Simple modeling of plasmon resonancesin Ag=SiO2 nanocomposite monolayers
Thiago Menegotto,* Marcelo B. Pereira, Ricardo R. B. Correia, and Flavio HorowitzPhysics Institute and Microelectronics Graduate Program, IF-PGMicro-UFRGS, Campus do Vale CP 15051,
In the past few years, composite metal-dielectricmedia have been the object of several studies due totheir peculiar optical properties. The main phenom-enon governing these properties is the collectiveoscillation of electrons at wavelengths near the reso-nance frequency, the so-called plasmon resonance,which microscopically is responsible for the extraor-dinary increase of the optical field in the vicinity ofmetallic structures [1,2].
This kind of system has been used for a wide rangeof applications. The locally enhanced field has at-tracted interest to increase the Raman scatteringsignal of suitable molecules and other nonlinearphenomena [2,3]. The dependence of the plasmon re-sonance frequency on the refractive index of the sur-rounding dielectric media has enabled the design ofsensors for several purposes (e.g., biosensing) [4].Also, the ability of metallic particles to scatter lighthas been used to improve light coupling into solarcells, enhancing their performance [5]. In associationwith this application, special interest is devoted tounderstand and model the optical properties of thinfilms with plasmonic nanoparticles [6].
For three-dimensional composite metal-dielectricmedia, such as glasses doped with metallic inclu-sions, the Maxwell Garnett theory is well known forattempting to describe optical properties [1]. Themodel assumes spherical inclusions, with the dielec-tric function εi and dispersed in a host medium, withthe dielectric constant εd. Those spheres have a ra-dius much smaller than their average spacing, whichis much smaller than the light wavelength. In thisapproach, the effective dielectric function of thecomposite material is given by
εeff − εdðωÞεeff þ 2εdðωÞ
¼ fεiðωÞ − εdðωÞεiðωÞ þ 2εdðωÞ
; ð1Þ
where f is the volume fraction of particles in the med-ium. The effective refractive index, usually complex,is given by neff ¼ ðεeff Þ1=2. The Maxwell Garnett mod-el has also been applied to describe the optical prop-erties of films with nanoparticles [7]. However, themodel applies for a diluted composition (f ≪ 1) ora symmetrically ordered arrangement of particles,in which interparticle effects can be neglected. Theinteraction between spherical particles of an arbi-trary size embedded in nonabsorbing media has alsobeen rigorously treated by solving the Maxwellequation [8]. This theory was later extended to take
into account the effects of absorbing embeddingmedia [9].
For noble metallic inclusions whose size allows abulklike response, the dielectric function is usuallywell described by the Drude–Lorentz model [10].However, in the frame of theMaxwell Garnett theory,the Drude–Lorentz model is no longer suitable whenthe diameter of the particle is reduced to below themean free path of electrons in the material. In orderto take this into account, Kreibig and co-workers pro-posed an extension to describe the optical response ofthe particles [1].
In this work, we investigate the applicability of theKreibig extension with the Maxwell Garnett theoryto calculate the optical properties of monolayer nano-composite films, in which Ag particles are prepared,either buried or unburied, in a SiO2 matrix. The needof a further extension to account for the observedplasmon resonance shifts is discussed and attributedto interparticle coupled oscillations.
2. Experimental
Radio frequency sputtering was used to fabricatefilms of Ag particles embedded in SiO2. The filmswere produced on glass and silicon substrates to en-able optical transmittance measurements aroundthe visible spectrum and transmission electron mi-croscopy, respectively. Film thicknesses, with massequivalent to that of the set of silver particles, weremonitored with a quartz crystal microbalance, whichallowed precision better than 5%.
Two types of samples with amonolayer sequence ofAg particles were prepared: (i) the buried type, inwhich Ag particles are totally embedded in SiO2,and (ii) the unburied type, in which Ag particles arenot covered by silica. Figure 1 shows a representa-tion of the buried type sample with its correspondingmicrograph in the top view, whereas the unburiedtype sample is shown in the cross section in Fig. 2.
Optical transmittance and reflectance measure-ments were performed in a Cary 5000 spectro-photometer. Micrographs were obtained from aJEM-1220 transmission electron microscope.
3. Theory
As a first step, the effective dielectric function andhence the optical properties can be calculated bythe Maxwell Garnett theory—whose input para-meters are the filling factor, the dielectric functionof the particles and of their surroundings—such asdescribed in Eq. (1).
For the dielectric function of the surroundings, theexperimental bulk dielectric function can be used.This is not always true for the particle inclusions, inparticular when their size decreases to smaller thanthe mean free path of electrons (around 50nm forAg) [1]. In this case, it is very useful to apply the the-oretical Drude–Lorentz model for the dielectric func-tion with the bulk relaxation constant Γ
∞replaced by
the phenomenological approach proposed by Kreibiget al. [11],
Γ ¼ Γ∞þ A
vFR
; ð2Þ
where vF is theFermi velocity of the electrons,R is theparticle radius, and A is a parameter related to thestrength of the electron-interface interaction withineach particle [11,12].
At this point, it is worth remembering that theDrude–Lorentz dielectric function is given by
εDLðωÞ ¼ ε∞þ ω2
P
ðω20 − ω2Þ þ iΓω ; ð3Þ
where ωP is the bulk plasmon frequency and ω0 is thenatural frequency of oscillation from the Hooke law(see, e.g., [2,10]).
Equations (1)–(3) are used to determine numeri-cally the optical transmittance of the system inwhich sputtered silver nanoparticles were buried insilica, shown in Fig. 1(a). Transmittance and reflec-tance curves are calculated through the characteris-tic matrix approach for multilayer films [13]. Theseresults are used as inputs to the merit function
F ¼Xj
½Texpðλj − TcalcÞðA; λjÞ�2; ð4Þ
Fig. 1. (a) Cross-sectional scheme and (b) top-view micrograph ofa sample with buried Ag particles in silica matrix; teff and ttotalstand for effective medium and total thicknesses, respectively. Ascale bar of 50nm is shown in (b).
whereTexp is the experimentally measured transmit-tance [14]. The merit function is minimized in orderto find the best fit as a function of A.
A similar procedure is used for the film withunburied Ag particles. In this case, by taking into ac-count that one part of the particles was exposed to airand another was in contact with silica, the dielectricfunction of the host medium is taken as
εdiel ¼ n2diel ¼ ½1=2ðnSiO2
þ nairÞ�2: ð5Þ
4. Results and Discussion
In Fig. 3, results are shown for the buried andunburiedAg cases. In each picture, the transmittanceis shown in theupper portion,whereas the reflectanceappears at the bottom. Corresponding values ofparameter A are 2:917þ 0:908i and 3:354þ 1:304i,respectively.
In the calculation, starting from teff ¼ 8nm inaccordance with the observed average particle size,optimized A parameters were pursued for best fittingto the measured transmittance curves.
By substituting the Γ parameter of Eq. (2) in theDrude–Lorentzmodel [Eq. (3)], it is easy to verify thatImfAg is responsible for shifting the bulk natural fre-quency of oscillation, whereas only RefAg accountsfor a true new relaxation constant for the particles.In Fig. 3, although the theoretical curves over-
estimate the transmittance tails above 650nm, a verygood fit was reached in the region around the absorp-tion dips caused by surface plasmon resonance, ineither case of buried or unburied Ag particles.
In addition, agreement was obtained between thetheoretical curves and the measured reflectancedata, well within their experimental uncertainties,for both types of composite film.
But what does ImfAg physically mean?Consider that when the wave electric field is par-
allel to the plane of the particles, which is the case fornormal incidence of light, coupling of charge oscilla-tion in the particles occurs, as shown qualitatively inFig. 4. The isolated particle natural frequency of os-cillation ω0 ¼ ðK0=mÞ1=2 is then reduced, due to theattractive force between charges in adjacent parti-cles, to ωC ¼ ½ðK0 − 2KCÞ=m�1=2, giving rise to a
Fig. 2. Cross-sectional (a) scheme and (b) micrograph of a samplewith unburied Ag particles in silica matrix; teff and ttotal stand foreffective medium and total thicknesses, respectively. A scale bar of20nm is shown in (b). Fig. 3. Experimental and theoretical data for samples with (a)
buried and (b) unburied Ag particles. At the insets, the recoveredreal and imaginary parts of the refractive index are shown in themeasured spectral range, with a vertical scale from 0 to 2.0.
Fig. 4. Scheme for description of the coupling between particleswhen the exciting electric field is parallel to the plane of particles.
redshift in the resonance frequency of oscillation, asexperimentally determined from 410 to 455nm. Alsothe deviation of the real part of A from unity, whosevalue was theoretically predicted [1], may be attrib-uted to the component of the damping frequency Γdue to the interparticle coupled oscillation of elec-trons in the particles.
From Eqs. (2) and (3), correspondence betweenImfAg and the coupling constant KC can beestablished:
ImfAg ¼�
RωvF
�2KC
m; ð6Þ
with frequency ω taken at resonance. This indicatesthe role of interparticle coupling in the observed plas-mon resonance redshift.
5. Conclusions
For monolayer films composed of almost sphericalsilver nanoparticles in a silica matrix, simple model-ing was shown to accurately and simultaneouslymatch the measured optical transmittance and re-flectance data around the surface plasmon resonanceregion, in cases of covered and uncovered Ag parti-cles. This indicates suitability to both cases of theMaxwell Garnett theory and of the Drude–Lorentzequation with the Kreibig extension with parameterA (related to the strength of the electron-interface in-teraction within each metal particle), with a furtherextension that accounts for the interaction betweenparticles distributed on a surface.
This interaction is responsible for the measuredredshifted plasmon resonances and for the asso-ciated role of the complex A parameter in this work.RefAg accounts for a modified relaxation constant forthe particles, whereas ImfAg is responsible for shift-ing the bulk natural frequency of oscillation. Thisimaginary part was shown to be directly related tothe interparticle coupling constant KC, and thusits physical meaning was here related to the coupledoscillation of electrons, beyond the Maxwell Garnetttheory and the Kreibig relaxation constant extensionin the Drude–Lorentz equation.
The results are also encouraging to the continuityof this work with multilayer nanocomposite films, be-cause they open the possibility of their analyticaltreatment in a similar manner.
This research was partially supported by theBrazilian agencies Conselho Nacional de Desenvolvi-mento Científico e Tecnológico (CNPq) and Coorde-nação de Aperfeiçoamento de Pessoal de NívelSuperior (CAPES).
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