APPLIED PHYSICS LETTERS
VOLUME 80, NUMBER 7
18 FEBRUARY 2002
Zero permittivity materials: Band gaps at the visible
N. Garcia,a) E. V. Ponizovskaya, and John Q. Xiao
Laboratorio de Fisica de Sistemas Pequeños y Nanotecnologia, (CSIC), Serrano 144, Madrid 28006, Spain
共Received 10 August 2001; accepted for publication 18 December 2001兲
We theoretically discuss the possibility of having materials with zero effective permittivity that
would create band gaps in a wide range of frequencies up to the visible. The physical realization of
these materials is also discussed in terms of embedding metallic nanoparticles and nanowires in a
dielectric medium. In the limit of long wavelengths, these composites will behave like a
homogeneous medium with zero permittivity that will completely reflect electromagnetic waves. We
present transmittivity calculations by using finite-difference time domain for periodic structures that
proves the concept and shows the validity of the long wavelength approximation. The striking result
is that the cutoff frequency ␻ c is determined by the lattice parameter of the composite. By properly
choosing the lattice constant of the composite and permittivity of metal and dielectric constituents,
we can have full band gaps at any frequency range but especially in the visible. © 2002 American
Institute of Physics. 关DOI: 10.1063/1.1449529兴
Considering an electromagnetic 共EM兲 wave entering
from the vacuum into a medium of permittivity ⑀, its velocity
in the medium is c/ 冑⑀ , where c is the speed of light. For
what follows, we take magnetic permeability unity and
imaginary ⑀⬙ zero. We will later remove the last condition.
This problem can be translated into a Schrodinger equation
with an energy dependent optical potential V⫽⫺k 20 ( ⑀ ⫺1),
as depicted in Fig. 1共a兲 in which the wave is impinging from
vacuum 共⑀⫽1兲, with frequency ck 20 共where k 0 ⫽2␲/␭兲, to a
medium that is delimited by a rough boundary.1 It can be
seen that if ⑀ is approximately zero the incident energy of the
wave k 20 is equal to the scattering potential energy, and the
wave is completely reflected, otherwise the speed of the
wave in the medium will be infinity. Full reflectivity will also
be obtained for ⑀⬍0. This is the case for metals. For frequencies below the plasma frequency, they are full band gap materials.
For metal there is a cutoff frequency 共plasma frequency兲
in the transmittivity T. This idea can be generalized to composite materials with metallic rods embedded in a dielectric
medium as sketched in Fig. 1共b兲. In this case there exists an
effective permittivity that can be given by the simple scalar,
or other more elaborate procedures.2 We consider, for discussion, the simple case of the scalar approximation where the
effective permittivity is expressed as
¯⑀ ⫽ f ⑀ m ⫹ 共 1⫺ f 兲 ⑀ i ,
共1兲
where f is the metal volume filling factor, ⑀ m and ⑀ i are the
permittivities for the metal and dielectric constituents, respectively. Metals have negative ⑀ m 共in principle for Drude
metals兲 at a frequency lower than the plasma frequency and
dielectrics have positive ⑀ i . It is clear that we can have a
medium with ¯⑀ ⫽0 for a proper volume fraction f . In other
words, we will have a new metal from the optical point of
view 共zero transmission兲 at a frequency much lower than the
plasma frequency for the pure metal, but the material is still
a兲
an insulator; i.e., we have an insulating optical metal. Of
course this will be true only for the long wavelength approximation 共LWA兲 where wavelengths of the EM radiation are
much larger than the intergrain separation and the effective
medium approximation becomes valid, i.e.; the system responds homogeneously to EM radiation. It should be noted
that in the case of k 0 approaching zero, transmission has to
rise to unity again because the energy dependent scattering
potential V also approaches zero. Consequently, transmission
is unity for k 0 ⬇0, then decreases to zero in the region where
Author to whom correspondence should be addressed; electronic mail:
[email protected]
FIG. 1. 共a兲 Scattering geometry with optical potential V(r)⫽⫺k 20 ( ⑀ ⫺1);
共b兲 schematics of 45° and 90° incident EM radiation onto composite materials consisting of cylindrical rods 共perpendicular to the plane兲 embedded in
an insulating matrix.
0003-6951/2002/80(7)/1120/3/$19.00
1120
© 2002 American Institute of Physics
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Appl. Phys. Lett., Vol. 80, No. 7, 18 February 2002
FIG. 2. The transmission of 90° incident EM radiation as a function
␻ a/(2 ␲ c) for composite materials of ⑀ m ⫽⫺8, ⑀ i ⫽8, f ⫽0.15 共dotted line兲
and ⑀ m ⫽⫺11, ⑀ i ⫽2, f ⫽0.5 共continuous line兲. Notice that the cutoff frequency takes place at the value ␻ b/(2 ␲ c)⬇1.4 larger than 0.5 due to the
fact that the values of r are not negligible in comparison to the lattice
constant a.
the LWA effective medium is valid, and increases again for
shorter wavelengths. Therefore there should be a large band
gap in the region where the LWA is valid.
To validate the above arguments, and search for the band
gap region where the LWA is valid, we use finite-difference
time-domain 共FDTD兲 methods3 to calculate the transmission
T and the reflectivity (1⫺T) of EM radiation from a periodic two-dimensional 共2D兲 composite in which a square lattice of metallic cylinders of radius r with a lattice constant a
is embedded in a dielectric matrix. The FDTD calculations
are well established and have been used for computing photonic crystals4 共our case兲 and elastic waves.5 We use periodic
boundary conditions in a 4⫻4 unit cell of cylinders and
Mur’s first order absorbing boundary conditions.3 The calculation should be done using energy dependent permittivity.
However, for the cases we are concerned with, the appearance of gaps, it is not necessary as will be shown below. We
proceed as follows: 共i兲 choose ⑀ m and ⑀ i for a metal and
dielectric medium, 共ii兲 then we determine the value for f
such that we will have ¯⑀ ⫽0. It should be emphasized that the
calculation is done by giving ⑀ m and ⑀ i to metal and dielectric, respectively 共not the ¯⑀ 兲, and calculate the transmission
and reflectivity of the medium. For these assumptions, we
predict there exists a cutoff frequency ␻ c below which the
¯⑀ ⬍0 and the transmission is zero, since ⑀ m has a much stronger frequency dependence than that of ⑀ i . In other words, we
predict a gap in transmission between near zero frequency,
where the scattering potential goes to zero, and the upper gap
frequency ␻ c , where the scattering potential equals the incoming EM kinetic energy. In the gap, the scattering potential is larger than the EM kinetic energy, and the EM wave is
completely reflected!
In Fig. 2 the value of T is plotted as a function of
␻ a/(2 ␲ c) in the case of the electric field of the incident
wave parallel to the rows of cylinders 共s polarization兲. For
⑀ m ⫽⫺8, ⑀ i ⫽8, and f ⫽0.15 共dotted line兲 we have an effective permittivity ¯⑀ ⫽5.6. It is clear, in this case, that the bands
exist in all frequencies and there is no gap. In the case of
⑀ m ⫽⫺11, ⑀ i ⫽2, f ⫽0.5, and thus ¯⑀ ⫽⫺4.5 共continuous line兲,
transmission T is practically zero everywhere, just like the
case in a metal.
In Fig. 3 we perform the same type of calculations with
¯⑀ ⫽0 and ⑀ m ⫽⫺8, ⑀ i ⫽8, f ⫽0.5 共dotted line兲 and ⑀ m ⫽⫺11,
⑀ i ⫽2, f ⫽0.15 共continuous line兲. Figures 3共a兲 and 3共b兲 are for
Garcia, Ponizowskaya, and Xiao
1121
FIG. 3. The transmission of 90° 共top panel兲 and 45° 共bottom panel兲 incident
EM radiation as a function ␻ a/(2 ␲ c) for composite materials of ⑀ m ⫽⫺8,
⑀ i ⫽8, f ⫽0.5 共dotted line兲 and ⑀ m ⫽⫺11, ⑀ i ⫽2, f ⫽0.15 共continuous line兲.
90° and 45° incident angles, respectively. A very wide gap
clearly exists in 0⬍␻ a/(2 ␲ c)⭐0.5 when rⰆa and more
general ␻ b/(2 ␲ c)⬍0.5 with b⫽a⫺2r. The entire first band
is forbidden because the value of ␻ a/(2 ␲ c)⫽0.5 corresponds to the first Brillouin zone edge 共ka⫽␲兲 where ␭
⫽2a. It is also clear that passing bands start to appear when
␻ a/(2 ␲ c)⬎0.5, or 2a⭓␭. Therefore ␭⬇2a sets the limit
above which the LWA is valid, and the gap is very large
covering the entire first Brillouin zone. When ␭⬍2a, or 2b
the LWA breaks down. The wavelength is short enough to
find a way through the dielectric medium between the cylinders. Notice that because the r is not much smaller than a the
cutoff frequency has to be corrected by the radius and the
condition is ␭⬍2b as is clearly manifest in the calculations.
The reason is that a kind of waveguide forms in the open
window between cylinder surfaces of size b We want to
stress that similar gap and bad structures exist in transmission if we perform calculations for a dispersive frequencydependent permittivity in the metal. In addition, we have also
performed calculations for an electric field perpendicular to
cylinders 共p polarization兲. The results are practically the
same although the gaps are somehow narrower by 10% then
that for s waves. In this case the physics is different because
we have not cutoff frequency but the injection of the wave
for ␭⬎2a is very inefficient and a gap appears at practically
the same frequency as that for s waves. To put numbers in
the values of T, while for s waves, for the four layers of
cylinders of Fig. 1, we have T⫽⫺40 dB at the transmitivity
minimum, and for the p wave we have ⫺25 dB. Also the
value of T reduces exponentially with the number of layers
of cylinders for s waves, while for p waves we have a strong
reduction for the first two layers and is practically constant as
the number of layers increases. This is proof that for p waves
although states exist at low frequency 共no cutoff frequency兲
the injection of the wave is practically impossible, a very low
injection efficiency. This is a problem that we have reiteratively mentioned for acoustic waves.5
There are two very important features in this calculation.
First the ␻ c 共thus ␭ c 兲 is controlled by the lattice constant a
and r. For example, if we choose the value of a to be 200
nm it will produce a ␭ c roughly around 400 nm, i.e., EM
radiation with ␭⬎␭ c will be fully reflected, and those with
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1122
Appl. Phys. Lett., Vol. 80, No. 7, 18 February 2002
␭⬍␭ c will be transmitted. Second, the metal and dielectric
medium has to be chosen with proper values of ⑀ m and ⑀ i
satisfying Eq. 共1兲 at ␻ c .
It should be emphasized that our approach for developing new gap materials is different than those used in photonic
band gaps4,6 and elastic waves.7,8 In those works the idea is
to have gaps for values of a⬇␭, in hopes that, with a lot of
multiple scattering, gaps will open. This happens in some
cases but with small gaps. Our approach is entirely different:
we produce gaps in regions of small scattering, but using
¯⑀ ⬍0 under the framework of LWA that is valid for a wide
range of frequencies as shown in our FDTD calculation. This
calculation is done for a periodic system, but LWA also holds
for disordered systems, as far as the lattice constant a is
much smaller than the wavelength. There has been a recent
article on band gap of air bubbles in water that shows results
similar to ours.9 In both cases we have a very high contrast
of speed propagation between the two constituents in the
composite.
Finally, we would like to briefly discuss the case where
the imaginary ⑀⬙ 共the losses兲 is not zero in metals. Its effect is
to reduce the overall value of T and may have bad consequences for real devices. An ideal metal will be Ag,10 which
has a negligible ⑀⬙ of about 0.3 between 1 and 3.7 eV. Consequently, the losses should be small. In addition, the losses
are further reduced by the filling factor in our case. For a
filling factor of 0.15, the ⑀⬙ is reduced to 0.045, practically
no loss! We have estimated this effect for our proposed structure and find the correction in the transitivity is less than 5%.
This will be discussed in more detail in a forthcoming article,
but for the moment it can be said that we have a very low
loss material. Therefore, we have materials with gaps in the
desired regions; easy to design, with no losses, that are ideal
for designing microstructured optical devices of all kinds.
Any type of structure consisting of a metallic constituent
embedded in an insulating matrix is a suitable band gap material, as long as the average intergrain separation is shorter
than the wavelength 共2a⬍␭兲. Granular materials11 and nanowire arrays12 are two examples that fall into this category.
The value of a can be controlled from a few nm to a few
hundred nm, easily covering the entire frequency spectra of
interest.
In summary we have demonstrated materials with zero
permittivity are true band gap materials. The theoretical arguments have been validated by calculating transmission of
Garcia, Ponizowskaya, and Xiao
EM waves using the FDTD method. The striking feature of
the calculation is the cutoff frequency ␻ c is controlled by the
lattice constant and the radius in composite materials that
should be designed according Eq. 共1兲 at ␻ c that will produce
a gap for the s wave. A similar gap is obtained for p waves
due to the surface impedance that produce a very low efficiency injection of the wave from the homogeneous to the
inhomogeneous media. This is especially enhanced in the
visible range where the photonic gaps are very interesting.
J.Q.X. on leave of absence from the University of Delaware acknowledges support from The Spanish Ministry of
Education. This work has been supported by the Spanish
through Project No. BFM2000-0599.
N. Garcia and M. Nieto-Vesperinas, Opt. Lett. 20, 949 共1995兲.
J. C. Garland and D. B. Tanner, Electrical Transport and Optical Properties of Inhomogeneous Media 共American Institute of Physics, New York,
1978兲.
3
A. Taflove, The Finite-Difference Time-Domain Method 共Artech House,
Boston, 1998兲.
4
C. T. Chan, Q. L. Yu, and K. M. Ho, Phys. Rev. B 51, 16635 共1995兲; M.
Sigalas et al., Microwave Opt. Technol. Lett. 15, 153 共1997兲.
5
M. Kafesaki, M. Sigalas, and N. Garcia, Phys. Rev. Lett. 85, 4044 共2000兲;
D. Garcia-Pablos, M. Sigalas, F. R. Montero de Espinosa, M. Torres, M.
Kafesaki, and N. Garcia, ibid. 84, 4349 共2000兲.
6
See, Photonic Band Gaps and Localization, edited by M. Soukoulis;
NATO ASI Series B, Physics, Vol. 308 共Plenum, New York, 1993兲; K. M.
Ko, C. T. Chan, and C. M. Soukoulis, Phys. Rev. Lett. 65, 3152 共1990兲; K.
M. Leung and Y. F. Liu, ibid. 65, 2646 共1990兲; M. M. Sigalas, E. N.
Economou, and M. Kafesaki, Phys. Rev. B 50, 3393 共1994兲; R. D. Meade,
K. D. Brommer, A. M. Rappe, and J. D. Joannopoulos, ibid. 44, 13772
共1991兲; 44, 10961 共1991兲; E. Yablonovitch, T. J. Gmitter, R. D. Meade, A.
M. Rappe, K. D. Brommer, and J. D. Joannopoulos, Phys. Rev. Lett. 67,
3380 共1991兲; A. D. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve,
and J. D. Joannopoulos, ibid. 77, 3787 共1996兲.
7
M. M. Sigalas and E. N. Economou, Solid State Commun. 86, 141 共1993兲;
Europhys. Lett. 36, 241 共1996兲; E. N. Economou and M. M. Sigalas, Phys.
Rev. B 48, 13434 共1993兲; M. M. Sigalas, J. Appl. Phys. 84, 3026 共1998兲;
8
M. S. Kushwaha, P. Halevi, L. Dobrzynski, and B. Djafari-Rouhani, Phys.
Rev. Lett. 71, 2022 共1993兲; M. S. Kushawaha, P. Halevi, G. Martı́nez, L.
Dobrznski, and B. Djafari-Rouhani, Phys. Rev. B 49, 2313 共1994兲; M.
Torres, F. R. Montero de Espinosa, D. Garcı́a-Pablos, and N. Garcı́a, Phys.
Rev. Lett. 82, 3054 共1999兲.
9
M. Kafesaki, R. S. Penciu, and E. N. Economou, Phys. Rev. Lett. 84, 6050
共2000兲.
10
P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 4370 共1992兲.
11
J. Q. Xiao, J. S. Jiang, and C. L. Chien, Phys. Rev. Lett. 68, 3479 共1992兲.
12
K. Ounadjela, R. Ferre, L. Louail, J. M. George, J. L. Maurice, L. Piraux,
and S. Dubois, J. Appl. Phys. 81, 5455 共1997兲; Z. Hao, Y. Shaoguang, N.
Gang, T. Shaolong, and D. Youwei, J. Phys.: Condens. Matter 13, 1727
共2001兲; T. M. Whitney, J. S. Jiang, P. C. Searson, and C. L. Chien, Science
261, 1316 共1993兲.
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