Dielectric Function

What is the Dielectric Function?

The dielectric function of a material describes the electrical and optical properties versus frequency, wavelength, or energy.  It describes the polarization (electric polarizability) and absorption properties of the material.  The dielectric function is commonly denoted by the Greek letter epsilon, ε, and consists of two parts called epsilon 1 and epsilon 2 written as ε1 and ε2, but often written simply as e1 and e2.  Since the quantities e1 and e2 vary with wavelength, frequency or photon energy it is common to write them as functional expressions such as e1(λ) & e2(λ), or e1(E) and e2(E), etc.

The quantity e1 represents how much a material becomes polarized when an electric field is applied due to creation of electric dipoles in the material (see Figure 1). When the applied field is oscillating between positive and negative such as with a light wave the sign of e1 can be either a positive or negative number depending on whether the induced dipoles are oscillating in phase or out of phase with the applied field.

Figure 1: In atoms by incident oscilator dipoles induced light wave,
resulting in polarization inside the material.

When the induced dipole oscillations in a material become large it is possible for the material to start absorbing energy from the applied field.  When absorption occurs the quantity e2 becomes important.  When a material is transparent e2 is zero, but becomes nonzero when absorption begins.  Thus e2 represents absorption in a material.

When written together e1 and e2 form what is called the complex dielectric function.  It is called complex because the two parts are considered together and written using complex number representation:

ε(λ) = ε1(λ) ± iε2(λ).

Note e2 is sometimes written as a positive quantity, or sometimes as a negative quantity, so the equation above contains ± to cover either case.

It is important to consider both e1 and e2 together since they affect each other, meaning the shape of e2 cause corresponding changes in the shape of e1 and vice-versa.  This is known as the Kramers-Kronig relation between the real (e1) and imaginary (e2) parts of the dielectric function.

In short, the dielectric function describes what an electric field such as an oscillating light wave does to material.

• ε1 = volume polarization termà Dipoles created.
• ε2 = volume absorptionà vibrational energy lost as heat in insulating materials, or electrons become free carriers in solar cells, for example.

The dielectric function is related to the refractive index of a material by the equation

ε = n²

Or as complex numbers:

(ε1 ± iε2) = (n ± ik)²

The following equations allow converting dielectric function & refractive index.  This must be done at each wavelength.  Woollam WVASE32 and CompleteEASE software does these conversions automatically!

Dielectric Function Conversion:

Dielectric Function of Typical Materials:

For more information on this topic please refer to the ellipsometry books here.

Permittivity - Wikipedia

Dielectric - Wikipedia