Dielectric function effective

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... 

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... 

The effective dielectric function of poly electrolyte solutions remains as a mystery, demanding a better understanding of structure of solvent surrounding poly electrolyte molecules. 

Mallik, B., Masunov, A., Lazaridis, T. Distance and exposure dependent effective dielectric function. J. Comput. Chem. 2002, 23, 1090-9. 

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

The separation of the field components, and the expression of anisotropic adsorbate dielectrics ( e x, e y, e z ) becomes useful when e is expressed in terms of meaningful microscopic quantities (eg. The dynamic charge of the adsorbate mode), and the Lorentzian oscillator [53] parameterisation of an adsorbate layer has proved useful [34] for this purpose. For any particular vibrational mode, the effective dielectric function 8(co) is given by ... 

Equation 23 can be significantly simplified in the limit of dilute media, that is, for low metal concentrations, since in this case the local field factor is the same for all inclusions and is given by Eq. (4). D. Ricard et al. have proposed a straightforward perturbation method to get the expression of x e// iii dilute medium approximation [79]. The optical Kerr effect in the material amounts to modifying the effective dielectric function under the action of the applied field Eq as... 

A key issue in nanostructured materials is the dipole coupling between nanocrystals which will cause the optical properties of a nanocrystal ensemble to become like those of the bulk material. There has been extensive investigation of the interactions between particles embedded within media for a range of boundary conditions. We have found that the effective dielectric function given by Eq. (10), based on the Maxwell-Garnett model [1] is very accurate for quite dense nanocrystal arrays. In practice, one measures the transmittance of a thin film of the dense nanoparticle based film. Conventional solutions are simply... 

For the calculations of the optical properties of polymer films with embedded nanoparticles, two routes can be selected. In the exact route, the extinction cross sections Cact(v) of single particles are calculated. The calculated extinction spectra for single particles—or, better, a summation of various excitation spectra for a particle assembly—can be compared with the experimental spectra of the embedded nanoparticles. In the statistic route, an effective dielectric function e(v) is calculated from the dielectric function of the metal e(T) and of the polymer material po(v) by using a mixing formula, the so-called effective medium theory. The optical extinction spectra calculated from the effective dielectric functions by using the Fresnel formulas can be compared with the experimental spectra. 

The most popular effective medium theories are the Maxwell Garnett theory [18], which was derived from the classical scattering theory, and the Bruggeman theory [19]. With these theories, an effective dielectric function is calculated from the dielectric functions of both basic materials by using the volume filling factor. At some extensions of these theories, a unique particle shape for all particles is assumed. There is also an other concept based on borders for the effective dielectric functions. The borders are valid for a special nanostructure. Between these borders, the effective dielectric function varies depending on the nanostructure of the material. The Bergman theory includes a spectral density function g(x) that is used as fit function and correlates with the nanostructure of the material [20]. 

For the following calculation, experimentally determined dielectric functions for silver [30] and for a plasma polymer [31] were taken. The effective dielectric functions e(v) were calculated with the Maxwell Garnett theory for parallel-oriented particles, equation (13). From the effective dielectric function, transmission or extinction spectra can be calculated by using the Fresnel formulas [10] for the optical system air-composite media-quartz substrate. As a further parameter, the thickness of the film with embedded particles and the thickness of other present layers that do not contain metal nanoparticles have to be included. The calculated extinction spectra can be compared with the experimental spectra. 

The formation of porous fihns, as in the case of the data shown in Fig. 6.16, may be regarded as an extreme case of surface roughening. However, if the scale of the porosity is much smaller than the wavelength, diffusion losses are negligible, and the fihn can be described as a homogeneous medium with an effective dielectric function. Other examples of films that may be described as an effective medium are nanoparticie electrodes or discontinuous metal deposits. Here we will first outline the principle of effective-medium descriptions, then show the various kinds of effects that suci composite fihns may cause. 

Effective-medium theories As long as the inhomogeneities of the medium are on a scale a much smaller than the IR wavelength 2, the optical properties of the medium are described by an effective dielectric function, which can in prin-... 

Effective medium theories characterize the frequency-dependent transport in systems with large-scale inhomogeneities such as metal particles dispersed in an insulating matrix [118,119]. An IMT in the effective medium model represents a percolation problem where a finite a c as T 0 is not achieved until metallic grains in contact span the sample. To understand the frequency dependence of the macroscopic material, an effective medium is built up from a composite of volume fraction /of metallic grains and volume fraction 1 — / of insulator grains. The effective dielectric function semaCw) and conductivity function (Tema(w) are solved self-consistently. 

The derivation of the effective dielectric function of a medium composed of small spheres and a vacuum will be presented [183], From elementary electrostatics [12], the polarization P of a sphere in a constant and uniform far field is uniform, and its polarizability a (1.3.2°) is given by... 

One can see that the effective dielectric function of the effective medium differs from the simple average of the dielectric functions of the constituents. Moreover, Eq. (1.122) is inherently asymmetric in the treatment of the two constituents the transformation of e -o- egm and / 1 - / will result in different effective dielectric functions. 

Figure 3.60. The TO and LO energy loss functions and real part of effective dielectric function, obtained using Maxwell-Garnett effective medium theory, are compared with corresponding experimental functions. Filling factor is 0.3 and pores are regarded as cylinders perpendicular to surface. Reprinted, by permission, from E. Wackelgard, J. Phys. Condens, Matter 8, 4289 (1996), p. 4297, Fig. 4. Copyright 1996 lOP Publishing Ltd.

Petzelt, J., Rychetsky, L Effective dielectric function in high-permittivity ceramics and films. Ferroelectrics 316, 89-95 (2005)... 

Composites can be divided into two subgroups statistical mixtures and matrix-inclusion type composites. The effective dielectric function of the first subgroup can be calculated by equations, which are symmetrical with respect to phase indices. Statistical mixtures exhibit the so called percolation Phenomenon which is extremely important in conductor-insulator composites. Percolation threshold is a critical... 

This possibility is extremely useful for the structural characterization of polymer blends. The optical properties of the polymer layer can be described by a so-caUed effective dielectric function, which is a suitable average of the dielectric functions of the two components. Three averaging effective medium approximation (EMA)s -the linear, Maxwell-Garnett and Bruggeman EMAs - are widely used for this purpose [9]. These approximations differ in their spectral densities for a given volume fraction. 

The conductivity of mesoscopic metals can be measured only by noncontact means. For this reason the particles were embedded in an insulating matrix. The manufacture of the (indium) particles was generally achieved by condensation from the gas phase in a rotating oil film [69]. This method yielded metal particles of about 20 nm that were (colloidally) dispersed in the oil matrix. By means of thermal coalescence, panicles with a diameter of up to several hundred nanometers were obtained. Thus the effective dielectric function (DF) of the heterogeneous oil-indium system was measured. At constant volumetric filling ratio it was possible to mea-... 

Any real surface contains a layer whose optical properties differ from those in the bulk crystal. That may be a thin film on the surface, in particular an oxide film, contamination, relaxed or reconstructed layer, or surface roughness. Therefore with the help of Eq. (5.1) an effective dielectric function, (e), is determined, which corresponds to an average over the region penetrated by the incident light. In order to extract the optical properties of a transition layer, the substrate contribution to (e) must be evaluated. This is usually performed by applying a three-phase model (see Section 3.1.3). Then the ellipsometric ratio, p, can be written using Eq. (3.40). The complex dielectric function (its real and imaginary parts) and the thickness of the transition layer (phase 2) are considered as the three unknown parameters. However, the measurements of the complex quantity p provide only two equations for them. To obtain the third one, it is necessary to invoke additional, physically reasonable restrictions. 

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