Plasmonic bandgap in random media
 Valentina V Zhurikhina^{1},
 Michael I Petrov^{2, 3},
 Oksana V Shustova^{1},
 Yuri P Svirko^{3}Email author and
 Andrey A Lipovskii^{1, 2}
DOI: 10.1186/1556276X8324
© Zhurikhina et al.; licensee Springer. 2013
Received: 6 June 2013
Accepted: 9 July 2013
Published: 16 July 2013
Abstract
We present a dispersion theory of the surface plasmon polaritons (SPP) in random metaldielectric nanocomposite (MDN) consisting of bulk metal embedded with dielectric inclusions. We demonstrate that embedding of dielectric nanoparticles in metal results in the formation of the plasmonic bandgap due to strong coupling of the SPP at the metalvacuum interface and surface plasmons localized at the surface of nanoinclusions. Our results show that MDN can replace metals in various plasmonic devices, which properties can be tuned in a wide spectral range. Being compatible with waveguides and other photonic structures, MDN offers high flexibility in the plasmonic system design.
PACS
78.67.Sc, 81.05.Ni, 71.36.+c
Background
The resonant coupling of light to oscillations of the free electron density near the metal surface, surface plasmons (SP), gave birth to a variety of advanced applications ranging from sensing to nonlinear optics. SPs are bound to the metallic surface, i.e., at the frequency of the surface plasmon resonance, light field exponentially decays in neighboring media. Since the decay length of SPs is two orders of magnitude smaller than the wavelength of the light in air, they can be employed for subwavelength localization of light. The guiding of light in plasmonic structures is possible via surface plasmon polaritons (SPP) that can propagate in periodical arrays of metal nanoparticles embedded in dielectrics. The multiple scattering of the SPPs off the periodic corrugation leads to the Bragglike plasmon modes [1, 2] and to the plasmonic band gaps [1, 3], i.e., they do not allow the SPP in a certain interval of wavelengths. When metal nanoparticles are placed into dielectric in a random fashion, e.g., in metal island films [4, 5], nanoporous metal films [6], and metaldielectric nanocomposite (MDN) [7–10], no SPP bandgaps have been observed. The optical properties of these materials dominated by SPs localized on individual metal nanoparticles are well studied [11, 12]; however, much less attention was paid to the behavior of SPP propagating at the MDNdielectric interface. In this paper, we present the theory of SPP in MDN based on noble metals with random distribution of the dielectric inclusions. We demonstrate that when the metal volume content is high, the coupling of propagating and localized at metalinclusion interface plasmon modes results in the formation of the SPP bandgap in such random media. By using Drude model for dielectric function of the metal, we develop dispersion theory of the SPP at the MDNvacuum surface. We demonstrate that in silver, bandgap persists when dielectric properties of the metal are described by experimental data. The presence of the SPP bandgap indicates that the MDN can replace metals in various plasmonic structures that will benefit from the tunability of the MDN properties.
Methods
One can observe from Equations 1 and 2 that SPP is allowed at Re(ϵ_{2}(ω) + ϵ_{1}) < 0 when Re(k_{ SPP }) ≠ 0 and Im(δ_{1,2}) = 0. The condition Re(ϵ_{2}(ω) + ϵ_{1}) = 0 corresponds to the excitation of the surface plasmon [1, 13]. If Re(ϵ_{2}(ω)) > 0, SPP is forbidden; however, a transversal bulk plasmon polariton (BPP) with wave vector ${k}_{\mathrm{BPP}}=\left(\omega /c\right)\sqrt{{\u03f5}_{2}\left(\omega \right)}$ can propagate at z > 0. If 0 > Re(ϵ_{2}(ω)) > − ϵ_{1}, no propagating electromagnetic perturbations are allowed, i.e., the energy of the incident light wave is transferred to the localized plasmons.
One can see from Equation 5 that the effective dielectric function has singularities at ω = 0 and ω = Ω_{TO}. The singularity at ω = 0 is a conventional ‘metal’ one, while the singularity at ω = Ω_{TO} corresponds to the collective oscillations of the conduction electrons at the surface of dielectric nanoparticles incorporated into the metal matrix, i.e., localized surface plasmon resonance at the metaldielectric interface. Frequency Ω_{LO} corresponds to the excitation of the longitudinal phonons in the GMN.
Results and discussion
The dispersion relation for propagating electromagnetic modes in Drude MDN with dielectric volume fraction g = 0.1 and ϵ_{d} = 3.42 is shown in Figure 1a. Figure 1b shows the map of collective excitations in Drude MDN in the ‘ωg’ plane at ϵ_{d}= 3.42. One can observe two SPP bands, the BPP band, and the forbidden gap separated by frequencies Ω_{LO}, Ω_{TO}, and ω_{SC1}. The upper limit of the higher SPP zone is ω_{SC2}. There also exists the second BPP frequency range for ω > ω_{p}. The width of both SPP and BPP bands increases with the increase of dielectric contained in MDN. The latter was earlier demonstrated by N. Stefanou and coauthors [15] for mesoporous metals. Our calculations also showed that the higher the permittivity of dielectric inclusions in MDN, the broader the upper SPP band and the bigger the downshift of the SPP forbidden gap.
When g → 0, the upper MDN surface plasmon frequency ${\omega}_{\mathrm{SC}2}\to {\omega}_{\mathrm{p}}/\sqrt{2}$, that is, the surface plasmon frequency at metalair interface, while Ω_{LO}, Ω_{TO}, and ω_{SC1} approach ${\omega}_{\mathrm{p}}\sqrt{2/\left(2+{\u03f5}_{\mathrm{d}}\right)}$, that is, the SP resonance of a single dielectric cavity in metal matrix [15]. At ϵ_{d} > 2, the frequencies Ω_{LO}, Ω_{TO}, and ω_{SC1} are lower than ω_{SC2}, and BPP zone and the conventional metal SPP band at ω < ω_{SC2} splits by two (see Figure 1b). At ϵ_{d} < 2, the Ω_{LO}, Ω_{TO}, and ω_{SC1} are higher than ω_{SC2}, and the conventional metal SPP band at ω < ω_{SC2} remains intact, however, the second SPP band appears at ω_{LO}< ω < ω_{SC2}. At ${\u03f5}_{\mathrm{d}}=2\text{and}g=0,{\mathit{\Omega}}_{\mathrm{LO}}={\mathit{\Omega}}_{\mathrm{TO}}={\omega}_{\mathrm{SC}1}={\omega}_{\mathrm{SC}2}={\omega}_{\mathrm{p}}/\sqrt{2}$.
It is worth noting that the dielectric dispersion should change the characteristic frequencies that will lead to the frequency shift of all bands and, in the case of strong dispersion, could possibly result in broadening or vanishing of the second SPP band. But for the most optically transparent dielectrics, their dispersion is negligible compared to the metal one. In this paper we neglect the dielectric dispersion that is valid, for example, for glasses in the visible and nearinfrared range.
Figure 3b shows the map of collective excitations in silverbased MDN on the ωg plane at ϵ_{d} = 3.42. One can observe that the shape and size of the gray area in which SPP is allowed is similar to that for Drude MDN (see Figure 1); however, the nonzero imaginary part of the dielectric permittivity of silver results in vanishing of the SPP bandgap at g < 0.03. Thus, only one surface plasmon polariton band exists at g < 0.03.
Conclusions
We demonstrate that SPP bandgap can exist not only in plasmonic crystals but also in MDN with low dielectric volume fraction, i.e., when dielectric nanoinclusions are distributed in a random fashion in metal host. In the MDN, the SPP bandgap arises due to strong coupling between SPP at the metaldielectric interface and plasmons localized on dielectric nanoinclusions allowing one to tailor the plasmonic properties by changing the dielectric content. By using MaxwellGarnett model, we calculated effective dielectric permittivity of the MDN using both Drude model and Johnson and Christy data for complex dielectric function of metal. We showed that dissipation caused by the scattering of conduction electrons in metal may result in vanishing plasmonic bandgap in noble metalbased MDN. However, at refractive index of dielectric inclusions n > 1.5, the plasmonic bandgap survives in Agbased MDN offering high flexibility in the plasmonic system design.
Authors’ information
VZh holds associate professor position at St. Petersburg State Polytechnical University. MP holds PhD degree at St. Petersburg Academic University. OSh is a PhD student at St. Petersburg State Polytechnical University. YuS holds DrSci degree and professor position at the University of Eastern Finland. AL holds DrSci degree and professor positions at St. Petersburg Academic University and St. Petersburg State Polytechnical University.
Abbreviations
 MDN:

Metaldielectric nanocomposite
 SP:

Surface plasmons
 SPP:

Surface plasmon polaritons.
Declarations
Acknowledgments
This study was supported by Russian foundation for Basic Research (project no. 120291664), Russian Ministry for Education and Science, Joensuu University Foundation, Academy of Finland (project nos. 135815 and 137859) and EU (FP7 projects ‘NANOCOM’ and ‘AN2’).
Authors’ Affiliations
References
 Zayats AV, Smolyaninov II, Maradudin AA: Nanooptics of surface plasmon polaritons. Phys Rep 2005, 408: 131–314. 10.1016/j.physrep.2004.11.001View ArticleGoogle Scholar
 Smith CLC, Desiatov B, Goykmann I, FernandezCuesta I, Levy U, Kristensen A: Plasmonic Vgroove waveguides with Bragg grating filters via nanoimprint lithography. Opt Express 2012, 20: 5696–5706. 10.1364/OE.20.005696View ArticleGoogle Scholar
 de Ceglia D, Vincenti MA, Scalor M, Akozbek N, Bloemer MJ: Plasmonic band edge effects on the transmission properties of metal gratings. AIP Adv 2011, 1(032151):1–15.Google Scholar
 Genov DA, Shalaev VM, Sarychev AK: Surface plasmon excitation and correlationinduced localizationdelocalization transition in semicontinuous metal films. Phys Rev B 2005, 72(113102):1–4.Google Scholar
 Chen W, Thoreson MD, Kildishev AV, Shalaev VM: Fabrication and optical characterizations of smooth silversilica nanocomposite films. Laser Phys Lett 2010, 9: 677–684.View ArticleGoogle Scholar
 Sardana N, Heyroth F, Schilling J: Propagating surface plasmons on nanoporous gold. J Opt Soc Am B 2012, 29: 1778–1783. 10.1364/JOSAB.29.001778View ArticleGoogle Scholar
 Stockman MI, Kurlayev KB, George TF: Linear and nonlinear optical susceptibilities of MaxwellGarnett composites: dipolar spectral theory. Phys Rev B 1999, 60: 17071–17083. 10.1103/PhysRevB.60.17071View ArticleGoogle Scholar
 Thoreson MD, Fang J, Kildishev AV, Prokopeva LJ, Nyga P, Chettiar UK, Shalaev VM, Drachev VP: Fabrication and realistic modeling of threedimensional metaldielectric composites. J Nanophotonics 2011, 5(051513):1–17.Google Scholar
 Lu D, Kan J, Fullerton EE, Liu Z: Tunable surface plasmon polaritons in Ag composite films by adding dielectrics or semiconductors. Appl Phys Lett 2011, 98: 243114–243117. 10.1063/1.3600661View ArticleGoogle Scholar
 Shi Z, Piredda G, Liapis AC, Nelson MA, Novotny L, Boyd RW: Surface plasmon polaritons on metaldielectric nanocomposite films. Opt Lett 2009, 34: 3535–3537. 10.1364/OL.34.003535View ArticleGoogle Scholar
 Kelly KL, Coronado E, Zhao LL, Schatz GC: The optical properties of metal nanoparticles: the influence of size, shape, and dielectric environment. J Phys Chem B 2003, 107: 668–677.View ArticleGoogle Scholar
 Kreibig U, Vollmer M: Optical Properties of Metal Clusters. Berlin: Springer; 1995.View ArticleGoogle Scholar
 Agranovich VM, Mills DL: Surface Polaritons. Amsterdam: NorthHolland Publishing Co; 1982.Google Scholar
 Maxwell Garnett JC: Colours in metal glasses and in metallic films. Philos Trans R Soc Lond A 1904, 203: 385–420. 10.1098/rsta.1904.0024View ArticleGoogle Scholar
 Stefanou N, Modinos A, Yannopapas V: Optical transparency of mesoporous metals. Solid State Commun 2001, 118(2):69–73. 10.1016/S00381098(01)000382View ArticleGoogle Scholar
 Johnson PB, Christy RW: Optical constants of noble metals. Phys Rev B 1972, 6: 4370–4379. 10.1103/PhysRevB.6.4370View ArticleGoogle Scholar
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