COMPARISON OF THE CALCULATION METHODS OF A DIELECTRIC FUNCTION



Comparison of the calculation methods of a dielectric function

Key words: dielectric function.

The properties of the dielectric permittivity 𝜀(𝜔)=𝜀′+𝑖𝜀′′ underlie of many physical phenomena and enable a common position to describe the interaction of electromagnetic radiation with matter.

As is known, solids, based on their electronic properties, can be divided into metals, semiconductors, and dielectrics. The main difference lies in the reflectivity, and respectively in current flows in the bulk. As for metals in a wide frequency range approaching an ideal conductor is fair, then a high reflectivity is possible. Impossibility of penetration of the electromagnetic radiation into metals causes the possibility of their use in microwave and infrared wavelengths for creating waveguides and resonators. In the process of reflection and absorption, the main role is played with the interaction of electromagnetic waves with free charge carriers, and it can be adequately described with the Drude classical model of free electrons. It is assumed that the electrons of the metal are the gas of some density, moving relatively positively charged ion cores. As the result of this approach is the analytic dependence of the dielectric constant from the frequency of the external field 𝜀(𝜔)=1−𝜔𝑝2𝜔2+𝑖𝛾𝜔 where 𝜔𝑝- a plasma frequency, 𝛾 - electron collision frequency because of the Coulomb interaction. During increasing of a frequency, the field penetration into the metal can be changed, which leads to increased absorption and approximation of an ideal conductor becomes not quite adequate. On ultraviolet frequencies, metals gain the dielectric properties. Since alkali metals such as sodium, have a response similar to the response of the free electrons, and become transparent to electromagnetic radiation. At the same time, the noble metals such as gold and silver become highly absorbing. Accordingly, the above approach turns on not entirely correct and requires clarification.

Changing of absorbance of radiation with precious metals in the optical range arises because of the deep lying excitation of the electrons, such as d-levels, which occur because of interband transitions. For gold and silver there are 4d-5sp and 5d-6sp transitions respectively, while the free electrons are in states 5s and 6s. For gold, the transition takes place when the light of wavelength is about 500 nm - (as it is visible region of the spectrum, gold has the color). For silver, the transition takes place at a wavelength about 320 nm (as the eye doesn't see the short wave, silver has no color).

Clarification of Drude model of free electrons in the optical range, is possible by means of an amendment, taking into account interband transitions, 𝜀∞ - high permittivity, and 1&lt;𝜀∞&lt;9 (the exact value for each metal is determined experimentally). This is the approach of quasi-free electrons, in which the expression for the complex dielectric function is 𝜀𝑠(𝜔)=𝜀∞−𝜔̃𝑝2𝜔(𝜔+𝑖𝛾)=𝜀∞−𝜔̃𝑝2𝜔2+𝛾2++ 𝑖𝛾𝜔̃𝑝2𝜔(𝜔2+𝛾2). It should be noted that in this case the plasma frequency  𝜔̃𝑝 different from conventional plasma frequency, by means of, it takes into account the effective mass of the electron 𝑚∗=𝑚𝑛0𝑛𝑒, where 𝑛0 – concentration of electrons in the valence band, and 𝑛𝑒 is concentration of free electrons.

To describe the optical properties of the metal, there is another way, developed by Lorenz at the beginning of the last century, in which electrons and ions substances regarded as being under the influence of forces caused by external electromagnetic fields, simple harmonic oscillators. Expressions for the real and imaginary parts of the permittivity are  𝜀′(𝜔)=1+𝜔𝑝2(𝜔02−𝜔2)(𝜔02−𝜔2)2+𝜔2𝛾2, 𝜀′′(𝜔)=𝜔𝑝2𝛾𝜔(𝜔02−𝜔2)2+𝜔2𝛾2,  where 𝜔0 is oscillator frequency.