Drude plasma frequency

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.

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. 

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. 

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. 

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

Measurements of the polarized reflectance in the NIR have frequently been used to obtain estimates for the transfer integrals. The method consists in fitting a reflectance model based on the Drude expression [Eq. (1)] to the experimental data. The Drude expression should be considered as a tool in estimating the plasmon frequency, ftp the background dielectric constants, e0 plasma frequency, (op and so on. The validity of the Drude analysis is limited to the conducting organic materials, with the electrical conductivity not less than a few S cm-1. 

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 Drude parameters in Table 4 have been derived by fitting the reflectance spectra with the Drude-like dielectric constant, Eq. (33). It has been stressed by Jacobsen [98] and Yamaji [68b] that in the limit of highly anisotropic one-dimensional conductors the ratio of transfer integrals is not proportional to the ratio of plasma frequencies but to its square instead. 

Unfortunately, there are few direct measurements of the complex conductivity at microwave frequencies, thereby avoiding the need for extrapolation from higher frequency data. Martens et al. (2001) present data in the range 8-600 GHz for PPy and PAni that can be described by the Drude model. However, the plasma frequencies are 7 meV, much lower than the 1 eV calculated from the known carrier density. This indicates that most carriers are localised by the disorder in the samples and that the density of delocalised states at the Fermi level is low. Prigodin and Epstein (2003) note that the low frequency response is provided by a very small fraction of the carriers that are highly mobile and have long scattering times. They develop a model for this... 

In analogy to the optical models for doped In203, the Drude model has been applied to undoped [187] and doped [155] tin oxide layers. In pure Sn02 films, the high frequency dielectric constant as well as the plasma frequency cOp, are strongly... 

From the plasma minimum (plasma frequency, wp), one can readily calculate t the electronic relaxation time by the classical Drude equation which is well established for metal films (11)- This gives a value of 1.9 X 10" sec for (SN). . From t and wp, one can calculate the dc conductivity of a metal by the well-known equation (11) ... 

Moreover, the microwave conductivity exhibits a maximum at 180 K which is a characteristic of traditional metals. The Drude model can be applied as previously shown for polyacetylene [34,58]. Calculations lead to a value of 0.015 eV for the plasma frequency. This relatively low value is put down to the... 

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. 

Actual calculations for the two limit cases above include also Drude-like intraband contributions, with a plasma frequency calculated self-consistently from the band structure [3]. As already discussed (cf. Eq. (2)), in the U = 0 limit this is the dominating contribution to the optical properties. However, intraband terms are also important in the large-U limit, where they account for the relevant contribution of the carriers thermally excited across the narrow gaps at high temperatures. 

The characteristic composite behavior of (t maM for medium consisting of spherical particles with volume fractions / of Drude conductor and 1 - / of insulator is shown in Figure 15.5. For a volume fraction / less than the percolation value (/ = 1/3 for spheres), (Tema (impurity band of localized plasmon-like excitations. As the system approaches the percolation threshold, the localized peak o-ema(w) shifts to lower frequency. Above the percolation threshold, a Drude peak corresponding to the carriers that have percolated through the composite structure occurs at low frequency. Only a fraction ( (3/— l)/2 [119]) of the full conduction electron plasma frequency appears in the Drude peak, depending on the proximity to the percolation threshold. The same percolating free electron behavior is observable in the dielectric response ema(w) for the system. 

---