Plasmonic focus * Plasmonic headlines * Metallic nanoparticles * Nanorice * Nanoshells * Photonic structures based on metal nanoparticles
Refreshing some of plasmonic headlines
A history of an alternative nanorice
Many of you might be familiar with the nanorice [1].
In form, the nanorice is similar to nanoshells [25], which tunability
properties have been first demonstrated in a beautiful analytical paper by
Neeves and Birnboim [2] (see Secs. 2A and 2B therein)
within the quasistatic approximation.
The results of Neeves and Birnboim were
later on, also within the quasistatic approximation,
generalized by Haus et al [3] to the case of an arbitrary
number of shells.
Both nanorice and nanoshells are made of a nonconducting core that
is covered by a metallic shell [15].
The nanorice research appeared in the April 12, 2006 issue of
Nano Letters [1]. The publication and its findings were announced
at a press conference at the
American Physical Society's 2006 March Meeting. After the announcement,
a flood of headlines has followed in world press. See just a few of them:
Yet only a few of you might
in connection with the nanorice be familiar with the work of a young Dutch scientist
Joan J. Penninkhof, who fabricated the "nanorice" 2 years before
its "official" birth at the APS 2006 March Meeting.
She reported preliminary results regarding what is now known as nanorice
in her talk at the EMRS Spring Meeting 2004 in
Strasbourg held on May 2428, 2004. See the contribution
A2/P.06 on p. 9 of the
abstract book [6].
Two journal publication soon followed [7,8], which both appeared
before the APS 2006 March Meeting.
A summary of her results you can find on the slides presented in her talk
W10.11 during
MRS Spring 2006 Meeting in
Symposium W: Colloidal Materials  Synthesis, Structure, and Applications
on April 20, 2006 (the full Symposium program is
here.)
Joan worked on her
PhD thesis in the group of Albert Polman at AMOLF Institute in Amsterdam
since August 2002. She succesfully defended her PhD thesis on 25th September 2006.
A difference between the Joan nanorice and the Rice nanorice is that
the latter is about 23 times smaller and has less multipolar contributions and absorption.
I participated in the modelling of the optical properties of Joans's nanorice.
Our results were submitted for a publication in Phys. Rev. B
(manuscript BV10382) in the second half of 2005. The referee judged the
theoretical part as trivial, and the experimental part was found
to be more suitable for a more specialised journal.
Because since 2004 none my work can appease Phys Rev,
I tried in vain to convince my collaborators
to submit our work to J. Phys. Chem. C. Unfortunately, they insisted
on resubmitting our amended paper back to PRB.
This was done after cca 6 months since the first report; another months then passed
before a second referee report arrived. Whereas one referee recommended
the manuscript publication, the other referee
did not; and a typically passive Phys. Rev.
editor refused the manuscript.
Eventually, the work was submitted to
J. Phys. Chem. C having a comparable impact factor to Phys. Rev. B,
where our manuscript was without any problems published within 4 months
[10].
References:
 H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas,
Nanorice: A hybrid plasmonic nanostructure,
Nano Lett. 6(4), 827832 (2006).
 A. E. Neeves and M. H. Birnboim,
Composite structures for the enhancement of nonlinearoptical susceptibility,
J. Opt. Soc. Am. B 6(4), 787796 (1989).
 J. W. Haus, H. S. Zhou, S. Takami, M. Hirasawa, I. Honma, and H. Komiyama,
Enhanced optical properties of metalcoated nanoparticles,
J. Appl. Phys. 73(3), 10431048 (1993).
 H. S. Zhou, I. Honma, H. Komiyama, and J. W. Haus,
Controlled synthesis and quantumsize effect in goldcoated nanoparticles,
Phys. Rev. B 50(4), 1205212056 (1994).
 S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas,
Composite structures for the enhancement of nonlinearoptical susceptibility,
Chem. Phys. Lett. 288(24), 243247 (1998).
 J. J. Penninkhof, C. Graf, A. Moroz, A. Polman,
A. van Blaaderen, Surface plasmons in anisotropic Aushell/silicacore colloids,
the contribution A2/P.06 on p. 9 in the book of abstracts
EMRS Meetings Strassbourg 2004.

J. J. Penninkhof, C. Graf, T. van Dillen, A. M. Vredenberg,
A. van Blaaderen, and A. Polman,
AngleDependent Extinction of Anisotropic Silica/Au Core/Shell
Colloids Made via Ion Irradiation,
Advanced Materials 17(12), 14841488 (2005).

J. J. Penninkhof, T. van Dillen, S. Roorda, C. Graf,
A. van Blaaderen, A. M. Vredenberg, and A. Polman,
Anisotropic deformation of metallodielectric coreshell colloids under MeV ion irradiation,
Proceedings of the 14th International Conference on Ion Beam Modification of Materials,
Nuclear Instruments and Methods in Physics Research Section B 242(12), 523529
(January 2006).

J. J. Penninkhof,
``Tunable plasmon resonances in
anisotropic metal nanostructures",
PhD thesis from 2006 [6.5 MB]  if for nothing else, have a look at the image gallery on
pp. 123125 of the thesis, which displays in the very last image "nanoufo's".
 J. J. Penninkhof, A. Moroz, A. Polman, and A. van Blaaderen,
Optical properties of spherical and oblate spheroidal gold shell colloids,
J. Phys. Chem. C 112(11), 41464150 (2008).
(Accompanying F77 code sphere.f. Read compilation and
further instructions here.)

J. J. Penninkhof
power point presentation at the MRS Spring Meeting, April 20, 2006.
Finite and infinite linear chains of nanoparticles
The work by Quinten et al [1] followed
by later experiments by Brongersma et al [2] have triggered a lot
of excitement regarding electromagnetic energy transfer and
switching in nanoparticle chain arrays below the diffraction limit.
A flurry of theoretical work attempting to describe
the propagation in the linear chains of coupled particles by various
approximative approaches has followed. Suprisingly enough,
it has often been forgotten that
the seminal work by Gérardy and Ausloos, represented
by a series of articles in Phys. Rev. B in early eighties,
provides an exact description of such chains, finite or infite [36],
with or without the inclusion of (i) nonlocal (spatial dispersion) effects
or (ii) highorder multipolar (electric and magnetic)
interaction effects.

For instance, in Ref. [3] the Laplace equation was solved exactly for
the system of N dielectric
spheres of arbitrary sizes, embedded in a dielectric matrix,
and general solutions was provided taking into account interactions
at all multipolar orders. Absorption and absorbed power spectra were calculated for
two (different or identical) spheres and for an
infinite linear chain of identical spheres for different field
directions.
 In Ref. [4] the same problem was reanalyzed in terms of the general
solution of Maxwell's equations for any arbitrary
cluster geometry and for any light (polarized or not) incidence
by usual expansion of the various fields. Usual boundary
conditions (including nonlocal effects) were used.
Highorder multipolar (electric and magnetic)
interaction effects were included. As an example,
the solution for an infinite linear chain was
presented in dipole approximation.
 In Ref. [5], the theory of [4] was extended for an arbitrary
distribution of heterogeneous spheres taking into account cluster
geometry, retardation effects, all multipolar (electric and magnetic)
order interactions, and any arbitrary light incidence.
 In Ref. [6] the following questions were answered in the dipolar approximation:
 When is a particle isolated?
 When can one not
distinguish one small particle near or in contact with a larger one?
The solution was then applied to binary
clusters of metallic particles and numerical results
were presented specifically for sodium. The dielectric constant contained plasmon
effects. A discussion of highorder polar effects was included.
References:

M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg,
Electromagnetic energy transport via linear chains of silver
nanoparticles,
Opt. Lett. 23, 13311333 (1998).

M. L. Brongersma, J. W. Hartman, and H. A. Atwater,
Electromagnetic energy transfer and switching in nanoparticle chain arrays
below the diffraction limit,
Phys. Rev. B 62, R16356R16359 (2000).
 J. M. Gérardy and M. Ausloos,
Absorption spectrum of
clusters of spheres from the general solution of Maxwell's equations.
The longwavelength limit,
Phys. Rev. B 22, 49504959 (1980).
 J. M. Gérardy and M. Ausloos,
Absorption spectrum of clusters of spheres from the general
solution of Maxwell's equations. II. Optical properties of
aggregated metal spheres,
Phys. Rev. B 25, 42044229 (1982).
 J. M. Gérardy and M. Ausloos,
Absorption spectrum of clusters of spheres from the general
solution of Maxwell's equations. III. Heterogeneous spheres,
Phys. Rev. B 30, 21672181 (1984).
 J. M. Gérardy and M. Ausloos,
Absorption spectrum of
clusters of spheres from the general solution of Maxwell's equations. IV.
Proximity, bulk, surface, and shadow effects (in binary clusters),
Phys. Rev. B 27, 64466463 (1983).
Plasmalike behavior in artificial dielectric composed of periodically
spaced lattices of metallic rods and plates
That an artificial dielectric composed of a
number of conducting plates of proper shape and spacing [1] or of periodically
spaced lattices of metallic rods, also known as wire grid, wire mesh, or rodded structure
[28], can exhibit a plasmalike behaviour [26,8]
and its index of refraction can be less than unity [1,2] have been discussed
in the literature much earlier than in the middle of nineties. See the references below.
References:
 W. E. Kock, MetalLens Antennas,
Proc. IRE 34, 828836 (1946).
 J. Brown, Artificial dielectrics having refractive
indices less than unity,
Proc. IEEE, Monograph no. 62R, vol. 100, Pt. 4, pp. 5162 (1953).

J. Brown and W. Jackson, The properties of
artificial dielectrics at centimeter wavelengths,
Proc. IEEE, paper no. 1699R, vol. 102B, pp. 1121 (1955).
 J. S. Seeley and J. Brown, The use of
artificial dielectrics in a beam scanning prism,
Proc. IEEE, paper no. 2735R, vol. 105C, pp. 93102 (1958).
 J. S. Seeley, The quarterwave matching of
dispersive materials,
Proc. IEEE, paper no. 2736R, vol. 105C, pp. 103106 (1958).
 A. Carne and J. Brown, Theory of reflections from the roddedtype
artificial dielectrics,
Proc. IEEE, paper no. 2742R, vol. 105C, pp. 107115 (1958).
 A. M. Model, Propagation of plane electromagnetic waves
in a space which is filled with plane parallel grids,
Radiotekhnika 10, 5257 (1955) (in Russian).
 W. Rotman, Plasma simulation by artificial dielectrics and
parallelplate media,
IEEE Trans. Antennas Propag. 10, 8295 (1962).
(For negative refraction
headlines see here.)
Effective medium properties, meanfield description,
homogenization, or homogenisation of photonic crystals
Let lambda be the wavelength
in the ambient medium, and the respective n and R
be refractive index and a characteristic dimension of a particle
embedded in the medium. One then speaks
of the static limit when both 2*pi*R/lambda << 1
and 2*pi*R*n/lambda << 1.
Obviously if the first condition is satisfied then is
also the second, provided that n is moderate.
However, if n is relatively large
(e.g., metallic particles at optical and infrared frequencies)
one can find a wavelength range so that
2*pi*R/lambda << 1 but not necessarily
2*pi*R*n/lambda << 1.
One then speaks of the quasistatic limit.
Note that unlike in the static limit,
the effective medium parameters in the quasistatic limit
are dispersive even for medium composed of nondispersive
components.
In the case of a cubic array of spherical particles,
the static case is satisfactorily covered by the
Maxwell Garnett theory [1] that can be viewed as a mean field theory.
In fact, the MaxwellGarnett formula effective medium is an exact static solution
for the case of coated spheres of all sizes which, without any void,
fill in the entire medium
(see Sec. IV of
[2]).
The static limit has also been discussed in depth by
D. J. Bergman and K.J. Dunn [3].
The usual assumptions of the
validity of the Maxwell Garnett type of composite geometry have been that:

the inclusions are assumed to be spheres
or ellipsoids of a size much smaller than the optical wavelength;

the distance between them is much larger than their characteristic size and
 much smaller than the optical wavelength.
Suprizingly enough, the Maxwell Garnett theory
works very well even if the hypotheses that 2*pi*R/lambda << 1
and f<< 1, where f is the volume fraction occupied by the spheres,
are no longer valid.
Indeed, for a moderate n, the Maxwell Garnett theory has been shown
to describe the effective medium properties of
cubic photonic crystals within a few percent of exact result
till the first stop gap, and that even in the closepacked case
(for dense dielectric spheres in air and for
inverted opals see Refs. [4,5]).
The latter also holds for core shell spherical particles
[4].
(A useful parametrization
of the Maxwell Garnett formula for the case of core shell spherical particles
can be found in Ref. [4].)
Effective medium properties (effective dielectric permittivity
and magnetic permeability tensors) of periodic arrays
of arbitrarily shaped particles
have been obtained in the quasistatic limit
by L. Lewin [6] and N. A. Khizhnyak [79,11].
Their work also contains the idea of
generating an effective magnetic response from a
composite comprising inherently nonmagnetic materials.
First Lewin [6] provided a quasistatic
extention of the Maxwell Garnett theory [1]
for the case of a cubic array of spherical particles.
(I wish I knew his work at the time of writing
my publications [4,5]).
Later on, Khizhnyak [79,11] generalized
the results of Lewin for cubic arrays
of spherical particles to the case of
arbitrary periodic arrays of arbitrarily shaped particles.
In Ref. [7], general formulas were obtained
for the respective tensors of dielectric permittivity and magnetic
permeability. Since Khizhnyak no longer
required that either the lattice be cubic or the particles be
spherical, such an artificial dielectric, or
effective medium, could in general have anisotropic properties,
that Khizhnyak analyzed in detail.
In Ref. [8], the respective tensors of dielectric
permittivity and magnetic permeability were investigated more closely
for an effective medium composed of
periodic arrays of spherical particles. Threedimensional
tetragonal, orthogonal, hexagonal,
and monoclinic arrays were studied.
On p. 2025, last paragraph the following observation was
made: ``From Eqs. (2) and (5) an interesting
property of artificial dielectrics follows. Even though
an artificial dielectric is made from inherently nonmagnetic materials,
at higher frequency it will exhibit both
dielectric and magnetic anisotropy."
In Ref. [9], the respective tensors of dielectric permittivity and magnetic
permeability were investigated in detail
for an effective medium composed of
periodic arrays of ellipsoidal particles. The role of multipole
interactions was assessed. A brief comparison with experimental results
by Kharadly and Jackson [10] was made.
Having in mind an application for the construction of
centimeterwave lens antennae,
Kharadly and Jackson [10] studied
(both theoretically and experimentally)
the effective permittivity of periodic
arrays of perfectly conducting elements of simple geometric
shapes (cylindrical rods, thin strips, spheres, and disks).
However, Kharadly and Jackson [10]
did not notice anything particular for centimeter waves
and their effective permittivity remained
to be equal or larger than one.
In 1959 Khizhnyak [11] obtained the
effective medium parameters (effective permittivity and permeability)
for artificial anisotropic dielectrics formed from
twodimensional square, rectangular, and rhombical
lattices of infinite circular and elliptical
bars and rods [11]. Sec. 5 of Ref. [11]
discussed dispersive properties of
the artificial dielectrics and the following observation
was made: ``Upon substituting Eqs. (29)(30) back
into Eq. (22) one may find that resonance frequencies are possible
at which the permittivity and permeability increase without bound."
At the very end, the case of a regular array of cylinders with
an axis inclined with respect to the twodimensional periodicty plane
was considered.
Relatively recent work by
Waterman and Pedersen [12] provided again very detailed
and explicit analytical and numerical results for the effective complex
dielectric constant and permeability in the quasistatic and infinitesimal lattice
(static) limits for several lattice geometries.
In this regard, Eq. (35) in Sec. IV of
P. C. Waterman and N. E. Pedersen [12]
yields corrections to the MaxwellGarnett formula (given by
Eq. (33) of [12]), in the form
of an expansion up to terms of order f**6,
where f is the volume fraction occupied by the spheres.
The results were shown to agree with existing static computations under
appropriate conditions.
According to the formula (35) of P. C. Waterman and N. E. Pedersen
[12], the results depend only weakly on the actual
periodic arrangmenet (e.g. whether simple cubic, bodycentered cubic, or
facecentered cubic lattice).
The formula (35) yields the effective medium properties within a few
percent up to volume fractions of about 90% of the maximum
allowable values (spheres touching).
A comparison with the MaxwellGarnett formula
confirmed that the MaxwellGarnett formula describes the effective medium
properties up to surprisingly large filling fractions.
See in this regard also Fig. 2 in the article by W. Lamb et al
[13].
Random arrays were also considered briefly, and the role of
singleparticle resonance effects was examined.
Although Waterman and Pedersen [12]
quoted the work by Kharadly and Jackson [10],
they were apparently unaware of the earlier results by Khizhnyak [79,11].
Yet even the Waterman and Pedersen work [12] appears to
be largely unknown in the photonic crystals community that
was formed merely a few years after the Waterman and Pedersen work [12]
had been published.
Some recent results obtained using a quasistatic extention of
the Maxwell Garnett theory for binary composites
(= two different types of spheres embedded in a matrix)
you can find in Ref. [14
(minor erratum)].
References:
 J. C. Maxwell Garnett, Colours in metal glasses and in metallic films,
Phil. Trans. R. Soc. London 203, 385420 (1904).

Z. Hashin and S. Shtrikman,
A Variational Approach to the Theory of the Effective
Magnetic Permeability of Multiphase Materials,
J.
Appl. Phys. 33, 31253131 (1962).

D. J. Bergman and K.J. Dunn,
Bulk effective dielectric constant of a composite with a periodic
microgeometry,
Phys.
Rev. B 45, 1326213271 (1992).
 A. Moroz and C. Sommers,
Photonic band gaps of
threedimensional facecentered cubic lattices,
J.
Phys.: Condens. Matter 11,
9971008 (1999).
[physics/9807057]
[story behind this article]
 A. Moroz, A simple formula for the Lgap width of a
facecenteredcubic photonic crystal,
J.
Opt. A: Pure Appl. Opt. 1,
471475 (1999).
[physics/9903022]
 L. Lewin,
The electrical constants of a material loaded with spherical particles,
Proc. Inst. Elec. Eng. 94, 6568 (1947).
 N. A. Khizhnyak, Artificial anisotropic dielectrics: I.,
Sov. Phys. Tech. Phys. 27, 20062013 (1957).
 N. A. Khizhnyak, Artificial anisotropic dielectrics: II.,
Sov. Phys. Tech. Phys. 27, 20142026 (1957).
 N. A. Khizhnyak,
Artificial anisotropic dielectrics: III.,
Sov. Phys. Tech. Phys. 27, 20272037 (1957).
 M. M. Z. Kharadly and W. Jackson,
The properties of artificial dielectrics comprising arrays
of conducting elements,
Proceedings IEE 100, 199212 (1953).

N. A. Khizhnyak,
Artificial anisotropic dielectrics formed from
twodimensional lattices of infinite bars and rods,
Sov. Phys. Tech. Phys. 29, 604614 (1959).

P. C. Waterman and N. E. Pedersen,
Electromagnetic scattering by periodic arrays of particles,
J. Appl. Phys. 59, 26092618 (1986).

W. Lamb, D. M. Wood, and N. W. Ashcroft,
Longwavelength electromagnetic propagation in heterogeneous media,
Phys. Rev. B 21, 22482266 (1980).
 V. Yannopapas and A. Moroz, Negative refractive index metamaterials
from inherently nonmagnetic materials for deep infrared
to terahertz frequency ranges,
J. Phys.: Condens. Matter. 17, 37173734 (2005)
(minor erratum)
(see also my accompanying F77 code EFFE2P).
Regular
void lattices in ion or neutron bombarded metals
J. H. Evans et al has demonstrated experimentally that,
under ion or neutron bombardement,
regular bodycenteredcubic void lattices form in metals, with lattice
constant being usually several tens
of nanometers  a truly metallic
photonic nanocrystal [1,2]!
Later on, this irradiation effect has been observed in various other metals.
A review on this subject has been compiled by Krishan [3].
References:
 J. H. Evans, Observations of a regular void array in high purity
molybdenum irradiated with 2 MeV nitrogen ions,
Nature 229, 403404 (1971).
 B. L. Eyre and J. H. Evans, Void Formed
by Irradiation of Reactor Materials,
eds. S. F. Pugh, M. H. Loretto, and D. I. R. Norris
(British Nuclear Energy Society, London, 1971), p. 323.
 K. Krishan, Ordering of voids and gas bubles in
radiation enviroments,
Radiat. Eff. 66, 121155 (1982).
Photonic crystals of metallic particles
Back at the end of the last "millenium", the dominant method in the
photonic band structure calculations was the plane wave method
and associated with it the
MIT PhotonicBands (MPB) package.
However, on using the above methods, it was impossible to deal with a dispersion
of the dielectric function. The first photonic band structure calculations
by taking into account the full material dispersion and using the real material data
for photonic crystals of metallic inclusions have been performed
on using 2D and 3D variants of the
photonic KKR method
in the articles listed below.
References:
 A. Moroz, Threedimensional complete photonic bandgap structures
in the visible,
Phys. Rev. Lett. 83, 52745277 (1999).
 W. Y. Zhang, X. Y. Lei, Z. L. Wang, D. G. Zheng, W. Y. Tam,
C. T. Chan, and Ping Sheng,
Robust Photonic Band Gap from Tunable Scatterers,
Phys. Rev. Lett. 84, 28532856 (2000).
 A. Moroz, Photonic crystals of coated metallic spheres,
Europhys.
Lett. 50, 466472 (2000).
[condmat/0003518]
[pdf]
 H. van der Lem and A. Moroz, Towards twodimensional complete
photonicbandgap structures below infrared wavelengths,
J.
Opt. A: Pure Appl. Opt. 2, 395399 (2000) [pdf].
2D metallodielectric photonic crystals
Although the work by Kuzmiak, Maradudin, and Pincemin [1]
was the first dealing with the properties
of 2D metallodielectric photonic crystals, their approach
allowed one to deal with only very small metal volume
fractions smaller than 1% (!!!).
The first calculations for 2D metallodielectric photonic crystals
without any limitation

on the metal volume filling fraction and
 on metal dispersion
have been preformed in
Ref. [2].
The calculations in [2]
were performed on using a 2D photonic KKR method.
2D metallodielectric photonic crystals of metal cylinders were investigated
for the Drude fit to silver data and
for the experimental silver data, for both square and triangular lattices.
In Drude/square and silver/triangle lattice cases
a plurality of complete photonic band gaps (CPBGs) were found, with
the largest CPBG having the gap width to midgap ratio
of 10% already for the host dielectric constant
of air. Below the complete photonic band gaps, a region of
flat bands was first observed.
The article [2]
was the first one to emphasize the importance of filling the pores
of a purely dielectric 2D air hole photonic crystal with silver, resulting in
a 2D lattice of metallic wires embedded in a dielectric matrix,
in order to obtain an elusive complete photonic bandgap (i.e. common for
both polarizations and for all propagation directions) in the visible.
A follow up for 2D photonic crystal of silver
coated silicon cylinders
and a comparison with the experimental reflection, transmission,
and absorption have been provided in
Ref. [3].
References:
 V. Kuzmiak, A. A. Maradudin, and F. Pincemin,
Photonic band structures of twodimensional systems
containing metallic components,
Phys. Rev. B 50, 1683516844 (1994).
 H. van der Lem and A. Moroz, Towards twodimensional complete
photonicbandgap structures below infrared wavelengths,
J.
Opt. A: Pure Appl. Opt. 2, 395399 (2000) [pdf].

V. Poborchii, T. Tada, T. Kanayama, and A. Moroz,
Silvercoated siliconpillar photonic crystals: enhancement
of a photonic band gap,
Appl. Phys. Lett. 82, 508510 (2003).
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December 14, 2008
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