Drude-Lorentz model

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)). 

By fitting a Lorentz-Drude model, e.g. by the method of least squares to a set of measured values for Tk respectively Rk, determined for a sufficiently big number of wavelengths, distributed over a wide wavelength range, a set of coefficients coj,... 

In electromagnetic theory it is shown that the real and imaginary part of e (co) are not independent of each other, but are connected by a pair of integral equations, the Kramers-Kronig relation (e.g. Bohren Huffman 1983). Equation (A3.10) satisfies these relations, i.e. using a Lorentz-Drude model fitted to the laboratory data automatically guarantees that the optical data satisfy this basic physical requirement. 

The Lorentz-Drude model is largely ad hoc [14,15] but still useful as starting points and for developing feelings for optical properties. Consider an atom with electrons boiuid to the nucleus in the same way as a small mass bound to a large mass by a spring. This is the Lorentz model and is applicable to a wide variety of materials (i.e., metals, semiconductors and insulators). The motion of an electron boiuid to the nnclens is then written as... 

We will now analyze the general optical behavior of a metal using the simple Lorentz model developed in the previous section. Assuming that the restoring force on the valence electrons is equal to zero, these electrons become free and we can consider that Drude model, which was proposed by R Drude in 1900. We will see how this model successfully explains a number of important optical properties, such as the fact that metals are excellent reflectors in the visible while they become transparent in the ultraviolet. 

Hall Effect in the Drude Model. The Drude treatment of the Hall effect starts from the Lorentz force on an electron ... 

Another success story for the Drude model was the explanation for the Wiedemann19 -Franz20 law of 1858, which stated the empirical observation that the ratio of the thermal conductivity k to the electrical conductivity absolute temperature—that is, that the so-called Lorentz number k/oT was independent of metal and temperature. The thermal conductivity k (> 0) is defined by assuming that the heat flow JH is due to the negative gradient of the absolute temperature T (Fourier s law) ... 

Metals are denoted as fi ee-electron metals if most of the electronic and optical properties are due to the conduction electrons alone. Examples are Al, Mg, and alkali metals. The dispersion of the optical constants of the fi ee-electron metals is described by the Drude model [72], which can be regarded as a special case of the Lorentz model (1.3.3°) when the restoring (bounding) force is equal to zero, and hence the resonance frequency of free carriers, a>o = -JWfm (here, m is electron mass), is also equal to zero, damping force y results... 

The Drude model for metals assumes that the electrons are free to move. This means that it is identical to the Lorentz model except that coq is set equal to zero. The real and imaginary parts of the metal s dielectric constant are then given by... 

The classical theory of the dielectric response in solids is frequently described by the Drude and Lorentz models. The Drude model is applicable to free-eiectron metals its quantum-meehanical analog includes intraband transitions, where intraband transitions are taken to mean all transitions not involving a reciprocal lattice vector. The Lorentz model is applicable to insulators its quantum-mechanieal analog ineludes all direct interband transitions, i.e., all transitions for whieh the final state of an electron lies in a different band but with no change in the k vector in the reduced-zone scheme. In the following discussion, both models will be surveyed and evaluated for real metals. 

The Lorentz and Drude models can be explained rigorously in relation to electronic band structure. Indeed, both models have quantum-mechanical analogs intraband transitions for the Drude model and direct interband transitions for the Lorentz model. To see the role of both models in describing real metals, consider the schematic band diagram as shown in Figure 149. TWo typical transitions are illustrated in Figure 149. The first of these, called an intraband transition, corresponds to the optical excitation of an electron from below the Fermi level (Ep) to another state above the Ef within the same band. There is no threshold energr for such transitions, and they can occur only in metals. 

To select suitable dispersion model, such Sellimeier model, Cauchy model, Lorentz model, Drude model, effective medium approximation (EMA) model etc., for each layer. Which dispersion model should be selected for a certain layer depends on the specific type of the film and we will discuss later in detail. In the model, some parameters are known and the others are unknown. The unknown parameters will be determined through mathematical inversion method. 

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.

To describe the optical properties of the crystals over a wide temperature range and to derive the basic phenomenological parameters of the electronic transition, we fitted the R(co) and a(co) spectra using the Drude-Lorentz model. The Drude part describes the intraband transition of free carriers,... 

The DOS at 0 K is shown in Figure 11.18 using the free electron model. However, the Drude-Lorentz model employs the classical equipartition of energy and does not take into account the fact that quantum mechanics places restrictions on the placement of the electrons (as a result of the Pauli exclusion principle). A revised theory, known as the Sommerfield model, allows for this modification. At temperatures above 0 K, the fraction, f(E), of allowed energy levels with energy E follows the... 

The classical theory of absorption in dielectric materials is due to H. A. Lorentz and in metals it is the result of the work of P. K. L. Drude. Both models treat the optically active electrons in a material as classical oscillators. In the Lorentz model the electron is considered to be bound to the nucleus by a harmonic restoring force. In this manner, Lorentz s picture is that of the nonconductive dielectric. Drude considered the electrons to be free and set the restoring force in the Lorentz model equal to zero. Both models include a damping term in the electron s equation of motion which in more modem terms is recognized as a result of electron-phonon collisions. 

A critical point in method A is the not exactly known refractive index n or, in terms of the Drude-Lorentz model, 800 of PEG in the infrared region, because it enters the spectrum calculation but cannot be determined with sufficient accuracy from the spectral fit. Changes of 800 have, other things being equal, opposite effects on ATR reflectivities calculated for bulk material and thin films on metal, respectively. In the present calculation, 8qo = 2.10 (n = 1.45) was used, somewhat smaller than the value np = 1.46 reported in the literature [18] in order to take into account the optical dispersion [19] in an approximate manner. 

The dielectric function (v) of the as-deposited Lai 3 Y3 alloys obtained from ellipsometry after correction for the substrate and the Pd over layer are shown in fig. 96(i). To analyze the data quantitatively, van Gogh et al. modeled (v) using a sum of one Drude and three Lorentz... 

---