Surface Plasmons on Metal Nanoparticles - UNAM

3808 J. Phys. Chem. C, Vol. 111, No. 10, 2007 Noguezrepresentation, finite differences methods, etc. 46 Once a numericalmethod is chosen, it is important to select a “realistic”dielectric function that better resembles the material propertiesof the system, in this case, particles of nanometric sizes. Here,we show the main characteristics that a realistic dielectricfunction should have to calculate the light extinction of NPs.A. Nanoparticle Dielectric Function. As starting point, wecan employ dielectric functions measured experimentally forbulk metals, ɛ exp (ω). These dielectric functions have contributionsfrom interband (inter) and intraband (intra) electrontransitions, which we can assume are additiveɛ exp (ω) ) ɛ inter (ω) + ɛ intra (ω) (2)Interband contributions are due to electron transitions fromoccupied to empty bulk bands separated by an energy gap. Theelectrons are bound by a restoring force given by the energydifference between ground and excited electronic states inmetals, usually at the ultra violet (UV) region. 41 Intrabandcontributions come from electron transitions at the Fermi levelin incompletely filled bands, or when a filled band overlaps inenergy with an empty band. These transitions also provide anabsorption mechanism but at lower energies. Electrons at theFermi level in metals are excited by photons of very smallenergies, such that, they are essentially “free’’ electrons. Thecontributions from free electrons to ɛ exp (ω) can be describedby the Drude model 41ω p2ɛ intra (ω) ) 1 -ω(ω + i/τ)where ω p is the plasma frequency and 1/τ the damping constantdue to the dispersion of the electrons. For most metals at roomtemperature, 1/τ is much less than ω p , and the plasma frequencyfor metals is usually in the visible and UV regions, with energiespω p from 3 to 20 eV. The theoretical values of ω p can disagreefrom those obtained experimentally, since the contribution fromthe bond electrons or interband transitions is not considered inthe Drude model. In fact, the bond electrons create a positivebackground that screens the free electrons and ω p is usuallypulled down. For instance, the Drude model predicts for silvera plasma frequency to be 9.2 eV, 41 but the measured one isactually 3.9 eV. 49The collision time τ determines a characteristic distance λ τ ,which plays a fundamental role in the theory of electronconduction. An electron picked at random at a given momentwill, on average, travel a distance λ τ before its next collision.This distance λ τ is also known as the mean free path of electrons,and at room temperatures is of the order of few nanometers.Here, we are interested in small NPs mostly at room temperature;hence, we have to consider that electrons can be alsodispersed by the NP surface, because the free electron’s meanfree path is now comparable or larger than the dimension ofthe particle. Therefore, it is necessary to include an extradamping term τ(a) toɛ exp (ω), due to the surface scattering ofthe “free’’ electrons. The surface dispersion not only dependson the particle size, but also on its shape. 50To include surface dispersion we need modify the intrabandcontributions by changing the damping term. From eq 2, weobtain the input of the bound charges by subtracting the freeelectron contribution from the bulk dielectric function. The freeelectron contributions are calculated with Drude and using thetheoretical values of ω p . Now, we include the surface dampingby adding the extra damping term τ(a) to the Drude model.Finally, we obtain a dielectric function, which also depends on(3)Figure 1. Interband contributions to the dielectric function of bulksilver.the NP size, and includes the contributions of (i) the freeelectrons, (ii) surface damping, and (iii) interband transitionsor bound electrons, given byɛ(ω, a) ) ɛ inter (ω) + ɛ NP intra (ω, a)) {ɛ exp (ω) - ɛ intra (ω)} +{2ω p1 -ω(ω + i/τ + i/τ(a))} (4)In this work, we will consider for all the cases the surfacedispersion of a sphere of radius a given by 1/τ(a) )V f /a, 51 whereV f is the Fermi velocity of the electron cloud. The smaller theparticle, the more important is the surface dispersion effect. Laterwe will show that surface dispersion effects do not change thelocation of the surface modes, but they affect the coupling ofsuch proper modes with the applied field, making the resonancepeaks wider and less intense. 18In Figure 1, we show the interband or bound electrons inputto the bulk dielectric function of silver as measured by Johnsonand Christy, 49 according to eq 2, and using Drude to model theintraband input. The main contribution to the imaginary part ofɛ inter (ω) is at wavelengths below 325 nm, whereas for largerwavelengths, the input is almost nil. On the other hand, thecontributions to the real part are always different from zero,being the most significant at about 315 nm, while between 350and 700 nm the real part is almost constant. We should keep inmind that interband electron transitions will absorb energy, butnot contribute to SPRs.Once the NP’s dielectric function is determined, we have tochoose a method to find the SPRs. In this work, we employ thespectral representation formalism and the discrete dipole approximation(DDA). In section III, we introduce the mainconcepts of the spectral representation formalism, and in sectionsIV and VI, we employ it to study the SPRs of ellipsoidal NPs,as well as supported NPs and 1D chains. For the case ofsuspended polyhedral NPs, we employ DDA, which is a wellsuited technique for studying scattering and absorption ofelectromagnetic radiation by particles of arbitrary shapes withsizes of the order or less than the wavelength of the incidentlight. 52 For a more complete description of DDA and itsnumerical implementation, DDSCAT, the reader can consultrefs 17 and 53-56.B. Large Nanoparticles versus Small Nanoparticles. Theimportance of the absorption and scattering process as a functionof the particle size can be studied for spherical particles usingthe Mie theory, see for example ref 18. It was found that for