Surface Plasmons on Metal Nanoparticles - UNAM

More documents

Recommendations

Info

Feature Article J. Phys. Chem. C, Vol. 111, No. 10, 2007 3809NPs of less than 40 nm of diameter the radiation processes arenegligible, and the particle only absorbs energy. On the otherhand, the scattering effects dominate the response of NPs of100 nm and larger.1. Small Particles (40 nm). When the size of the particleincreases, the radiation effects become more and more important.The displacement of the electronic cloud is no longer homogeneouseven for spherical particles, and once more, highmultipolar charge distributions are induced. This fact can beseen from the Mie theory, where the extinction and scatteringcross sections are expressed in series expansion of the involvedfields, which are described in terms of spherical harmonicfunctions, such that, the different multipolar excitations and theircontribution can be easily identified. 43 For instance, in a recentexperiment, up to hexa-multipolar charge distributions have beenobserved in large spherical NPs. 57 Furthermore, the acceleratedelectrons produce an additional polarization field that dependson the ratio between the size of the particle and the wavelengthof the incident light. 58 Because of this secondary radiation, theelectrons lose energy experiencing a damping effect, whichmakes wider the SPRs. 18 This field reacts against the quasistaticpolarization field and shifts the position of the modes to largerwavelengths. Thus, the radiation damping reduces the intensityand makes broader and asymmetric the SPR peaks, which arered-shifted.In summary, the optical signature of small particles is givenby the SPRs that depend on the NPs shape, which are associatedto different surface charge distributions that are explained interms of different multipolar moments. The SPRs are influencedby the NP size, such that, for particles of few nanometers theresonances do not change their position or frequency, but theybecome wider because of surface dispersion effects. When thesize increases the SPRs are now affected by the secondaryradiation, which moves their position to smaller frequencies andmakes broader the peaks. Besides, light scattering is also present,which induces the excitation of new SPRs of higher multipolarorder.In conclusion, to describe the optical response of NPs, it iscrucial to understand the number, position, and width of theSPRs as a function of the NP shape, size, and environment.III. <strong>Surface</strong> <strong>Plasmons</strong>: General TheoryWhen conduction electrons oscillate coherently, they displacea electron cloud from the nuclei giving rise to a surface chargedistribution. The Coulomb attraction between positive andnegative charges results in restoring forces, characterized byoscillation frequencies of the electron cloud with respect to thepositive background, which are different from those of theincident EM wave. Each collective oscillation with differentsurface charge distribution is known as surface plasmonresonance (SPR). The number of such modes as well as theirfrequency and width are determined by the electron density,effective mass, particle’s shape, size, dielectric function, andits environment. In this work, we are interested in the surfaceplasmon response of metal NPs; therefore, we consider particleswhose sizes are much smaller than the wavelength of theincident light, where the quasi-static limit is still valid.A. <strong>Surface</strong> <strong>Plasmons</strong>. Let us consider a NP composed of ahomogeneous, isotropic, and nonmagnetic material with dielectricfunction ɛ(ω). The NP is embedded in a nonmagnetic hostwith dielectric constant ɛ h . Under the action of an incident EMwave, the free charges are displaced producing a polarizationfield PB. The light absorption is given by the proper modes ofthe NP, which are specified by PB. The proper modes responsiblefor the optical absorption satisfy ∇ ‚PB )∇×PB )0, inside theparticle, but ∇ ‚PB*0, on the surface. These surface modes areevanescent electromagnetic waves, which are not necessarilylocalized near the surface, but they are accompanied by apolarization charge ∇ ‚PB on it. We should recall that boundelectrons do not participate in the collective motion of theelectron cloud, thus, SPRs are quite independent of the interbandcontributions to the dielectric function, except that they can actas a positive background changing in some way the environmentof the free electrons.In the existing theories, the frequencies of the resonant propermodes and their coupling strength to the applied field cannotbe immediately calculated, because they involve a procedurethat usually requires taking the nondissipation limit, which callsfor a vast amount of numerical effort. However, we can constructa theory that yields both the frequencies of the proper modesand the size of their coupling strength to the applied field bybuilding a spectral representation of the effective polarizabilityof the system. In this representation, the effective polarizabilityis expressed as a sum of terms with single poles, such that, thelocation of the poles is associated with the frequencies of thenormal modes through a spectral variable, and their strengthwith the coupling of these modes to the applied field. Thespectral representation has advantages over other theories,because it separates the contribution of the dielectric propertiesfrom the geometrical ones. The latter allows us to perform asystematic study of the optical response of NPs: once a shapeis chosen, the frequencies of the plasmon resonance of particlesof different size and dielectric properties can be calculated witha minimum numerical effort. Next we present a brief introductionto the spectral representation formalism.
3810 J. Phys. Chem. C, Vol. 111, No. 10, 2007 NoguezB. Spectral Representation Formalism. The spectral representationformalism was first introduced by Fuchs to studythe optical properties of ionic crystal cubes. 37 In this seminalpaper, Fuchs showed that the particle’s polarizability can bewritten as a sum over normal modes. Later, Bergman 38 andMilton 39 showed that the effective local dielectric function ofany two-phase composite in 3D can always be written in termsof a spectral representation. The main advantage of thisrepresentation is that the proper modes do not depend on thedielectric properties of the components but only on the geometryof the system. Moreover, from the explicit expressions of thespectral representation, we can easily obtain the strength of thecoupling of different optically active modes with the appliedfield, whose frequency is determined by the poles of thepolarizability.Within the spectral representation, the NP’s polarizability R-(ω) is expressed as a sum of terms with single poles 374πV R(ω) )- ∑nC(n)s(ω) - s n(5)where C(n) is the spectral function, which gives the strength ofeach eigenvalue s n , and V is the NP volume. The spectral variables(ω) is defined in terms of the dielectric properties of thecomponents as 371s(ω) )(6)1 - ɛ(ω)/ɛ hThe nth SPR given by the nth pole of eq 5 has a complexfrequency Ω n ) ω n + iγ n , which satisfies s(Ω n ) ) s n , wherethe real frequency, ω n , gives the location of the proper mode,and γ n gives its relaxation rate. Furthermore, it has been shown 38that the eigenvalues are real numbers between 0 es n e1, andthe spectral function satisfies the sum rule 37∑C(n) ) 1nwhich means that the total strength of all modes is conserved.Now, substituting the dielectric function from eq 4 into thespectral variable expression in eq 6, the frequency Ω n of thenth eigenmode is given byΩ n ) ω n + iγ n )-iΓ + -Γ 2 + ω p 2 s n A n (7)with A n ) [s n (ɛ inter + 1 - ɛ h ) + ɛ h ] -1 and 2Γ ) 1/τ + 1/τ(a).From here, we can infer some general behavior of the SPRs.For instance, if the relaxation time decreases (Γ increases), thefrequency of the SPR is always red-shifted. However, this shiftis very small since for typical metals Γ , ω p . Now, when theparticle is immersed in vacuum ɛ h ) 1, and assuming that thereare not contributions from interband transitions (ɛ inter ) 0), weobtain that A n ) 1, such that, ω n ) ω p s n . For a homogeneoussphere, it has being found that there is a single mode with s n )1/3 and C(n) ) 1, such that, by substituting in eq 5, one obtainsthe well-known expression of the sphere polarizability.Otherwise A n < 1, such that, the SPRs are always red-shiftedwith respect to vacuum. Furthermore, we find that the shift isnot just by a constant because it depends on the proper modeitself: the smaller the eigenvalue s n is, the larger the red shiftis. Therefore, we found that the location of the surface plasmonresonances is sensitive to the dielectric environment, and as therefraction index increases, the spectrum suffers a red shift andbecomes wider. This is because the red-shift is larger forresonances at greater wavelength. However, the number ofresonances is still the same independently of the refraction indexof the host medium.The main advantage of the spectral representation is that thelocation of the poles and their strength are independent of thesize and dielectric properties of the particle and depend onlyon its shape. To show the capabilities of the spectral representationformalism, we will study in detail the case of ellipticalparticles, where an explicit expression of the spectral representationof the polarizability is easily found. This case is useful toillustrate how the eigenvalues s n depend on the geometricalparameters, and how the frequencies of these proper modes canbe easily obtained. Although the proper modes and their strengthare independent of the NPs size and dielectric properties, explicitvalues of these are difficult to obtain. Until now, the spectralrepresentations of different NPs have been found only forspherical, ellipsoidal, and cubic geometries, as well as forcolloidal suspensions of spherical particles, and supportedellipsoidal NPs, which we will employ in section VI.IV. Elongated NanoparticlesMetal nanoellipsoids possess three plasmon resonances correspondingto the oscillation of electrons along the three axesof the NP. The resonance wavelength depends on the orientationof the electric field relative to the particle. By changing theaxes length, the plasmon resonance frequencies of the nanoellipsoidcan be tuned systematically. The possibility of tuningthe optical response of NPs has attracted the attention ofscientists, and a large variety of new synthesis methods havebeen developed to fabricate elongated NPs. 59-71 When theseNPs are dispersed in a matrix (solid or liquid), the randomorientation leads to an average absorption spectrum containingthe three plasmon resonances. On the other hand, when the axesof the nanoellipsoids are oriented in the same direction, it ispossible to distinguish between the different resonances by usingpolarized light. 70,71In this section, we employ the spectral representation to obtainthe SPR of nanoellipsoids as a function of their geometricalparameters and independent of the dielectric properties. Thisallows us to do a systematic study of the SPRs of gold andsilver elongated NPs with different aspect ratios and immersedin different media. Finally, we study the case of alignedspheroids and the dependence of the SPRs on light polarization.The reader can consult recent reviews by Murphy et al. 72 andPérez-Juste et al., 73 which provide a nice overview of thesynthesis and properties of elongated NPs.A. Spectral Representation of Ellipsoidal Nanoparticles.Let us consider an ellipsoidal NP under the action of an uniformexternal electric field. The ellipsoid has a volume V e )(4π/3)abc, where a, b, and c are its semiaxes. The componentsof the ellipsoid’s dipolar polarizability is 74R γ (ω) ) V e ɛ(ω) - ɛ h4π ɛ h + L γ [ɛ(ω) - ɛ h ]where γ denotes x, y, orz and L γ are functions of a, b, c, andthe eccentricity e. Notice that L γ are independent of the materialproperties of the ellipsoid and depend only on the geometricalparameters. Using the definition of the spectral variable in eq6, we can rewrite each component γ of the ellipsoid’s polarizability,as(8)4π1RV γ (ω) )-(9)e s(ω) - L γ
Page 3: 3808 J. Phys. Chem. C, Vol. 111, No
Page 7 and 8: 3812 J. Phys. Chem. C, Vol. 111, No

Surface Plasmons on Metal Nanoparticles - UNAM

Create successful ePaper yourself

Delete template?

Save as template?