Complex dielectric function

Dielectric functions, complex indices of refraction and Maxwell s equations... 

A quantitative analysis of the electromagnetic response begins with the evaluation of the optical constants of a system (complex dielectric function, complex conductivity, etc.) from the experimentally accessible quantities such as reflectance, transmission or absorption. It is convenient to discuss the response of metals and superconductors in terms of the complex conductivity ), defined from J(o)) = o(ci))E(co) where J is the current vector and E is the electric field vector. All optical constants are interrelated through simple analytical expressions and contain the same physical information. The advantage of the conductivity notation is that o(co) has no singularity at ft) = 0. Also, an extrapolation of ai(tft) to ft) —> 0 yields the dc conductivity of a metal. 

It should be noted that low-loss spectra are basically connected to optical properties of materials. This is because for small scattering angles the energy-differential cross-section dfj/dF, in other words the intensity of the EEL spectrum measured, is directly proportional to Im -l/ (E,q) [2.171]. Here e = ei + iez is the complex dielectric function, E the energy loss, and q the momentum vector. Owing to the comparison to optics (jqj = 0) the above quoted proportionality is fulfilled if the spectrum has been recorded with a reasonably small collection aperture. When Im -l/ is gathered its real part can be determined, by the Kramers-Kronig transformation, and subsequently such optical quantities as refraction index, absorption coefficient, and reflectivity. 

Spectroscopic dlipsometry is sensitive to the dielectric functions of the different materials used in a layer stack. But it is not a compositional analytical technique. Combination with one of the compositional techniques, e. g. AES or XPS and with XTEM, to furnish information about the vertical structure, can provide valuable additional information enabling creation of a suitable optical model for an unknown complex sample structure. 

As shown in Fig. 7, a large increase in optical absorption occurs at higher photon energies above the HOMO-LUMO gap where electric dipole transitions become allowed. Transmission spectra taken in this range (see Fig. 7) confirm the similarity of the optical spectra for solid Ceo and Ceo in solution (decalin) [78], as well as a similarity to electron energy loss spectra shown as the inset to this figure. The optical properties of solid Ceo and C70 have been studied over a wide frequency range [78, 79, 80] and yield the complex refractive index n(cj) = n(cj) + and the optical dielectric function... 

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 reflectivity of bulk materials can be expressed through their complex dielectric functions e(w) (i.e., the dielectric constant as a function of frequency), the imaginary part of which signifies absorption. In the early days of electroreflectance spectroscopy the spectra were often interpreted in terms of the dielectric functions of the participating media. However, dielectric functions are macroscopic concepts, ill suited to the description of surfaces, interfaces, or thin layers. It is therefore preferable to interpret the data in terms of the electronic transitions involved wherever possible. 

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... 

There are two sets of quantities that are often used to describe optical properties the real and imaginary parts of the complex refractive index N = n + ik and the real and imaginary parts of the complex dielectric function (or relative permittivity) e = c + ie". These two sets of quantities are not independent either may be thought of as describing the intrinsic optical properties of matter. The relations between the two are, from (2.47) and (2.48),... 

We must reemphasize that the real and imaginary parts of the complex dielectric function (and the complex refractive index) are not independent. Arbitrary choices of c and <" (or n and k) do not necessarily correspond to... 

The conditions (12.5) have been stumbled upon from time to time and then dismissed as unphysical n cannot be 0 But the reader who has faithfully waded through Chapters 9 and 10 should by now be somewhat hardened to refractive indices less than 1—or even 0. Indeed, one of our objectives in Chapter 9 was to clear the way for the introduction of (12.5), knowing full well that it is often unpalatable. Prejudices about what the dielectric function can or cannot be are not nearly so deeply rooted as those surrounding the refractive index thus, (12.3) can be cast in a more palatable form in terms of the complex dielectric function of the particle c = t + it" ... 

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. 

Dielectric relaxation (DR) measures the complex dielectric function, e(u>), that can be decomposed into real and imaginary parts as,... 

As material science progresses, the size and complexity of the created structures increase. The pursuit of analytical solutions for LELS of these objects becomes very cumbersome and the future could well be in numerical solutions. The Boundary Element Method (BEM) has recently [37] been proven successful in calculating relativistic spectra of any-shape objects. Although anisotropic dielectric functions are not yet implemented, the LELS region, thanks to these new computing tools, should grow in use in a very near future. Indeed, the spectra are material-shape dependent which should be seen, not as a drawback, but as a very valuable piece of additional information. 

It is useful to consider the solution of Maxwell s Equations (5.1) for plane electromagnetic waves in the absence of boundary conditions, which can be written as exp[i(/ 2 — u>t) assuming propagation in z-direction of cartesian coordinates. The quantity / is the complex propagation constant of the medium with dominant real part for dielectrics and dominant imaginary part for metals. The impedance of the medium, Z, defined as ratio of electric to magnetic field is related to / by Z = ojp,0/f3 with /x0 = 1.256 x 10 6 Vs/Am. As it can be derived from Maxwell s equations, the impedance is related to the conductivity/dielectric function by the following expression ... 

The technique of complex-valued dielectric functions was originally applied to solvation problems by Ovchinnikov and Ovchinnikova [2] in the context of the electron transfer theory. They reformulated in terms of s(k, to) the familiar golden rule rate expression for electron transfer [3], This idea, thoroughly elaborated and extended by Dogonadze, Kuznetsov and their associates [4-7], constitutes a background for subsequent nonlocal solvation theories. 

For given to value the apparent charge density cr(r, to) is available in terms of the extended PCM procedure with a complex-valued dielectric function s, namely, e(w) = s1((o) + is2((o) where e w) = 1 +4ttXi(oj) and s2(oj) = 4ttx2(m) with complex-valued susceptibilities defined in Equation (1.127). The complication that both a(r, to) and 0(r, to) become complex is inevitable. However, after applying the inverse Fourier transform, they become real in the time domain. This is warranted by the symmetry properties,... 

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