Drude collision time

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 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 fact that the ratio of the thermal conductivity to the electrical conductivity of any metal is a constant times the absolute temperature was observed by Wiedemann and Franz and this relationship is known as the Wiedemann-Franz ratio. This relationship works because the collision time t for the electron carriers is the same in both models and cancels out when taking the ratio of the two conductivities. From Chapter 17, the classical electronic thermal conductivity was found in Equation 17.33 to be K = 4nl(fT/m n)T. The classical electrical conductivity from the Drude model is given by Equation 18.15 and the Wiede-mann-Eranz ratio becomes... 

Before we leave the Drude model, we may derive the conductivity in terms of the collision theory. Let n electrons be moving in a certain direction at time t = 0 dn of these make a collision at time t, with constant probability P, within the time window dt. We obtain... 

In an external electric field S the force on an electron is e S. The Drude model assumes that we can apply classical mechanics to the electrons. Classical mechanics cannot successfully be applied to individual electrons, but it can sometimes be an adequate approximation for some average properties. From Newton s second law the electric field produces a constant acceleration equal to -e-if/m. If the electron does not undergo a collision, the change in velocity of an electron in time t is... 

Note that the conductivity a has the dimensionality of inverse time. The Drude model is characterized by two time parameters r, that can be thought of as the time between collision suffered by the electron, and a. Typical values of metallic resistivities are in the range of 10 f2cm, that is, a = 10 (f2cm) = 10 s . Using this in (4.69) together n 10 cm , e 4.8 x 10 esu and m 9 X 10 g leads to r of the order 10 " s. Several points should be made ... 

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