Dielectric Function of Metals

Finally we note that without sources in an homogeneous space, Eq. (1.97) is equivalent to Eq. (1.22), and thus a plane-wave solution can be expressed using the magnetic vector as  

The electric field can be obtained from Eq. (1.111) and the magnetic field from Eq. (1.107), while the the Po)mting vector in the vacuum is  

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Under this assumption, the effective dielectric functions of metal alloy, 6, , are given as ... 

Despite of its simplicity the Drude model correctly describes the optical properties of simple metals. In Fig. 1.4 we report the experimental dielectric constant, the refractive index and the reflectance for aluminum. Figure 1.4 closely resembles Fig. 1.3 with (Op 15 eV. The best agreement of the dielectric function of metal with that obtained in the framework of the free electron model can be obtained for the alkali metals (Li, Na, K, Cs, Rb), whose response seems to be weakly affected by the contribution from the core electrons. Notably, alkali metals, such as sodium, have an almost free-electron-like response and thus, in accordance... 

The optical properties of metal nanoparticles have traditionally relied on Mie tlieory, a purely classical electromagnetic scattering tlieory for particles witli known dielectrics [172]. For particles whose size is comparable to or larger tlian tire wavelengtli of the incident radiation, tliis calculation is ratlier cumbersome. However, if tire scatterers are smaller tlian -10% of tire wavelengtli, as in nearly all nanocrystals, tire lowest-order tenn of Mie tlieory is sufficient to describe tire absorjDtion and scattering of radiation. In tliis limit, tire absorjDtion is detennined solely by tire frequency-dependent dielectric function of tire metal particles and the dielectric of tire background matrix in which tliey are... 

The first act consists of removing a small part of the insulator (e,) and replacing it by a small amount df of metal (E,n)- Thereafter with Eq.(6), we calculate Ef,fj( ). For the first step, there is no difference with MG. If we now add another amount df2 of metallic particles (e, ) in the brand new system (e l)), we can again calculate the new effective dielectric function with Eq.(6). Instead of using / for the dielectric function of the host, we now use ej (l) obtained by the previous step. Since we removed some insulating material and replaced it with metal, we have to replace the filling factor/by dfil -//-]).//-i is the amount of metal already in the material and /// the metal we add at step i. The... 

With the proper definitions of ex and k0, this equation is applicable to the metal as well as to the electrolyte in the electrochemical interface.24 Kornyshev et al109 used this approach to calculate the capacitance of the metal-electrolyte interface. In applying Eq. (45) to the electrolyte phase, ex is the dielectric function of the solvent, x extends from 0 to oo, and x extends from L, the distance of closest approach of an ion to the metal (whose surface is at x = 0), to oo, so that kq is replaced by kIo(x — L). Here k0 is the inverse Debye length for an electrolyte with dielectric constant of unity, since the dielectric constant is being taken into account on the left side of Eq. (45). For the metal phase (x < 0) one takes ex as the dielectric function of the metal and limits the integration over x ... 

The dielectric function of a metal can be decomposed into a free-electron term and an interband, or bound-electron term, as was done for silver in Fig. 9.12. This separation of terms is important in the mean free path limitation because only the free-electron term is modified. For metals such as gold and copper there is a large interband contribution near the Frohlich mode frequency, but for metals such as silver and aluminum the free-electron term dominates. A good discussion of the mean free path limitation has been given by Kreibig (1974), who applied his results to interpreting absorption by small silver particles. The basic idea is simple the damping constant in the Drude theory, which is the inverse of the collision time for conduction electrons, is increased because of additional collisions with the boundary of the particle. Under the assumption that the electrons are diffusely reflected at the boundary, y can be written... 

By means of this combination of the cross section for an ellipsoid with the Drude dielectric function we arrive at resonance absorption where there is no comparable structure in the bulk metal absorption. The absorption cross section is a maximum at co = ojs and falls to approximately one-half its maximum value at the frequencies = us y/2 (provided that v2 ). That is, the surface mode frequency is us or, in quantum-mechanical language, the surface plasmon energy is hcos. We have assumed that the dielectric function of the surrounding medium is constant or weakly dependent on frequency. 

Figure 12.9d shows the dielectric function of several metals that either have been discussed in Chapter 9 or will be discussed in connection with small particle extinction in Section 12.4. The energy dependence of the dielectric function is given in the form of trajectories in the complex e plane, similar to ihe Cole-Cole plots (1941) that are commonly used for polar dielectrics the numbers indicated on the trajectories are photon energies in electron volts. 

Here, s(co) is the dielectric function of the metallic sphere. Near a surface dipole plasmon resonance in Cn, e(co) acquires the following form [43]... 

N/A represents the surface coverage by species, fi is the magnitude of the dipole moment perpendicular to the surface, Q is the vibrational frequency, co is the frequency of the incident radiation and G (a) is a reflectivity factor containing the angle of incidence, a, and the dielectric function of the metal, e ... 

Figure 13.10 The dispersion relations of curves two layer Si02 / binary alloy structures. For simplicity, only the real parts of dielectric functions of different metal alloys are considered [37].

In order to excite SPP in a metallic tip, the metal is required to have an optical property that has a negative real and minimal imaginary part in the dielectric function at the excitation wavelength [98] as discussed in Sect. 4.2. As a plasmonic material in the UV region, aluminum can be used instead of silver and gold. The dielectric function of aluminum shows a reasonably small imaginary part while the real part keeps negative in the UV. 

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