can serve as a good candidate for two- or three-gap
material. For example the 4V structure (r 1 ⫽0.036, r 2
⫽0.072, r 3 ⫽r max) offers three gaps with: f (2 – 3) ⬇5.5,
f (8 – 9) ⬇7.7, and f (24– 25) ⬇1. But 3V structures are slightly
better as for example: f (2 – 3) ⬇5.4, f (8 – 9) ⬇8 and f (24– 25)
⬇1.2 for the 3V structure characterized by r 1 ⫽0.072a, r 2
⫽r max , and ␩ ⫽13.
The 5V structures do not present wider gaps than the
three or four subvolume samples.
Babin, Garstecki, and Hołyst
FIG. 3. Dispersion relations ␻共k兲 computed for the three subvolume dfcc
lattice with ␧ matrix⫽␧ low (r 1 ⫽0.036a, r 2 ⫽r max) along the path connecting
high symmetry points X, U, L, ⌫, X, W, K of the first Brillouin zone 共see
for example Ref. 13兲. All of the frequences are given in units of 2 ␲ c/a
where a is the lattice spacing of the conventional cubic unit cell and c is the
speed of light. Three full gaps 共2–3兲, 共8 –9兲, and 共24–25兲 are clearly visible.
On the right the corresponding density of states ␳ s ( ␻ ) is shown with the gap
relative widths indicated. Please note that in some directions the gap widths
are significantly larger. For example for k⫽X: f (2 – 3) ⫽20, f (8 – 9) ⫽19, and
f (24– 25) ⫽9.
IV. CONCLUSIONS
Coating the spherical homogeneous particles with layers
does not lead to the increase of the photonic band gap width
in case of the fcc and bcc lattices. In case of the dfcc arrangement the best structure for applications is the low dielectric
constant core covered by the high dielectric constant shell
and embedded in the low dielectric constant matrix. For the
latter system the core-shell architecture of particles changes
the photonic properties: first of all the single gaps become
FIG. 4. Photonic band gap map of the 3V dfcc lattices with ␧ matrix⫽␧ low .
The lines show the boundaries between structures exhibiting different number of gaps. The solid lines lie between computed points, the dashed lines
are drawn in regions of low density of samples tested. The dotted line shows
approximately the boundaries between 8 –9, 24–25 and 2–3, 24–25 gap
structures.
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J. Appl. Phys., Vol. 94, No. 7, 1 October 2003
wider than in the case of homogeneous particles, and moreover, in some cases photonic band spectra has multiple gaps.
Most of the explored structures exhibit interesting photonic band spectra for high dielectric contrast ( ␩ ⬎10) only.
Therefore, a fabrication of organic-inorganic hybrid materials is probably required to implement such structures in practice. The semiconductors which are already used19 such as
gallium-arsenide offer dielectric constant ␧⬎10. The biggest
challenge is in obtaining materials with low dielectric constant (␧ low⬇1) which could still serve as structural elements
of the system. A possible solution, proposed by the pioneer
in the field,1 is to use foams or gels. They can be obtained for
example in the sol-gel emulsion systems20 or in the silica
polymerized bicontinous structures.21
ACKNOWLEDGMENTS
This work has been supported by the KBN Grant No.
2P03B00923 共2002–2004兲 and KBN Grant No.
5P03B01121. P.G. acknowledges a fellowship from The
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