Drude frequency

The ZE is the high-friction (Smoluchowski) limit of eqn (13.9), in which 7b B, while 0b/1 b = 7d which assumes the Drude frequency cutoff parameter remains finite. It corresponds the bath spectral density in eqn (13.3) (and also in eqn (13.2)) to the form ... 

We now want to study the consequences of such a model with respect to the optical properties of a composite medium. For such a purpose, we will consider the phenomenological Lorentz-Drude model, based on the classical dispersion theory, in order to describe qualitatively the various components [20]. Therefore, a Drude term defined by the plasma frequency and scattering rate, will describe the optical response of the bulk metal or will define the intrinsic metallic properties (i.e., Zm((a) in Eq.(6)) of the small particles, while a harmonic Lorentz oscillator, defined by the resonance frequency, the damping and the mode strength parameters, will describe the insulating host (i.e., /((0) in Eq.(6)). 

Fig. 7. Model calculations for the reflectivity (a) and the optical conductivity (b) for a simple (bulk) Drude metal and an effective medium of small metallic spherical particles in a dielectric host within the MG approach. The (bulk) Drude and the metallic particles are defined by the same parameters set the plasma frequency = 2 eV, the scattering rate hr = 0.2 eV. A filling factor/ = 0.5 and a dielectric host-medium represented by a Lorentz harmonic oscillator with mode strength fttOy, 1 = 10 eV, damping ftF] = I eV and resonance frequency h(H = 15 eV were considered for the calculations.

The filling factor is in good agreement with estimation from electron microscopy [6]. A filling factor of about 0.6 was obtained in all cases. The filling factor sensitively determines the position of the resonance at 0), which indeed shifts in frequency for different specimens. Moreover it is important to observe that / is already quite large and close to the boundary value for a percolation limit (which is -0.7 for spheres and -0.9 for cylinders). The realisation of such a limit would lead to a low frequency metallic Drude-like component in ai(to) for the composite. At present, this possibility seems to be... 

Thus, the Drude model predicts that ideal metals are 100 % reflectors for frequencies up to cop and highly transparent for higher frequencies. This result is in rather good agreement with the experimental spectra observed for several metals. In fact, the plasma frequency cop defines the region of transparency of a metal. It is important to realize that, according to Equation (4.20), this frequency only depends on the density of the conduction electrons N, which is equal to the density of the metal atoms multiplied by their valency. This allows us to determine the region of transparency of a metal provided that N is known, as in the next example. 

Drude s theory characterizes the electron as a harmonic oscillator with a resonance frequency w. Adopting this model, Bohr found that the Coulomb... 

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. 

Impurities in semiconductors, which release either free electrons or free holes (the absence of an electron in an otherwise filled sea of electrons), also give rise to optical properties at low energies below the minimum band gap (e.g., 1.1 eV for Si) that are characteristic of the Drude theory. Plasma frequencies for such doped semiconductors may be about 0.1 eV. 

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. 

Below the plasma frequency at about 15 eV the only appreciable deviation from Drude theory occurs near 1.5 eV, where interband electronic transitions produce a peak in t" and associated structure in c with this exception, c for aluminum goes monotonically toward negative infinity and c" toward positive infinity as the energy approaches zero. 

We showed in the preceding section that for solids with strong vibrational bands the position of features in absorption spectra can be shifted appreciably in going from the bulk to particulate states. Metallic particles can deviate even more markedly from the behavior of the bulk parent material they can have absorption features over broad frequency regions where none appear in the bulk. For a simple metal—one that is well described by the Drude formula... 

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... 

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. 

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 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) ... 

Hamada et al. 1992 Saito et al. 1992), this increase of absorption is caused by high-frequency conductivity of the free carriers in metallic nanotubes. Relative intensities of the spectra of Fig. 11.7 we have found as a result of the diffuse reflection measurements of powders at low wavenumbers. The discussed above Drude approximations of the low-energy part of the absorption spectra are shown by dashed curves in Fig. 11.7. Comparison of the spectra 1 and 2 shows that hydrogenation decreases high-frequency conductivity of the SWNTs by one order of magnitude. 

By using special forms of the so-called spectral density J(w) it is possible to treat memory effects in QMEs. The spectral density J(w) contains information on the frequencies of the environmental modes and their coupling to the system. Tanimura and coworkers [18,20,26] were the first to do calculations along the lines described here using spectral densities of Drude shape. This spectral densities lead to bath correlation functions with purely exponential... 

The chemical nature of the metal appears in Equation (2.333) via the frequency-dependent permittivity emet(w), evaluated at the imaginary frequency iw. For simple metals, the Drude form is often reasonable ... 

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