Drude conductivity

For materials near the insulator-metal (locahzation-delocalization) transition the optical conductivity is suppressed at low frequencies relative to the usual Drude conductivity [115]. The suppression is usually strong for frequencies up to a critical-frequency (Oc D/L where D is the diffusion coefficient and L is the localization length for the electron. This conductivity suppression occurs because the carriers would diffuse a distance greater than the localization length within the period of the AC wave for... 

From the experimental data which exist for current systems, estimates of the intrinsic conductivity also can be made [197]. The intrinsic Drude nature of metallic carriers has been identified using both microwave and optical techniques. Both of these techniques have identified the presence of a group of carriers which demonstrate Drude behavior with a long scattering time (t 10-"s). The Drude conductivity for traditional metals is given by... 

Localization Modified Drude Model a model for conduction electrons which includes suppression of the Drude conductivity at low frequencies due to finite localization lengths for the electrons. 

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. 

Among many fascinating properties, quasicrystals with high structural quality, such as the icosahedral AlCuFe and AlPdMn alloys, have unconventional conduction properties when compared with standard intermetallic alloys. Their conductivities can be as low as 450-200 (Qcm) [7]. Furthermore the conductivity increases with disorder and with temperature, a behaviour just at the opposite of that of standard metal. In a sense the most striking property is the so-called inverse Mathiessen rule [8] according to which the increases of conductivity due to different sources of disorder seems to be additive. This is just the opposite that happens with normal metals where the increases of resistivity due to several sources of scattering are additive. Finally the Drude peak which is a signature of a normal metal is also absent in the optical conductivity of these quasicrystals. 

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... 

The sp-valent metals such as sodium, magnesium and aluminium constitute the simplest form of condensed matter. They are archetypal of the textbook metallic bond in which the outer shell of electrons form a gas of free particles that are only very weakly perturbed by the underlying ionic lattice. The classical free-electron gas model of Drude accounted very well for the electrical and thermal conductivities of metals, linking their ratio in the very simple form of the Wiedemann-Franz law. However, we shall now see that a proper quantum mechanical treatment is required in order to explain not only the binding properties of a free-electron gas at zero temperature but also the observed linear temperature dependence of its heat capacity. According to classical mechanics the heat capacity should be temperature-independent, taking the constant value of kB per free particle. 

The classical theory for electronic conduction in solids was developed by Drude in 1900. This theory has since been reinterpreted to explain why all contributions to the conductivity are made by electrons which can be excited into unoccupied states (Pauli principle) and why electrons moving through a perfectly periodic lattice are not scattered (wave-particle duality in quantum mechanics). Because of the wavelike character of an electron in quantum mechanics, the electron is subject to diffraction by the periodic array, yielding diffraction maxima in certain crystalline directions and diffraction minima in other directions. Although the periodic lattice does not scattei the elections, it nevertheless modifies the mobility of the electrons. The cyclotron resonance technique is used in making detailed investigations in this field. 

This relation holds for a completely isolating crystal. When conduction electrons are present, the so-called Drude-term has to be added on the right-hand side and we have... 

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. 

Fig. 1. Optical conductivity spectra of AXC60 (x = 0, 3, 4, and 6) [7]. K3C60 is a metal, which shows a Drude-like behavior at low energy region, while K4C6o is an insulator, which does not show such a behavior.

Figure 3. Optical conductivity spectra of p -BEDO-TTF)5[CsHg(SCN)4]2 for E L L and E L at 300, 200, 100 and 10 K (L is BEDO-TTF stack direction). The fit with Drude-Lorenz model for T=10 K is shown by thin solid line.

The Drude model applies the kinetic theory of gases to metal conduction. It describes valence electrons as charged spheres that move through a soup of stationary metallic ions with finite chance for scattering. 

Fig. 15 Optical conductivity of K3C60 and K4C60 illustrating the disappearance of the Drude term in K4C60. The peak at 1 eV is due to interband transitions. The strong peak at 0.5 eV in K4C60 corresponds to the gap seen by transport. (From [38])...

The decay of the nanoparticle plasmons can be either radiative, ie by emission of a photon, or non-radiative (Figure 7.5). Within the Drude-Sommerfeld model the plasmon is a superposition of many independent electron oscillations. The non-radiative decay is thus due to a dephasing of the oscillation of individual electrons. In terms of the Drude-Sommerfeld model this is described by scattering events with phonons, lattice ions, other conduction or core electrons, the metal surface, impurities, etc. As a result of the Pauli exclusion principle, the electrons can be excited into empty states only in the CB, which in turn results in electron-hole pair generation. These excitations can be divided into inter- and intraband excitations by the origin of the electron either in the d-band or the CB (Figure 7.5) [15]. 

---