I was told that it is possible to describe frequency dependent properties of metals in terms of the equations of motions of the charge carriers. In order to do so the Drude model can be exploited and the following relationship can be derived:
$$\sigma _n = \frac{Nq^2}{m(\tau^{-1} – i\omega)}$$
where $\sigma _n$ is the complex conductivity, $N$ is the number of charge carriers per unit volume, $q$ the charge of each carrier, $\tau$ the average scattering time between collisions, and $\omega$ is the frequency of the electric field.
Where does this result come from and how can it be derived? Also, what is its significance?
Best Answer
This complex conductivity is a simple result of the classical Drude model for the metal conduction. In the Drude model it is assumed that in the metal there is a concentration $n$ of electrons and that an electron has a probability $\frac {1}{\tau}$, $\tau$ is called scattering or relaxation time, for scattering with the lattice for losing its momentum. The motion of the electrons can be described by Newton's law $$m\frac {d v}{d t}=-eE-\frac {mv}{\tau} \tag 1$$ where $m$ is the (effective) electron mass, $v$ the mean electron velocity in field direction, and $E$ is the electric field strength. The last term on the RHS of eq. (1) resembles a "friction term" in the equation of motion. Assuming sinusoid variation of the electric field $$v \propto E \propto \exp(-i\omega t) \tag 2$$ Thus from eq.(1) follows $$v=-\frac{eE}{m}\frac {1}{1/\tau-i\omega}\tag 3$$ This gives the current density $$j=-nev=\frac{ne^2E}{m}\frac {1}{1/\tau-i\omega} \tag 4$$ and thus the complex metal conductivity $$\sigma=\frac{ne^2}{m}\frac {1}{1/\tau-i\omega} \tag 5$$