In formula (3) it is assumed $rot\text{\hspace{0.05em}}rot\text{\hspace{0.05em}}E=grad\text{\hspace{0.05em}}div\text{\hspace{0.05em}}E-{\nabla}^{2}E$ , and $grad\text{\hspace{0.05em}}div\text{\hspace{0.05em}}E=0$ . Thus, we can have several solutions for (3). The most acceptable and appropriate solution of (3) is to obtain an equation for the intensity of the electric field in the form

$E={E}_{0}\mathrm{exp}i\omega \left(t-\frac{z}{v}\right).$ (4)

and this expression is a wave spreading in the z direction with the velocity v and angular frequency ω. [8] . The same can be obtained for the magnetic field strength. In order to obtain (4) as the final equation it is necessary to fulfill the conditions of (3) corresponding to the integrated complex index of refraction

${v}^{-2}=\mu {\epsilon}_{1}{\mu}_{0}{\epsilon}_{0}-i\frac{\sigma {\mu}_{0}\mu}{\omega}$ (5)

In general, to satisfy Equation (4), it is better to take into account that firstly, the square of the propagation speed of light in a vacuum with ${\u0441}^{2}={\left({\mu}_{0}{\epsilon}_{0}\right)}^{-1}$ and secondly, in the optical range of most semiconductors, in particular silicon, the magnetic properties are weak. It can be taken as $\mu \approx 1$ . Also, taking into account the rate between the refractive index and the absorption index and the dielectric constant, the specific electrical conductivity can be written as

$\epsilon ={\left(n-ik\right)}^{2}={\epsilon}_{1}-i\frac{\sigma}{{\epsilon}_{0}\omega}={\epsilon}_{1}-i{\epsilon}_{2}$ (6)

In (6) the speed of light spreading is taken ${c}^{2}={\left({\mu}_{0}{\epsilon}_{0}\right)}^{-1}$ . The expression for complex refraction, after separation of the real and imaginary parts, can be written in the following form of the system of equations

$\begin{array}{l}{n}^{2}-{k}^{2}={\epsilon}_{1}\left(\omega \right),\\ 2nk=\frac{\sigma}{{\epsilon}_{0}\omega}={\epsilon}_{2}\left(\omega \right).\end{array}\}$ (7)

Here ε(ω) is the complex dielectric constant, ε_{1}(ω) is the real part, and ε_{2}(ω) is the coefficient of the imaginary part.

On the one hand, the rate between the complex refractive index and (6), as well as the dielectric permittivity on the other hand allows us to conclude that the optical constants of the substance are equivalent in their generality. This covers the macroscopic parameters characterizing the interaction of the electromagnetic wave and the absorbing medium.

${n}_{*}=n-ik$ (8)

It was shown that, based on the causality condition, one can write formulas connecting n and k with each other [9]

$n=1+\frac{2}{\text{\pi}}{\displaystyle {\int}_{0}^{\infty}\frac{\omega k}{{\omega}^{2}-{\omega}_{0}^{2}}\text{d}\omega}$ (9)

$k=-\frac{2{\omega}_{0}}{\text{\pi}}{\displaystyle {\int}_{0}^{\infty}\frac{n}{{\omega}^{2}-{\omega}_{0}^{2}}\text{d}\omega}$ . (10)

The Equation (9) in our case is basic and important, because according to this expression n can be calculated at any frequency ω_{0} in the interval from zero to infinity, and therefore, based on the selected selective absorption spectrum, the spectrum of the refractive index can be calculated and vice versa. So, the above method allows us to write relationships that connect
${n}^{2}-{k}^{2}={\epsilon}_{1}\left(\omega \right)$
and
$2nk={\epsilon}_{2}\left(\omega \right)$
:

${n}_{1}\left({\omega}_{0}\right)={\left({n}^{2}-{k}^{2}\right)}_{{\omega}_{0}}=1+\frac{2}{\text{\pi}}{\displaystyle {\int}_{0}^{\infty}\frac{\omega {\epsilon}_{2}\left(\omega \right)}{{\omega}^{2}-{\omega}_{0}^{2}}\text{d}\omega},$ (11)

${\epsilon}_{2}\left({\omega}_{0}\right)={\left(2nk\right)}_{{\omega}_{0}}=-\frac{2{\omega}_{0}}{\text{\pi}}{\displaystyle {\int}_{0}^{\infty}\frac{\omega {\epsilon}_{2}\left(\omega \right)}{{\omega}^{2}-{\omega}_{0}^{2}}\text{d}\omega},$ (12)

Thus, to the wave Equation (4) substituting (5) and (8) it turns out

$E={E}_{0}\mathrm{exp}\left(-\frac{\omega kz}{c}\right)\mathrm{exp}i\omega \left(t\frac{nz}{c}\right).$ (13)

This shows that the main absorption coefficient k characterizes the attenuation of an electromagnetic wave in a semiconductor.

As for the semiconductor band structure, for a start, we can consider the simplest band structure. It is a spherical non-degenerate and energetic surface for electrons and holes with a quadratic dispersion law. Analytically such an energy structure can be expressed by the equation

$\left(E-\frac{{\hslash}^{2}{k}^{2}}{2{m}_{n}}\right)\left(E+{E}_{g}+\frac{{\hslash}^{2}{k}^{2}}{2{m}_{p}}\right)=0.$ (14)

where $\hslash $ is Planck’s constant divided by 2π. The value of $\hslash =1.05\times {10}^{-34}\text{\hspace{0.17em}}\text{J}\cdot \mathrm{sec}$ .

The roots of Equation (14) are

${E}_{c}=\frac{{\hslash}^{2}{k}^{2}}{2{m}_{n}};\text{\hspace{0.17em}}\text{\hspace{0.17em}}{E}_{v}=-{E}_{g}-\frac{{\hslash}^{2}{k}^{2}}{2{m}_{p}};$ (15)

However, the silicon diagram of the conduction band differs from the simplest band structure [10] [11] . The difference in band structures exists even with germanium. In silicon, the absolute minimum is located at a different brillouin zone in accordance with a different number of ellipsoids (valleys) in the zone. In addition, the quantitative rates between the energies at some points are significantly different.

The dependences of the energy on the wave number near the absolute minimum for the case when the coordinate axes coincide with the direction of the main axes of the mass ellipsoids for multi-valley semiconductors can be represented as

$E\left(k\right)=\frac{{\hslash}^{2}}{2}\left(\frac{{k}_{x}^{2}}{{m}_{l}}+\frac{{k}_{y}^{2}+{k}_{z}^{2}}{{m}_{t}}\right),$ (16)

where m_{l} and m_{t} are the longitudinal and transverse components of the mass ellipsoid.

3. The Design of a Selective Photo Thermo Generator with a Fixed Slit Trench

Selective photo thermo generator given in [1] energy performance exceeds FTG with equal numbers of cascades [12] . However, when using them as power plants of even low power, there are some problems associated with cluttering up their overall structure. Because the presence of lenses, other optical glasses and their attachment mechanisms, to some extent obscures the front surface of converters. In addition, the choice of photoactive radiation from the total illumination does not guarantee a clear distribution of light exactly into two parts, which without difficulty will be possible to direct to the surface of the photoconverter and the thermal converter.

For a positive solution to this problem, design of a photo thermo generator with a fixed gap was designed by us (Figure 1).

Optical Diagram of the Position Selective Photoconverter

This design provides for the arrangement of silicon photoelectric and thermoelectric converters based on semiconductor materials from the ternary compounds BiTeSe (n-type) and BiTeSb (p-type) so that, depending on the photo-activity of the FEP on a certain radiation spectrum, after the dispersion of the light beam into the optical an axis is placed on a photoconverter, then a TEP is mounted around the FEP. This method of installation and connection is almost always necessary. This is explained by the following. First, the light passing through the optical system distributes the color sequence based on the electrical and optical properties of the material of the optical glass, and secondly, there are still no specific research results recommending highly efficient separation of light into two parts; Often the required spectrum may be in the middle of divided colors

Figure 1. Design of a photo thermo generator with a fixed gap. a) 1-prism, 2-photoelectrical converter (PhC), 2-p- and n-branches of thermo converter (ThC), 3-hot switching plates (TEP), 4-cold switching plates of (TEP), b, c, d: Accordingly, three variants of arrangement of (FB) and (TB) on section A-a: I, II, III-Variant.

Thirdly, a solar cell made of the same semiconductor material with different technologies has a spectral sensitivity in different color areas [13] . Fourth, the solar cell is not always made of silicon. It can be made of other semiconductor materials both by homo and heterostructure. It can be seen from the figure that the previously measured and determined photoactive radiation through a special slit of increasing light loss, the slit through which the selected radiation passes, must be strictly in terms of the geometric dimensions of the FEP surface. A thermoelectric converter is placed around the photoelectric converter, depending on the location of other radiations [14] . In general, it is possible to assemble both converters on the same plane. At the same time, according to the results of theoretical studies, the coefficient of absorption and refraction will obey the well-known laws of physics.

The front surfaces of the converter can be assembled with different configurations (see Figure 1, I, II, III options). It depends on the geometric shape of the photoelectric converter and the size of the area of the divided light spot [15] . Attaching the switching plates of the hot junctions of the thermocouple into one single ring-shaped or right-angled common surface (for example, from the beryllium oxide material BeO, since the temperature coefficients of expansion of BeO and the hot switching plates of TE are close).

4. Experimental Study

Testing the practicality of using a selective photo thermo generator of light radiation showed that the introduction of this design will lead to simplification of assembly and assembly work in the construction of power plants. The loss of light energy due to the space between the converters is about 3% (for the case of covering the surface of a thermal converter of a continuous heat-receiving BeO). The efficiency of a thermoelectric converter is almost equal to the values given in domestic and foreign literature [16] , that is, when the temperature of the cold junction is equal to the ambient temperature T_{x} = 20˚C and hot junction Т_{г} = 1200С η = 6.0%. The temperature of the photoconverter is practically unchanged [17] . When using twenty-fold solar radiation, it rose only 1.10С. This can apparently be explained by the error and measurement of the light intensity during photometry and the lack of a clear boundary between the spectral wavelengths.

5. Conclusion

Based on the scientific and theoretical design decisions, it can be concluded that the photo thermoelectric converter of light energy into electrical energy with a fixed slit trench is one of the original technical solutions in the use of alternative sources of electrical energy. It allows to use photo converter and thermo converter from a semiconductor of any structure with least radiation losses and prevention of blockage makes it an efficient energy source. It also would lead to simplification of assembly and its work in the construction of power plants. Of course, the use of relatively efficient converters leads to the most tangible results.

Acknowledgements

The author would like to acknowledge Mr. Muhammad Ibrahim and Mrs. Riffat Tahira for their guidance and support, also, would like to thank Miss Nayab Fatima Noor for technical support.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

Cite this paper

Kasimakhunova, A.M., Olimov, Sh.A., Mamadalieva, L.K., Norbutaev, M., Nazirjanova, Sh.S. and Laraib, S.R. (2019) Photo Thermal Generator of Selective Radiation Structural and Energetic Features. Journal of Applied Mathematics and Physics, 7, 1263-1271. https://doi.org/10.4236/jamp.2019.76086

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