Download SAMPLE ARTICLE FOR ACTA ELECTROTECHNICA ET

, 1–2
1
ELECTROMAGNETIC RESPONSE OF SYSTEMS OF METALLIC NANOPARTICLES
∗
Tomáš Váry∗ , Peter Markoš∗
Department of Physics, INPE FEI STU Bratislava, Slovakia, Ilkovičova 3, 812 19 Bratislava, Slovakia
E-mail: [email protected], [email protected]
ABSTRACT
We study numerically the response of periodic arrays of nanosized metallic particles to the incident electromagnetic waves. We
identify localized plasmons excited in individual particles and describe how the resonance frequency depends on particle size and the
structure geometry. Numerical data for the absorption enable us to distinguish plasmon resonance from interference effects observed
outside the resonance region.
Keywords: plasmons, transfer matrix, effective permittivity
1. INTRODUCTION
Most of optical properties of metals are determined by
free electrons which are very sensitive to the external electric field. Oscillations of electrons due to the incident electromagnetic (EM) field determine the frequency-dependent
metallic permittivity of metal which can be expressed by
the Drude formula [1]
εm = 1 −
f p2
,
f ( f + iγ)
structure of a typical “atom” is shown and described in Fig.
1. Antiresonant coupling of two plasmonic oscillations was
used for the experimental proof of the possibility to construct material with negative frequency dependent effective
magnetic permeability at THz frequencies [11].
(1)
with the plasma frequency f p and the absorption parameter
γ. Since f p ∼ 2000 THz for typical metals, the real part of
the permittivity (1) is negative for the visible light.
The model of the metal as a material with negative permittivity enables us to describe and explain various optical properties of metal without discussing microscopic processes inside the metal [2, 3]. Typical textbook examples
are the reflection amplitude of the EM field incident to the
metallic surface, the skin depth or the transmission of the
field through the thin metallic slab [4].
The negative permittivity model enables us also to describe quantitatively the plasmonic excitations in metals. At
the interface between the metal and the dielectrics the surface plasmon [5] can be excited, which concentrates the entire energy of the incident electromagnetic field to the thin
two-dimensional layer parallel to the metallic surface [6,7].
While the surface plasmon cannot be excited by the incident electromagnetic wave (the dispersion relation of the
surface plasmon lies outside the light cone [4]), localized
plasmons in small metallic nanoparticles [7] are excited
directly by incident light. Therefore, localized plasmons
could be used for the manipulation of the electromagnetic
response of nano composites. The size and concentration of
the particles influences the frequency dependent reflectivity
and absorption, and opens a wide variety of technical and
commercial applications [8]. Physically, the most interesting is the use of plasmonic excitations for the design and
construction of metamaterials, which are structures with
required frequency-dependent effective electric permittivity and magnetic permeability. Metamaterial [9] consists
of a periodic lattice of artificial “atoms” (unit cells of the
size much smaller than the wavelength of the interacting
EM field). Each “atom” is constructed from metallic components which create resonant structure [10]. Schematic
Fig. 1 Left: A typical structure of the metamaterial “atom”
consists of two metallic elements (two identical nanorods are
shown in the figure). Normally incident EM fields induces the
oscillations of electrons inside metallic nanoparticles. Owing to
the coupling of the field induced in two neighboring
nanoparticles, the system possesses two resonant frequencies
similar to the two eigenmodes of coupled mechanical pendulums
shown in the right panel. The symmetric resonance creates
parallel currents in nanorods, the antisymmetric resembles the
circular current in loop with two cuts at the ends of nanorods.
When periodic lattice of identical “atoms” is fabricated, then two
resonances create frequency bands with negative permittivity and
permeability [10]. Thus, local currents inside each individual
“atom” determine the effective permittivity and permeability of
the metamaterial. If two bands overlap, the macroscopic
metamaterial has negative effective refractive index.
Metallic nanosphere represents the simplest shape of
nanoparticle. Thanks to the spherical symmetry, the scattering process of the EM wave on single nanosphere can
be analyzed exactly with the use of the Mie theory [3, 12].
However, scattering on the more complicated systems of
nanoparticles must be analyzed numerically.
In this paper we present results of our numerical simulations of scattering of the EM plane wave incident on various
geometrical configurations of metallic nanospheres. The
numerical program is based on the transfer matrix method
described in [13]. Here we only summarize the main ideas.
In the program, it is assumed that the sample has a periodic
structure lying in the xy plane. Therefore, only one unit
c 2010 FEI TUKE
ISSN 1335-8243 2
Electromagnetic response of systems of metallic nanoparticles
cell, shown in Fig. 2, is considered and periodic boundary conditions are used in two transversal (x, y) directions.
An incident monochromatic plane wave with the frequency
f propagates along the z direction, scatters at the sample,
and the transmission and reflection amplitudes (t and r) are
calculated. The absorption is found from the energy conservation requirement, A = 1 − R − T , where R = |r|2 and
T = |t|2 are the reflection and absorption coefficients.
Fig. 2 Left: The geometry used in the transfer matrix numerical
method. The monochromatic EM wave incoming from the left is
scattered on the sample and reflected and transmitted waves are
calculated. The fields are calculated only in the finite volume
a × b × L where a and b are size of the sample in the x and y
direction, respectively and L is the length of the sample along the
z direction. Periodic boundary conditions are used in the x and y
directions. Right: An example of the discretization of the
computed area and corresponding shape of the spherical particle
used in numerical simulations.
(a)
(c)
particle is much smaller than the wavelength λ , we find that
the scattering cannot be described within the static limit of
the Mie theory.
To demonstrate the role of the interaction of plasmon
resonances excited in neighboring particles, we discuss in
sect. 3 the shift of the plasmonic resonance excited in
the long one-dimensional chain of near-spaced nanospheres
and confirm the dipole-dipole character of the interaction
between neighboring plasmons. In sect. 4 we demonstrate
the split of the absorption peak due to the interaction of
plasmons exited in two neighboring linear chains. In sect.
5 we discuss the multilayer structures composed of very
small (with diameter d = 6 nm) nanoparticles. We find a
broad maximum in the absorption spectrum, which correspond to plasmonic excitation. Conclusion and discussion
are given in sect. 6.
(b)
Fig. 4 Absorption (A), transmission (T ) and reflection (R)
coefficients of the light incident on the square lattice of metallic
nanoparticles. The diameter of spheres is d = 96.6 nm. The
distance of neighboring particles a = 400 nm is large enough to
prevent any interaction of plasmon resonances excited on
neighboring particles The present data can be interpreted as a
spectrum of a single nanoparticle.
(d)
2. SINGLE SPHERICAL NANOPARTICLE
2.1. Numerical results
Fig. 3 Structures simulated in the paper: (a) Isolated metallic
particle of diameter d. Shown is also the size a of the unit cell
(equal to the distance between neighboring particles) and
discretization of the unit cell, used in the program. (b) Linear
chain along the y direction. The parameter b determines the
distance between neighboring particles, a b is a size of the unit
cell. We assume a b (c) Two parallel chains of nanoparticles.
(d) Layered structure of metallic particles coated by thin
dielectric layer which prevents conductive contact between
neighboring particles.
Four structures shown in Fig. 3 will be investigated. In
Sect. 2 we study the response of single metallic nanoparticle to the incident EM wave and identify the position of the
plasmonic resonance. We discuss the role of the permittivity of metallic components, which differs from the permittivity of the bulk material. Although the diameter d of the
As was discussed in Sect. 1, our numerical program
assumes that the sample has a periodic structure in the x
and y directions. To investigate the scattering of the EM
wave on isolated particle the spatial period a must be sufficiently large so that the interaction between plasmons excited on neighboring particles can be neglected. We found
numerically that this condition is fulfilled when the distance
a ∼ 4d.
Figure 4 shows numerical data for the frequency dependence of the transmission, absorption and reflection coefficient of the EM wave incident on a single nanoparticle with
diameter d = 96.6 nm. (The structure is shown in Fig. 3a.)
Resonant plasmon frequency
λr ≈ 570 nm
(2)
is identified from the position of the deep minima in the
transmission coefficient and maxima of the absorption.
c 2010 FEI TUKE
ISSN 1335-8243 3
Note that outside the resonant region, the sample is almost
transparent: the absorption of the EM inside the nanoparticles is negligible and the transmission is close to unity.
we use in numerical simulations the experimental data for
thin film and fit them to the function
2
f02
,
(3)
εr = 1 −
f 2 + Γ2
with fitting parameters f0 = 1300 THz and Γ = 400 THz.
The real part of the permittivity is zero at the frequency
f˜p = 1236 THz, which is much smaller than the plasma frequency ( f p = 2160 THz) given by the Drude formula.
Fig. 5 The absorption coefficient A as a function of the
frequency for particles of different size d. To avoid mutual EM
interaction between neighboring particles, we keep the same
distance/diameter ratio a = (29/7)d. Absorption maxima are
red-shifted when the particle size increases. The metallic
permittivity is given by the Drude formula (1).
Figure 5 shows how the absorption peak depends on the
size of the particle. The resonant wavelength decreases
from 570 nm for the particle diameter d = 96.6 nm to
545 nm when the diameter decreases to d = 24.1 nm. These
results are in good agreement with experiment [14] which
reports λr ∼ 606 nm for d = 100 nm and λr ∼ 550 nm for
the particle size d = 20 nm. The difference between numerical and experimental data is given by the inaccuracy
of both methods. In numerical simulations, the accuracy of
obtained results is limited by the discretization of the space
which introduces numerical inaccuracy due to approximation of derivatives and deforms the shape of the spheres as
is shown in right Fig. 2. Another reason for the inaccuracy
of numerical results is that the permittivity of small metallic
particles differs from the bulk permittivity given by Eq. (1).
On the other hand, experimental data cannot be associated
with a given size of a particle, because the size of particles
varies considerably within a given sample. Also, particles
are not distributed periodically, so that the distance between
neighboring particles fluctuates.
2.2. The permittivity of small metallic particles
The most important input parameter for numerical simulations is the permittivity of small metallic components.
This is different from the Drude formula, given by Eq. (1)
since the scattering of electrons at the surface of the particle
causes an increase of the imaginary part of the permittivity. To show the importance of the finite size effects on the
permittivity, we compare in Fig. 6 the experimental data
for the permittivity [15] of metallic films of the thickness
30 nm with the permittivity of the bulk metal. Since there
is no analytical model for the permittivity of small particles
Fig. 6 Real part of the permittivity of thin silver film [15]. Solid
line is the fit given by Eq. (3). Dashed line is the bulk
permittivity given by Drude formula, Eq. (1). Inset: Imaginary
part of the permittivity [15] compared with Drude formula
(dashed line). Thin solid line is the imaginary Drude permittivity
increased by +0.15. An increase of the imaginary part of the
permittivity for larger frequencies is typical for the absorption
peak at to the “plasma frequency” f p ≈ 1236 THz where real part
of the permittivity is zero.
2.3. The static limit
The scattering of the plane EM wave on spherical objects can be solved analytically within the Mie theory [12].
In the static limit, when the wavelength of the incident EM
field is much larger than the diameter of the sphere, λ d,
the resonant plasmon frequency can be found from the simple equation [7]
εr + 2εd = 0
(4)
where εd is the permittivity of surrounding media. Using
the Drude permittivity (1) and εd = 1 we obtain that the
plasmon is excited at the frequency
√
(5)
fr ≈ f p / 3
which corresponds to the wave length λr ≈ 236 nm. With
the use of the permittivity given by Eq. (3) we obtain that
λr ≈ 332 nm, which is still much less than the numerically obtained resonance (2). Therefore, the static limit is
not sufficient for the description of plasmonic excitations in
nanoparticles.
3. LINEAR CHAIN OF PARTICLES
Consider now the one-dimensional linear chain of particles. shown in the Fig. 3b. We formed it by placing the particle in rectangular cell with the unit cell size a × b, where
c 2010 FEI TUKE
ISSN 1335-8243 4
Electromagnetic response of systems of metallic nanoparticles
b is the particle-to-particle distance along the chain, and a
is large enough to avoid interchain interactions. The EM
wave propagates perpendicularly to the chain.
Because of the linear structure of the sample, the polarization of the incident wave is important. The electric field
is perpendicular (parallel) to the chain for the p (s) polarized wave, respectively.
Figure 7 shows the resonant plasmon frequency as a
function of the distance b between particles for both polarizations. For the longitudinal (s) polarization, the resonant frequency decreases when b decreases, while for the
transversal (p) polarization we observe an opposite behavior. The shift of resonant frequency exhibits for both polarization the 1/b3 - dependence in agreement with measurements of Maier et al. [16]. This confirms the dipole character of the electric interactions between neighboring particles. Dipoles oscillate with higher frequency when they are
oriented perpendicular to chain axis (p polarization).
larger side and b for shorter. Distance between two parallel
chains is c.
Figure 9 shows the absorption spectra for such structure for two different distances c between chains. For sufficiently small c = 33.3 nm the absorption peak splits into
two peaks (red curve). Interaction between plasmons on
particles causes frequency of resonance behave similar to
coupled harmonic oscillators. Two resonant frequencies, fr
and fr + ∆ f appears, where ∆ f is proportional to the coupling between neighboring particles and converges to zero
when the distance c increases.
As shown in Fig. 7, coupling between two plasmons
excited in two neighboring particles is much stronger when
the electric field is parallel to the line connecting particles.
Therefore, we expect that coupling is stronger when electric field is parallel to the x axis (Fig. 3c). In this case, the
frequency difference ∆ f is negative (Fig. 7). This observation explain our results shown in Fig. 9. For the p polarization (electric field parallel to the x axis), the absorption
peak splits into two peaks when c is sufficiently small. The
second peak with a larger wavelength corresponds to the
interchain interaction.
On the other hand, coupling is weak for the s polarized
wave (electric field parallel to the x axis). Consequently,
we observed only one absorption peak for both values of
the interchain distance c.
Fig. 7 Resonant frequency for various linear chains of metallic
nanoparticles as a function of the particle distance b. Solid lines
show fits which confirm the dipole character of particle
interactions ∼ 1/b3 . The diameter of the particles d = 50 nm and
the distance between chains a = 300 nm.
Fig. 8 Representation of the nanoparticle by electric dipoles for
the transversal p (top) and longitudinal s (bottom) polarization of
the EM field.
Fig. 9 Absorption coefficient for two parallel nanoparticle
chains. For the s polarization, the absorption peak does not
depend on the distance between chains. For the p polarization
(the electric field is perpendicular to chains), two absorption
peaks occurs due to the dipole-dipole interaction between two
neighboring particles. The right peak disappears when the
distance between chains increases. Parameters of the structure
are as follows: a = 100 nm, b = 33.3 nm, particle diameter
d = 29 nm. Drude formula for the permittivity, Eq. (1) was used
in simulations.
5. SMALL METALLIC NANOPARTICLES
HEXAGONAL LAYERED STRUCTURE
4. TWO PARALLEL CHAINS OF METALLIC
NANOPARTICLES
Next structure is composed from two parallel linear
chains of nanoparticles along the y direction (Fig. 3c).
Again the unit cell is rectangular with dimensions a for
IN
In this section, we simulate the scattering of the EM
wave on the multilayer structure of very small nanoparticles. The individual layer is shown in Fig. 3d. Small
spheres are self-assembled to the regular hexagonal lattice.
To prevent the conductive contact between particles, each
c 2010 FEI TUKE
ISSN 1335-8243 5
particle is coated by thin dielectric layer. The size of particles d = 6 nm, the thickness of the dielectric layer is only
1 nm.
The present structure can be easily prepared experimentally [17] and the reflection coefficient can be measured for
various number of layers [18, 19]. Numerical results can
be therefore directly compared with available experimental
data. This is important for the estimation of the frequencydependent permittivity of nanoparticles. We have namely
observed that experimental data for the permittivity shown
in Fig. 6 are not more relevant for metallic particles of the
size of a few nanometers. To find more realistic frequency
dependence of the permittivity, we simulated the scattering of the EM wave at a single layer of particles for various
values of the permittivity and compare obtained results with
experiment [19]. We found the best (although not perfect)
coincidence when the real part of the permittivity follows
Eq. (3) and the imaginary part εi = 1 [20].
for 15 layers, and εeff = 8.7 for 25 layers of nanoparticles). Thus, composite medium which contains materials
with permittivity εr ∼ 2.25 (coating) and εr ∼ −15 (silver)
behaves as weakly absorbing dielectrics with high refractive index.
Fig. 11 Absorption and transmission coefficients for multilayer
hexagonal nanoparticles structure. Number of layers is 10, 15,
20, 25 and 30. Arrows show the direction of increasing of the
number of layers. Note oscillations of the transmission
coefficient in the region outside the absorption peak. Reflection
coefficient for the structure with 30 layers is also shown.
6. CONCLUSIONS
Fig. 10 Reflection spectra for the normal incident plane wave to
multilayer of hexagonal close-packed silver nanoparticles.
Besides absorption peak located between 400-500 nm,
oscillations of the reflection coefficient are observed for larger
wavelengths. Real part of permittivity εr of silver is given by Eq.
(3 and the imaginary part is constant εi = 1.0. The permittivity of
dielectric layer is εd = 2.25. The diameter of nanoparticles
d = 6 nm, the thickness of dielectric coating is 1 nm.
Reflection coefficient for multilayer structures is shown
in Fig. 10. Corresponding data for the transmission and absorption coefficients are given in Fig. 11. Well pronounced
maximum of the reflection coefficient and absorption located for 400 < λ < 500 nm corresponds to the excitation
of localized plasmons in individual metallic particles. Location of this peak is independent of number of layers. Note
that the transmission is very small and decreases when the
number of layers increases.
Contrary, oscillations of the reflection coefficient, observed for larger wavelengths, are accompanied with oscillations of the transmission coefficient and very low absorption. Therefore, these peaks do not correspond to the
excitation of plasmon, but rather with interference of the
EM wave in thin layer. Comparing the position of reflexion
maxima with that for the homogeneous layer of the same
width and permittivity εeff , we find that the multilayer structure of nanoparticles behaves as the dielectric layer with
rather high effective permittivity (for instance, εeff = 3.6
Localized plasmons play crucial role in various physical phenomena. It is therefore important to understand
their response to incident electromagnetic field. This can
be obtained numerically. We demonstrated the application
of the numerical transfer matrix method for the calculation
of the reflection coefficient and absorption for a few configurations of metallic nanoparticles. Numerical simulation
enable us to calculate the transmission, reflection and absorption coefficients therefore can be useful for the interpretation of experiment which usually measured only the
reflection coefficient.
Our data confirm that plasmonic excitations cannot be
studied within the framework of the static limit, in spite of
the fact, that the diameter of metallic spheres is order of
magnitude smaller than the wavelength of incoming light.
The crucial problem of theoretical and numerical analysis of the plasmonic resonances is in the estimation of
the permittivity of small metallic particles. Both experimental and numerical data indicate that for relatively large
nanoparticles (d > 25 nm, the permittivity of Johnson and
Christy [15] can be used [14] and experimental data agree
very well with the analytical theory Mie. The estimation
of the permittivity of smaller nanoparticles is still an open
problem. Physically relevant model must consider the scattering of electrons on the surface of the finite volume of the
particle [21]. For extremely small particles, the permittivity
is influenced by the spectrum of the quantum eigenstates of
the confined electrons [22].
c 2010 FEI TUKE
ISSN 1335-8243 6
Electromagnetic response of systems of metallic nanoparticles
ACKNOWLEDGEMENT
work was supported by the Slovak Research and Development Agency under the contract No. APVV-0108-11.
Authors thank J. Chlpı́k for valuable discussions. This
[14] C. Sönnichsen et al. Phys. Rev. Lett. 88, 077402 (2002).
[15] P. B. Johnson and R. W. Christy, Phys. Rev. B 6, 43704379 (1972).
REFERENCES
[1] Ch. Kittel, Introduction to Solid State Physics, John
[16] S. A. Maier et al., Phys.Rev. B 65, 193408 (2002).
Willey and Sons (2005).
[2] J. D. Jackson, Classical Electrodynamics, John Wiley [17] M. Weis, private communication.
& Sons, Inc. (1999).
[18]
[3] M. Born and E. Wolf, Principles of Optics, Cambridge
University Press (2003).
[19]
[4] P. Markoš and C. M. Soukoulis, Wave Propagation:
[20]
From Electrons to Photonic Crystals and Left-Handed
Materials, Princeton Univ. Press (2008).
[21]
[5] E. N. Economou, Phys. Rev. 182, 539 (1968).
M. H. Lin, H. Y. Chen, S. Gwo, J. Am. Chem. Soc 132,
11259-11263 (2010).
J. Chlpı́k, private communication
T. Váry, PhD Thesis, FEI STU Bratislava (2012) (in
Slovak).
U. Kreibig, L. Grenzel, Surf. Science 156, 675 (1985).
[22] J. A. Scholl, A. I. Koh, J. A. Dionne, Nature 483, 421
(2012).
[7] S. A. Maier, Plasmonics: Fundamentals and Applications, Springer Science (2007).
[6] A. V. Zayats et al., Phys. rep. 408, 131 (2005).
[8] E. Hutter and J. A. Fendler, Advanced Materials 16,
1685 (2004).
[9] A. K. Sarychev, V. M. Shalaev, Electrodynamics of
Metamaterials, World Scientific (2007).
[10] V. P. Drachev et al., Laser Phys. Lett. 3. 49 (2006).
[11] A. N Grigorenko et al.. Nature 438, 335 (2005)
[12] H. C. van de Hulst, Light Scattering by Small Particles,
Dover Publications, New York (1957).
[13] P. Markoš and C. M. Soukoulis, Phys. Rev. E 81,
036622 (2002).
BIOGRAPHIES
Tomáš Váry (1984) graduated (MEng) in 2008 at the Department of Physics of the Faculty of Electrotechnical Engineering and Information Technology at Slovak University
of Technology in Bratislava. In 2012 he defended the PhD
Thesis “Surface electromagnetic waves on interfaces”. He
is Assistant at the STU.
Peter Markoš (1958) is Assistant Professor at the FEI
STU. His research interests include Anderson localization
and electromagnetic properties of composite metamaterials.
c 2010 FEI TUKE
ISSN 1335-8243