Drude theory

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. 

Figure 9.11 Measured reflectance of aluminum compared with the Drude theory. The dielectric (unction and refractive index are from Hagcmann et al. (1974).

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. 

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

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. 

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... 

Sommerfeld modified the Drude theory by introducing the laws of quantum mechanics. According to quantum mechanics, electrons are associated with a wave character, the wavelength A being given by A = /i/p where p is the momentum, mv. It is convenient to introduce a parameter, k, called the wave vector, to specify free electrons in metals the magnitude of the wave vector is given by... 

P. Drude, Theory of Optics, Dover Publishing New York, NY (1959) 350. 

P. Drude, Theory of Optics, Dover Reprint (1959) R.M.A. Azzam, N.M. Bashara. Ellipsometry and Polarized Light. North Holland (1977). 

Next consider the motion of these electrons. It was already mentioned that in addition to their density, metallic electrons are characterized, at this level of theory, by a relaxation time t. In the Drude theory this enters via a simple friction force by assuming that under a given force f (i) the electron moves according to... 

Various other electronic transitions are possible upon light excitation. Besides the band-band transitions, an excitation of an electron from a donor state or an impurity level into the conduction band is feasible (transition 2 in Fig. 1.9). However, since the impurity concentration is very small, the absorption cross-section and therefore the corresponding absorption coefficient will be smaller by many orders of magnitude than that for a band-band transition. At lower photon energies, i.e. at ph < g, an absorption increase with decreasing ph has frequently been observed for heavily doped semiconductors. This absorption has been related to an intraband transition (transition 4 in Fig. 1.9), and is approximately described by the Drude theory [4]. This free carrier absorption increases with the carrier density. It is negligible for carrier densities below about 10 cm ... 

As clusters become much larger in size, the collective mode can be tracked from the quantum limit to the bulk [712], so that the evolution of the many-body resonances is known over an enormous range. In the bulk limit, it has been shown that the resonances tend towards the surface plasmon of the solid, except that, since the solid does not have spherical symmetry, the -y/3 factor of the Mie-Drude theory does not appear and... 

In the classical Drude theory of metals, the Maxwell-Boltz-mann velocity distribution of electrons is used. It states that the number of electrons per unit volume with velocities in the range of dv about any magnitude v at temperature T is... 

The dc conductivity parallel to the conducting chain, (7 , is calculated from the parameters of the simple one electron Drude theory using Eq, 17 with 0) set equal zero... 

Microwave measurements for PPy samples provide independent confirmation of the results reported at IR frequencies. Figure 15.46 compares (TdciT) with o-MwiT) for PPy(PF6) [29], PPy(TsO) [29], and PPy(S-PHE) [136]. The absolute values and the temperature dependence of cr in the dc and microwave frequency ranges for PPy(PF6) are nearly identical, in agreement with the Drude theory. The stronger temperature dependence of for PPy(TsO) and PPy(S-PHE) in comparison with cr wC T) is expected because the sample is in the localized regime [181]. 

As a rule, the experimental (Section 3.2) frequency dependence of the free-carrier absorption (or the Drude absorption) disagrees with the law predicted by the Drude model [Eq. (1.43a)]. The actual dependence of the decay constant a follows the co p law [40, 42, 46], where is a constant over the range < p < A. The constant p depends on the semiconductor, the frequency range, the temperature, and the concentration of impurities and free carriers. The quantum-mechanical extension of the Drude theory [73] shows that... 

At the same time, according to the Drude theory, a structure of a metal consists of cations surrounded by electronic gas, where the cation charge is equal to the atomic valence and its valence electrons belong to the whole crystal. However, atoms in the solid metals cannot be ionized to more than h-1, because of high magnitudes of the consequent ionization potentials. This limit is evident from the work functions, which... 

Figure 1.16 Imaginary part of the complex polarizability a o)) for an Na cluster with N = 198 in units of Effective single-pair excitations, as well as the surface plasmon and the volume plasmon, are clearly resolved. For comparison the result of the local Drude theory is also given. In this case there is only one mode of excitation, the classical surface-plasmon polariton or Mie-resonance at coply/3. Because the frequency is scaled with this frequency the Drude curve peaks trivially at 1. For more explanation see text. Reproduced with permission from Reference [5]. Copyright 1985 by the American Physical Society...

Considering its electrical and magnetic properties, ScSe like ScS and ScTe may be regarded as a monovalent metal in which the conduction electron density n corresponds to one electron per formula unit. Theoretically, n = 2.54x 10 cm" for nominal stoichiometric ScSe, Zhuze et al. [3]. The carrier density n = 1 x 10 cm" was calculated from the minimum In the diffuse reflection spectrum (at 2.26 eV) of ScSe obtained by the metal hydride method. Similarly, the effective free carrier mass m = 1.00 mo, the mobility of free carriers = 2.5 cm V" s" and the relaxation time t = 5.3x 10" s were obtained for this sample, Obolonchik et al. [5]. The reflection spectrum for a Bridgman-single crystal gives t = 2.2 x 10" s calculated with use of the Drude theory. The optical effective mass was estimated to be m = 2.9 mo [3]. 

The Drude model is a crude model, but it contains the accepted mechanism for electrical resistance in solids, which is the effect of collisions with the cores of the crystal. There are a number of more sophisticated theories than the Drude theory. However, the results of these theories are similar in their general form to Eq. (28.4-9). The major differences are in the interpretation of the quantities r, and m One problem with the Drude theory is that the conductivities of most common metals are found experimentally to be approximately inversely proportional to the temperature, instead of being inversely proportional to the square root of the temperature, as in Eq. (28.4-11). One can rationalize this by arguing that the mean free path should decrease as the temperature rises, because of the increased vibrational amplitude of the cores, making them into targets with larger effective sizes at higher temperature. 

Recall the other serious difficulty discussed in Chapter 17 that arises from the fact that the classically predicted heat capacity of the electrons is not observed even though they are the major contributor to both the thermal and electrical conductivity of metals. We will find yet another problem with the classical theory when we take up the topic of paramagnetism and find that the electronic contribution expected from classical theory is not observed. Despite the success of the classical Drude theory of the free electron gas in being able to describe many of the observed properties of metals, it was these discrepancies between the classical theory and observation that prompted theorists to reexamine the classical theory of the electron and to apply the quantum mechanical treatment that had been developed to explain the electronic structure of atoms and molecules to describe the behavior of electrons in metals. 

Now that we are better equipped to handle some of the quantum theory necessary to understand the behavior of electrons in metals, let us review some of the problems we encountered in the simple Drude theory. 

The Drude theory treats the electrons in a metal as a classical monatomic gas. This simple theory is able to explain many observed properties of metals such as why they are good... 

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