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 [2-5], 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 quasi-static approximation. The results of Neeves and Birnboim were later on, also within the quasi-static approximation, generalized by Haus et al [3] to the case of an arbitrary number of shells. Both nanorice and nanoshells are made of a non-conducting core that is covered by a metallic shell [1-5]. 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 E-MRS Spring Meeting 2004 in Strasbourg held on May 24-28, 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 2-3 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:

  1. H. Wang, D. W. Brandl, F. Le, P. Nordlander, and N. J. Halas, Nanorice: A hybrid plasmonic nanostructure, Nano Lett. 6(4), 827-832 (2006).
  2. A. E. Neeves and M. H. Birnboim, Composite structures for the enhancement of nonlinear-optical susceptibility, J. Opt. Soc. Am. B 6(4), 787-796 (1989).
  3. J. W. Haus, H. S. Zhou, S. Takami, M. Hirasawa, I. Honma, and H. Komiyama, Enhanced optical properties of metal-coated nanoparticles, J. Appl. Phys. 73(3), 1043-1048 (1993).
  4. H. S. Zhou, I. Honma, H. Komiyama, and J. W. Haus, Controlled synthesis and quantum-size effect in gold-coated nanoparticles, Phys. Rev. B 50(4), 12052-12056 (1994).
  5. S. J. Oldenburg, R. D. Averitt, S. L. Westcott, and N. J. Halas, Composite structures for the enhancement of nonlinear-optical susceptibility, Chem. Phys. Lett. 288(2-4), 243-247 (1998).
  6. J. J. Penninkhof, C. Graf, A. Moroz, A. Polman, A. van Blaaderen, Surface plasmons in anisotropic Au-shell/silica-core colloids, the contribution A2/P.06 on p. 9 in the book of abstracts EMRS Meetings Strassbourg 2004.
  7. J. J. Penninkhof, C. Graf, T. van Dillen, A. M. Vredenberg, A. van Blaaderen, and A. Polman, Angle-Dependent Extinction of Anisotropic Silica/Au Core/Shell Colloids Made via Ion Irradiation, Advanced Materials 17(12), 1484-1488 (2005).
  8. J. J. Penninkhof, T. van Dillen, S. Roorda, C. Graf, A. van Blaaderen, A. M. Vredenberg, and A. Polman, Anisotropic deformation of metallo-dielectric core-shell 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(1-2), 523-529 (January 2006).
  9. 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. 123-125 of the thesis, which displays in the very last image "nano-ufo's".
  10. 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), 4146-4150 (2008).
    (Accompanying F77 code sphere.f. Read compilation and further instructions here.)
  11. 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 [3-6], with or without the inclusion of (i) nonlocal (spatial dispersion) effects or (ii) high-order multipolar (electric and magnetic) interaction effects.

References:

  1. M. Quinten, A. Leitner, J. R. Krenn, and F. R. Aussenegg, Electromagnetic energy transport via linear chains of silver nanoparticles, Opt. Lett. 23, 1331-1333 (1998).
  2. 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, R16356-R16359 (2000).
  3. J. M. Gérardy and M. Ausloos, Absorption spectrum of clusters of spheres from the general solution of Maxwell's equations. The long-wavelength limit, Phys. Rev. B 22, 4950-4959 (1980).
  4. 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, 4204-4229 (1982).
  5. 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, 2167-2181 (1984).
  6. 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, 6446-6463 (1983).


Plasma-like 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 [2-8], can exhibit a plasma-like behaviour [2-6,8] 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:

  1. W. E. Kock, Metal-Lens Antennas, Proc. IRE 34, 828-836 (1946).
  2. J. Brown, Artificial dielectrics having refractive indices less than unity, Proc. IEEE, Monograph no. 62R, vol. 100, Pt. 4, pp. 51-62 (1953).
  3. J. Brown and W. Jackson, The properties of artificial dielectrics at centimeter wavelengths, Proc. IEEE, paper no. 1699R, vol. 102B, pp. 11-21 (1955).
  4. J. S. Seeley and J. Brown, The use of artificial dielectrics in a beam scanning prism, Proc. IEEE, paper no. 2735R, vol. 105C, pp. 93-102 (1958).
  5. J. S. Seeley, The quarter-wave matching of dispersive materials, Proc. IEEE, paper no. 2736R, vol. 105C, pp. 103-106 (1958).
  6. A. Carne and J. Brown, Theory of reflections from the rodded-type artificial dielectrics, Proc. IEEE, paper no. 2742R, vol. 105C, pp. 107-115 (1958).
  7. A. M. Model, Propagation of plane electromagnetic waves in a space which is filled with plane parallel grids, Radiotekhnika 10, 52-57 (1955) (in Russian).
  8. W. Rotman, Plasma simulation by artificial dielectrics and parallel-plate media, IEEE Trans. Antennas Propag. 10, 82-95 (1962).

(For negative refraction headlines see here.)

Effective medium properties, mean-field 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 quasi-static limit. Note that unlike in the static limit, the effective medium parameters in the quasi-static 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 Maxwell-Garnett 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:

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

  2. the distance between them is much larger than their characteristic size and

  3. 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 close-packed 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 quasi-static limit by L. Lewin [6] and N. A. Khizhnyak [7-9,11]. Their work also contains the idea of generating an effective magnetic response from a composite comprising inherently nonmagnetic materials.

First Lewin [6] provided a quasi-static 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 [7-9,11] 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. Three-dimensional 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 centimeter-wave 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 two-dimensional 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 two-dimensional 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 Maxwell-Garnett 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, body-centered cubic, or face-centered 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 Maxwell-Garnett formula confirmed that the Maxwell-Garnett 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 single-particle 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 [7-9,11]. 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 quasi-static 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:

  1. J. C. Maxwell Garnett, Colours in metal glasses and in metallic films, Phil. Trans. R. Soc. London 203, 385-420 (1904).
  2. Z. Hashin and S. Shtrikman, A Variational Approach to the Theory of the Effective Magnetic Permeability of Multiphase Materials, J. Appl. Phys. 33, 3125-3131 (1962).
  3. D. J. Bergman and K.-J. Dunn, Bulk effective dielectric constant of a composite with a periodic microgeometry, Phys. Rev. B 45, 13262-13271 (1992).
  4. A. Moroz and C. Sommers, Photonic band gaps of three-dimensional face-centered cubic lattices,
    J. Phys.: Condens. Matter 11, 997-1008 (1999). [physics/9807057] [story behind this article]
  5. A. Moroz, A simple formula for the L-gap width of a face-centered-cubic photonic crystal,
    J. Opt. A: Pure Appl. Opt. 1, 471-475 (1999). [physics/9903022]
  6. L. Lewin, The electrical constants of a material loaded with spherical particles, Proc. Inst. Elec. Eng. 94, 65-68 (1947).
  7. N. A. Khizhnyak, Artificial anisotropic dielectrics: I., Sov. Phys. Tech. Phys. 27, 2006-2013 (1957).
  8. N. A. Khizhnyak, Artificial anisotropic dielectrics: II., Sov. Phys. Tech. Phys. 27, 2014-2026 (1957).
  9. N. A. Khizhnyak, Artificial anisotropic dielectrics: III., Sov. Phys. Tech. Phys. 27, 2027-2037 (1957).
  10. M. M. Z. Kharadly and W. Jackson, The properties of artificial dielectrics comprising arrays of conducting elements, Proceedings IEE 100, 199-212 (1953).
  11. N. A. Khizhnyak, Artificial anisotropic dielectrics formed from two-dimensional lattices of infinite bars and rods, Sov. Phys. Tech. Phys. 29, 604-614 (1959).
  12. P. C. Waterman and N. E. Pedersen, Electromagnetic scattering by periodic arrays of particles, J. Appl. Phys. 59, 2609-2618 (1986).
  13. W. Lamb, D. M. Wood, and N. W. Ashcroft, Long-wavelength electromagnetic propagation in heterogeneous media, Phys. Rev. B 21, 2248-2266 (1980).
  14. V. Yannopapas and A. Moroz, Negative refractive index metamaterials from inherently non-magnetic materials for deep infrared to terahertz frequency ranges, J. Phys.: Condens. Matter. 17, 3717-3734 (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 body-centered-cubic 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:

  1. J. H. Evans, Observations of a regular void array in high purity molybdenum irradiated with 2 MeV nitrogen ions, Nature 229, 403-404 (1971).
  2. 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.
  3. K. Krishan, Ordering of voids and gas bubles in radiation enviroments, Radiat. Eff. 66, 121-155 (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 Photonic-Bands (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:

  1. A. Moroz, Three-dimensional complete photonic bandgap structures in the visible, Phys. Rev. Lett. 83, 5274-5277 (1999).
  2. 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, 2853-2856 (2000).
  3. A. Moroz, Photonic crystals of coated metallic spheres, Europhys. Lett. 50, 466-472 (2000). [cond-mat/0003518] [pdf]
  4. H. van der Lem and A. Moroz, Towards two-dimensional complete photonic-bandgap structures below infrared wavelengths, J. Opt. A: Pure Appl. Opt. 2, 395-399 (2000) [pdf].

2D metallo-dielectric photonic crystals

Although the work by Kuzmiak, Maradudin, and Pincemin [1] was the first dealing with the properties of 2D metallo-dielectric photonic crystals, their approach allowed one to deal with only very small metal volume fractions smaller than 1% (!!!).

The first calculations for 2D metallo-dielectric photonic crystals without any limitation

have been preformed in Ref. [2]. The calculations in [2] were performed on using a 2D photonic KKR method. 2D metallo-dielectric 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:

  1. V. Kuzmiak, A. A. Maradudin, and F. Pincemin, Photonic band structures of two-dimensional systems containing metallic components, Phys. Rev. B 50, 16835-16844 (1994).
  2. H. van der Lem and A. Moroz, Towards two-dimensional complete photonic-bandgap structures below infrared wavelengths, J. Opt. A: Pure Appl. Opt. 2, 395-399 (2000) [pdf].
  3. V. Poborchii, T. Tada, T. Kanayama, and A. Moroz, Silver-coated silicon-pillar photonic crystals: enhancement of a photonic band gap,
    Appl. Phys. Lett. 82, 508-510 (2003). [pdf] [try MS Windows executable rta1in2k.exe]


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