Drude dielectric function

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. 

The free-electron contribution to the dielectric function in Fig. 9A2b is obtained from the Drude theory with parameters determined from the low-... 

Frohlich and Pelzer (1955) determined the frequencies of longitudinal waves in matter described by the three simple dielectric functions—Lorentz, Drude, and Debye—discussed in this chapter. 

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... 

As an example of extinction by spherical particles in the surface plasmon region, Fig. 12.3 shows calculated results for aluminum spheres using optical constants from the Drude model taking into account the variation of the mean free path with radius by means of (12.23). Figure 9.11 and the attendant discussion have shown that the free-electron model accurately represents the bulk dielectric function of aluminum in the ultraviolet. In contrast with the Qext plot for SiC (Fig. 12.1), we now plot volume-normalized extinction. Because this measure of extinction is independent of radius in the small size... 

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 12.6 Calculated absorption spectra of aluminum spheres, randomly oriented ellipsoids (geometrical factors 0.01, 0.3, and 0.69), and a continuous distribution of ellipsoidal shapes (CDE). Below this is the real part of the Drude dielectric function.

Extinction calculations for aluminum spheres and a continuous distribution of ellipsoids (CDE) are compared in Fig. 12.6 the dielectric function was approximated by the Drude formula. The sum rule (12.32) implies that integrated absorption by an aluminum particle in air is nearly independent of its shape a change of shape merely shifts the resonance to another frequency between 0 and 15 eV, the region over which e for aluminum is negative. Thus, a distribution of shapes causes the surface plasmon band to be broadened, the... 

The observed darkening of the indium slides results from a shift of the absorption peak because of the coating on the particles. Because of the cumbersomeness of the expressions for coated ellipsoids (Section 5.4) this shift can be understood most easily by appealing to (12.15), the condition for surface mode excitation in a coated sphere. For a small metallic sphere with dielectric function given by the Drude formula (9.26) and coated with a nonabsorbing material with dielectric function c2, the wavelength of maximum absorption is approximately... 

The optical constants of a metal are determined to a large degree by the free electrons. According to the Drude model, the contribution of the free electrons to the frequency-dependent dielectric function is expressed as follows (16) ... 

Some applications of the method will be shown for the ternary salt trimethylammonium (TMA)-TCNQ-I. Tanner et al. [43] have obtained a best fit with a Drude-Lorentz dielectric function with a>p = 5290 cm-1, -y = 2560 cm"1, ec = 2.65, , is the frequency and T is the relaxation rate of the second oscillator. The average value of the frequency-dependent conductivity below 200 cm"1 should be 19 IS- cm"1. The energy gap of 0.10 to 0.14 eV, the effective quantity of electrons participating in the electric transport Ncfi = 0.67 per molecule, and the effective mass of the carriers m = 5.2 m were found from the plasma frequency. 

The temperature-dependent Purcell factor has been determined to explain the PL data. By taking into account the dependence of electron-electron and phonon-electron scatterings on temperature, Drude model is used fcff calculating the temperature-dependent dielectric functions of Al and the results are shown in Fig. 13.6(a) indicating the functions are rather insensitive to temperature [22]. On the other hand, for ZnO, reflectivity measurements have been performed and the reflectance spectra are shown in Fig. 13.6(b) for different temperatures. To extract... 

The dielectric function, Ei(w), also deviates from the functional dependence expected for a Drude metal. Even for the highest conductivity and most metallic sample (A), E (a)) is positive in the IR, increasing to larger values for 0x2500 cm , and remaining positive at frequencies at least as low as o) = 8 cm see Fig. VI-5b). The frequency-independent a(oj) implies that e,(to) will remain positive as w approaches 0 K-K consistency requires that for Ei(o>) to go negative below 8 cm , o(o)) would have to sharply increase as o) approaches 0. The excellent agreement between a(o>"0) and o ,. (300 K), evident in the inset to Fig. VI-5a confirms the accuracy and precision of the R(oj) measurements and indicates that such an increase in ct(o)) as w approaches 0 does not occur. 

In recent years, great attention has been paid to the analysis of the dependence of the properties of metal-insulator composites on frequency [91-109], which is related to the difficulties in describing the anomalous behavior of dielectric properties in the low-frequency limit. The nature of the anomalous behavior of the frequency dependence of the dielectric properties can be clarified if we consider a model medium consisting of small spherical metallic particles described by the Drude dielectric function... 

However, the complex local conductivity for the metallic phase with Drude dielectric function (262) was determined as... 

As several works devoted to the nonlinear optical properties of metal nanoparticles include a size dependence of the linear dielectric function, it seems to us relevant to introduce and briefly comment now the most widespread approach used to describe such a dependence. It consists in modifying the phenomenological collision factor F in the Drude contribution (Eq. 2) as ... 

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 region near the plasma edge in Id conductors /any gap uj < interband region/ is known to approximately obey the Drude formula for the dielectric function ... 

Theoretical modelling shows that this configuration is more favourable for the achievement of a full photonic band-gap, provided that there is a dielectric contrast of at least 2.8 [30]. This requirement is hard to achieve, and it has only been reported for silicon [31] and germanium [32] inverse opals. An alternative can be the use of metals with a Drude-like behavior of the dielectric function. When the dielectric contrast of the photonic crystal becomes extremely... 

Many metals exhibit a strong dependence of their UV/Vis/NIR absorption on the behavior of their free electrons up to the so-called bulk plasma frequency (located in the UV). The simple Dmde model describes the dielectric response of the metals electrons (24). Thus, the dielectric function e (a) can be written as a combination of an interband term e/g(([Pg.545]

In the case of many metals, the region of absorption up to the bulk plasma frequency (in the UV) is dominated by the free electron behaviour, and the dielectric response is well described by the simple Drude model. According to this theory [145], the real and imaginary parts of the dielectric function may be written as. 

The real part of the dielectric function Slmdm( ) corresponding to the localization-modified Drude model can be calculated using the Kramers-Kronig relations, giving... 

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