When an electromagnetic wave excites a material, it induces polarization effects as well as the displacement of the conduction electrons. These processes constitute the optical response of the material and can be characterized by the dielectric function which is determined by adopting the dipole approximation.
1. The complex dielectric function ε (ω)
The dielectric function ε (ω) is used to describe the linear response of the material to electromagnetic radiation which is linked to the interaction of photons with electrons.
The dielectric function is determined by the electronic transitions between the valence band and the conduction band. It is calculated by evaluating the matrix elements in representation of the impulse. It brings into play a real part (the dispersive part) and an imaginary part (absorptive part), using the formalism of Ehrenreich and Cohen, in the form of the following formula:
The imaginary part ε 2 (ω) is given by the following relation:
with
q is the charge of the electron and m is its mass,
represents the matrix elements for the transitions between the
valence band and the conduction band with e is the polarization vector of the electric field and uck (r) is the periodic part of the Bloch wave function for a state of the band conduction and wave vector k,
ωcv(k) = Eck – Evk is the energy of excitement.
The summation over the Brillouin irreducible zone in the equation is calculated using tetrahedral interpolation. The imaginary part ε2 (ω) of the dielectric function depends on the electronic transition at the origin of the absorption. Direct inter-band transitions can be derived from identification with the energy band structure.
The real part ε1 (ω) of the dielectric function can be obtained from the imaginary part ε2 (ω) using the following Kramers-Kronig relation:
P represents the main value of the integral
A difficulty comes from the fact that the gap is systematically underestimated by the DFT method, which poses a deficiency when using the Kramers-Kronig relation. This deficiency can be overcome by the use of a "scissor operator" which shifts the bands above the Fermi level with Δc energy. Although the GGA underestimates the value of the gap, we did not want to correct it by the value of the scissor because this correction does not influence the shape of the properties but only it causes a shift in the axis of the energies .
It is also possible to add a Lorentzian enlargement which accounts for the experimental enlargement. This makes it easier to compare the spectra obtained with EELS (Electron Energy Loss Spectroscopy) data. The adopted value for enlargement is 0.1 eV which is the default value.
Knowledge of the real part and the imaginary part makes it possible to calculate the other optical functions such as the reflectivity, the optical conductivity and the refractive index.
To calculate the optical response from the band structure, a photon energy range of 0 - 30 eV was chosen. The optical properties have been shown for a photon energy interval of 0 - 14 eV for two reasons:
Show the optical behavior of the material within the visible light spectrum,
Illustrate the limit of the anisotropy of the optical properties of the material
Since Bismuth Sulfide has an orthorhombic structure, which is considered optically as a biaxial system, the linear dielectric tensor has three independent components which are the diagonal elements of this tensor.
The evolution of the imaginary (absorptive) part of the dielectric function is shown in the following figure:
The main characteristic of the absorptive part is the broad peak. The peaks of ε2x, ε2y and ε2z correspond to the optical transitions from the valence band to the conduction band and are in agreement with the other theoretical results. The maximum peak values for ε2x, ε2y and ε2z are around 2.81, 3.19 and 2.87 eV respectively. There is an optical anisotropy for all directions, and despite this anisotropy the three components have a similarity. Note that a single peak does not correspond to a single inter-band transition since several direct or indirect transitions can be found with an energy corresponding to the same peak.
The evolution of the real (dispersive) part of the dielectric function is shown in the following figure:
The components ε1x, ε1y and ε1z begin to increase until peaks around 1.78, 2.16 and 1.78 eV respectively. Then they start to decrease until they cancel out at around 4.15, 3.28 and 3.39 eV, respectively.
The following table gives the static values of ε1.
2. The optical conductivity σ (ω):
The occupied states are excited towards the unoccupied states above the Fermi level by the absorption of photons. This inter-band transition is called "optical conduction" and photon absorption is called "inter-band absorption". The term "optical conduction" means electrical conduction in the presence of the electric field included in the light.
The real part of the optical conductivity is calculated according to the following relation:
The variation of the optical conductivity is shown in the following figure.
From the figure, optical conduction starts from the energies of about 1.23 ev, 1.15 ev and 1.07 ev for the directions X, Y and Z, respectively. These values represent the optical energy gaps that must be overcome for the transition to begin. The optical conduction begins to increase from these energies and arrives at a maximum level for 4.1, 3.25 and 3.25 eV for the directions X, Y and Z, respectively. It is noted that the optical conduction reaches a maximum value in the region of the visible spectrum. It is this important optical conduction that allows the material to be used in photovoltaic applications.
3. The absorption coefficient α (ω):
The inter-band absorption coefficient α (ω) characterizes the part of energy absorbed by the solid. It determines how far light, of a particular wavelength, can penetrate a material before it is absorbed. In a material with a low absorption coefficient, the light is only weakly absorbed, and, if the material is thin enough, it appears transparent for this wavelength. The absorption coefficient depends on the material and also on the wavelength of the light that is absorbed. It can be defined according to the extinction coefficient K (ω) by the following relation:
Where λ represents the wavelength of light in the vacuum.
The absorption coefficient can be calculated via the dielectric function by the following relationship:
Photon absorption is at the origin of the interband optical transition. It will not take place only for determined energies which represent the energies of transition between the states of energies; and that’s why we find energies that are absorbed and others that are not. Note that absorption will not take place when the photon energy is less than the energy of the optical gap.
The evolution of the absorption coefficient is shown in the following figure.
From the figure, we notice that the absorption starts from the energies 1.21, 1.18 and 1.07 eV for the directions X, Y and Z, respectively. The absorption begins to increase until reaching the maximum for energies 5.15 ev, 3.85 and 4.69 ev for the directions X, Y and Z, respectively; then it begins to decrease until reaching a minimum for an energy of about 2.25 eV for all directions. It begins to increase again to arrive at another maximum greater than the first at energies 11.08, 11.0 and 10.68 eV for the directions X, Y and Z, respectively. We note that a maximum absorption corresponds to a maximum conduction and to a minimum dispersion, that is to say to a minimum value of ε1.
4. The optical constants n (ω) and K (ω):
The refraction causes the propagation of light waves with a velocity smaller than that in a free space (for example: air). The reduction in velocity leads to bending of the light rays at the interfaces described by Snell's law of refraction. Refraction, in itself, does not affect the intensity of light during its propagation.
The propagation of a light beam through a translucent medium is described by the refractive index n. The latter is defined by the relationship between the velocity of light in free space c and that in the medium Ê‹ according to the relationship:
The refractive index depends on the frequency of the light beam. This effect is called: dispersion. The refractive index n (ω) is calculated by the following relation:
The extinction coefficient represents the energy loss (due to absorption and diffusion) of the medium in the magnitude of the complex refractive index ñ (ñ = n + iK) which characterizes any medium. So, it is linked to the absorption coefficient by the relationship described above.
The extinction coefficient K (ω) is given by the following relation:
At low frequencies (ω = 0), the relations of n and K become:
The refractive index represents the dispersion behavior of the material.
The evolution of the refractive index is shown in the following figure:
From the figure, the static values of the refractive index are 3.65 for the X direction and 4.03 for the Y and Z directions; then it begins to increase until reaching its maximum at 2.46 eV for the X direction and at 2.29 eV for the Y and Z directions. The refractive index begins to decrease to a minimum at energies 6.73, 6.05 and 6.16 eV for the X, Y and Z directions, respectively. At this minimum, the refraction phenomenon disappears since the refractive index becomes almost equal to "1", and the material behaves like a free space. From this minimum, the variation in the refractive index is small and therefore the dispersion is very low. We note that the dispersion phenomenon is very important in the visible spectrum region. This significant variation in refractive index in the visible spectrum region does not allow the material to be used in the manufacture of optical fibers.
The evolution of extinction coefficient K is shown in the following figure.
The extinction or attenuation coefficient represents the absorption phenomenon in the complex refractive index and is directly linked to the absorption coefficient. So the extinction coefficient only starts to increase from a threshold which represents the optical gap. This threshold is equal to 1.29 ev for the directions X and Y and 1.12 eV for the direction Z. It begins to increase to reach a maximum at energies 4.34, 3.36 and 3.49 eV for the directions X, Y and Z, respectively.
5. The reflection coefficient R (ω):
The reflection phenomenon that occurs at the front and rear surfaces is described by the reflection coefficient which represents the ratio between the energy reflected and the energy incident on the surface.
The reflection coefficient R (ω) is given by the following relation:
The modulus of the complex function R (ω) is given by the following relation:
The evolution of the reflection coefficient is shown in the following figure:
The reflection coefficient represents the amount of energy reflected in relation to the incident energy. The static values of the reflection coefficient are 0.32 for the X direction and 0.36 for the Y and Z directions. The reflection coefficient begins to increase until reaching maximum values of 0.52 to 4.88 eV for the X direction, 0.61 to 3.41 eV for the direction Y and O.58 to 3.55 eV for the direction Z. The average values of the coefficient of reflection in the region of the visible spectrum show that the material is semi transparent in this region of energy. We note that the maximum reflection occurs when ε1 (ω) goes towards negative values.
The following table gives the static values of n and R.
6. The EELS function:
EELS (Electron Energy Loss Spectroscopy) spectroscopy is a tool for investigating different aspects of materials. It has the advantage of covering the entire energy range including non-scattered and elastically scattered electrons (Zero loss). At intermediate energies (typically between 1 and 50 ev), the energy losses are mainly due to a complex mixture of simple electronic excitations and collective excitations (Plasmons). The positions of the simple electronic excitation peaks are related to the joint density of the states between the valence band and that of conduction; while the energy required for collective excitations depends mainly on the electron density in the solid. Here, the electrons that excite the electrons of atoms in the outer layer are called "Valence Loss: Valence Loss" or "interband valence transitions". At higher energies (typically a few hundred bps), edges can be noticed in the spectrum indicating the start of excitations of the various internal atomic layers towards the conduction band. In this case, the fast electrons excite the electrons of the internal layer (Loss of Heart: Core Loss) or induce excitations of the level of heart of the structure of the near edge ELNES (Level Excitation of Near Edge Structure) and XANES (X- rays Absorption Near Edge Structure). Edges are characteristics of particular elements and their energy and height can be used for elemental analysis.
In the case of interband transitions which consist mainly of plasmon excitations, the probability of diffusion for volume losses is directly linked to the EELS function. We can calculate the EELS spectrum of the following relation:
The EELS function is an important factor that describes the energy loss of a fast electron passing through the material. In this case, we represent the EELS function for an energy interval [0 - 30 eV].
The variation of the EELS function is shown in the following figure.
The peaks correspond to the plasma resonance and the corresponding frequency is called the plasma frequency ωp. The resonant energy loss occurs at an energy of about 19 eV for all directions. The figure shows that the energy loss is the same for all directions.
My name is Dr Abderrahmane Reggad. I'm a 54 year old and I am a researcher in materials' sciences. 
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