Metals dielectric function

One question that has not been discussed so far concerns the metal dielectric functions used in the modelling. It appears in fact that the dielectric functions of the bulk metals, either gold or silver dielectric functions as taken from P.B. Johnson and R.W Christie for example, do not yield a good agreement of the models with the experimental data [57]. Hence, the models used in Figures 9 and 10 are calculated with the use of Drude type dielectric functions of the form (< >) = , (w) + such that [58] ... 

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... 

Fig. 6. MG model the metal with dielectric function (e ,((0)) particles are surrounded by an insulator ( ((0)) (left). The jnixture results in an effective medium e,.jy(right).

Let us consider small metallic particles with complex dielectric function e /jfco) embedded in an insulating host with complex dielectric function e/fco) as shown in Fig. 6. The ensemble, particles and host, have an effective dielectric function = e j i(co) -I- We can express the electric field E at any point... 

Fig. 10. In the first step few small metallic particles are dispersed in an insulating host. This modifies the medium which now has a dielectric function e,jy(0)) instead of e,(M). We repeat iteratively this process (in n consecutive steps) of adding metallic particles until we reach a filling/.

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... 

If V is localized, say, near the origin, then for locations far from the origin, this behaves like j 2kFr)/r2, which means as cos(2kFr)/ r3. These damped oscillations of frequency 2kF are the Friedel oscillations, which always arise when an electron gas is perturbed the frequency of oscillation comes from the kink in the dielectric function at 2kF. We see the Friedel oscillations (in planar rather than in spherical geometry) for the electron gas at a hard wall [Eq. (12) et seq.] and for the electron density at the surface of a metal. 

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 ... 

One has to solve for (x) with o) = V, (oo) = 0, and and D continuous at z = 0. Since the effect of the metal electrons is incorporated into the dielectric function, there are no free charges to consider in the metal, so that D is constant inside the metal, and the equation becomes... 

The calculations were subsequently extended to moderate surface charges and electrolyte concentrations.8 The compact-layer capacitance, in this approach, clearly depends on the nature of the solvent, the nature of the metal electrode, and the interaction between solvent and metal. The work8,109 describing the electrodesolvent system with the use of nonlocal dielectric functions e(x, x ) is reviewed and discussed by Vorotyntsev, Kornyshev, and coworkers.6,77 With several assumptions for e(x,x ), related to the Thomas-Fermi model, an explicit expression6 for the compact-layer capacitance could be derived ... 

One possibility to obtain a relatively small leakage into the substrate is to introduce a thin film of metal or absorbing layer such as a polymer or a dye with a complex dielectric function, or a thin layer of low refractive index material... 

Electrons in metals at the top of the energy distribution (near the Fermi level) can be excited into other energy and momentum states by photons with very small energies thus, they are essentially free electrons. The optical response of a collection of free electrons can be obtained from the Lorentz harmonic oscillator model by simply clipping the springs, that is, by setting the spring constant K in (9.3) equal to zero. Therefore, it follows from (9.7) with co0 = 0 that the dielectric function for free electrons is... 

An elementary treatment of the free-electron motion (see, e.g., Kittel, 1962, pp. 107-109) shows that the damping constant is related to the average time t between collisions by y = 1 /t. Collision times may be determined by impurities and imperfections at low temperatures but at ordinary temperatures are usually dominated by interaction of the electrons with lattice vibrations electron-phonon scattering. For most metals at room temperature y is much less than oip. Plasma frequencies of metals are in the visible and ultraviolet hu>p ranges from about 3 to 20 eV. Therefore, a good approximation to the Drude dielectric functions at visible and ultraviolet frequencies is... 

The reflectance, dielectric functions, and refractive indices, together with calculations based on the Drude theory, for the common metal aluminum are shown in Fig. 9.11. Aluminum is described well by the Drude theory except for the weak structure near 1.5 eV, which is caused by bound electrons. The parameters we have chosen to fit the reflectance data, hu>p = 15 eV and hy = 0.6 eV, are appreciably different from those used by Ehrenreich et al. (1963), hup = 12.7 eV and hy = 0.13 eV, to fit the low-energy (hu < 0.2 eV) reflectance of aluminum. This is probably caused by the effects of band transitions and the difference in electron scattering mechanisms at higher energies. The parameters we use reflect our interest in applying the Drude theory in the neighborhood of the plasma frequency. 

Note that there is no bulk absorption band in aluminum corresponding to the prominent extinction feature at about 8 eV. Indeed, the extinction maximum occurs where bulk absorption is monotonically decreasing. This feature arises from a resonance in the collective motion of free electrons constrained to oscillate within a small sphere. It is similar to the dominant infrared extinction feature in small MgO spheres (Fig. 11.2), which arises from a collective oscillation of the lattice ions. As will be shown in Chapter 12, these resonances can be quite strongly dependent on particle shape and are excited at energies where the real part of the dielectric function is negative. For a metal such as aluminum, this region extends from radio to far-ultraviolet frequencies. So the... 

Up to this point we have considered only the conditions for resonances in the cross sections of small spherical particles of various kinds we have said nothing quantitative about their strengths and the frequencies at which they might occur other than brief introductory remarks about ionic crystals in the infrared and metals in the ultraviolet. To determine if a resonance is realizable, where it occurs, and its strength, we need to know how the dielectric function varies with frequency. Therefore, in the following sections we shall examine some of the preceding resonance conditions in the light of simple, but realistic, dielectric functions. 

There is one clear exception to the rule that bulk dielectric functions tend to be applicable to very small particles in metal particles smaller than the mean free path of conduction electrons in the bulk metal, the mean free path can be dominated by collisions with the particle boundary. This effect has been... 

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... 

Near the plasma frequency in metals 2 y2 therefore, to good approximation, the imaginary part of the Drude dielectric function (9.26) is... 

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