Two limits for : and . The first one refers to the collective excitations of the Fermi sea, which is related to the plasma, and the latter describes the electrostatic screening of the electron with electron, lattice and impurity interactions in crystals.
To obtain plasma frequency , we can proceed with the equation of motion of a free electron in an electric field:
x is the position of electron. Assuming electron moves according to the behaviour of simple planar wave, we need to introduce on both side,
Further, we have polarization: the dipole moment per unit volume,
Since we are dealing with polarization, we can’t forget electric field, E, which both of them expressed in the electric displacement, D,
That is, we find the characteristic of each material depicted by specific mass, m. Thus, this is the plasma frequency ,
The dielectric function can be re-written as follows
In the next section, we will deal with the implication of this equation.
The main concept from Kittel’s book: Introduction to solid state physics, 8th edition.