Thickness Of SiO2 Layer Research Articles

Large-scale high purity and brightness structural color generation in layered thin film structures via coupled cavity resonance

Abstract Structural colors, resulting from the interaction of light with nanostructured materials rather than pigments, present a promising avenue for diverse applications ranging from ink-free printing to optical anti-counterfeiting. Achieving structural colors with high purity and brightness over large areas and at low costs is beneficial for many practical applications, but still remains a challenge for current designs. Here, we introduce a novel approach to realizing large-scale structural colors in layered thin film structures that are characterized by both high brightness and purity. Unlike conventional designs relying on single Fabry–Pérot cavity resonance, our method leverages coupled resonance between adjacent cavities to achieve sharp and intense transmission peaks with significantly suppressed sideband intensity. We demonstrate this approach by designing and experimentally validating transmission-type red, green, and blue colors using an Ag/SiO2/Ag/SiO2/Ag configuration on fused silica substrate. The measured spectra exhibit narrow resonant linewidths (full width at half maximum ∼60 nm), high peak efficiencies (>40 %), and well-suppressed sideband intensities (∼0 %). In addition, the generated color can be easily tuned by adjusting the thickness of SiO2 layer, and the associated color gamut coverage shows a wider range than many existing standards. Moreover, the proposed design method is versatile and compatible with various choices of dielectric and metallic layers. For instance, we demonstrate the production of angle-robust structural colors by utilizing high-index Ta2O5 as the dielectric layer. Finally, we showcase a series of printed color images based on the proposed structures. The coupled-cavity-resonance architecture presented here successfully mitigates the trade-off between color brightness and purity in conventional layered thin film structures and provides a novel and cost-effective route towards the realization of large-scale and high-performance structural colors.

Simulation of silicon conical field effect GAA nanotransistors with stack SiO2/HfO2 dielectric of gate

The issues of modeling the electrophysical characteristics of a silicon conical field effect GAA nanotransistor are discussed. An analytical model of the drain current of a transistor with a fully enclosing conical gate with a stack sub-gate oxide SiO2/HfO2 has been developed, taking into account the effect of the charge of the interphase trap at the Si/SiO2 interface. To simulate the potential distribution in a conical working area under the condition of constant trap density, an analytical solution of the Poisson equation was obtained using the method of parabolic approximation in a cylindrical coordinate system with appropriate boundary conditions. The potential model was used to develop an expression for the GAA drain current of a nanotransistor with a stack gate oxide. The key electrophysical characteristics are numerically investigated depending on the density of traps and the thicknesses of SiO2 and HfO2 layers.

Towards water-resistant perovskite solar cells: Electron-beam deposited SiO2 layers for highly stable perovskite solar cells

Metal halide perovskite solar cells (PSCs) incorporating organic–inorganic compounds exhibit promising potential for future photovoltaic devices. One of the main obstacles to commercializing perovskite photovoltaic technology is oxygen and water-induced degradation in ambient conditions. Herein, we demonstrate a general approach to protect PSCs from water-induced degradation using electron-beam evaporated silicon dioxide (SiO2) barrier coating. A number of small (0.25 cm2) and large-area (2.0 cm2) PSCs have been fabricated to obtain statistically significant data. An optimal 1-μm thick SiO2 layer coated over RbCsMAFAPb(IBr)3-PSCs provided remarkable protection even after fully immersing the devices in a water-filled container. After immersing the PSC in water for 1 min, the device efficiency showed only a small reduction in efficiency from 14.3 % to 12.0 %, highlighting the excellent water protection offered by the barrier layer. We show that SiO2 barrier layers can improve the overall stability of the cells. Perovskite devices coated with 1 µm-SiO2 protective layers maintained more than 85 % of their initial power conversion efficiency (PCE) values after 2000 h in ambient storage. The devices coated with the optimum SiO2 layer thickness of 1 µm also showed excellent stability during outdoor sunlight testing for 2000 h, maintaining ∼ 80 % of their initial efficiency values.

Communication—A Powerful Method to Improve Dielectric/GaN Interface Properties: A Dummy SiO2 Process

We report a simple and effective method for improving dielectric/GaN interface properties. In the process, a 5 nm thick SiO2 layer was deposited onto a GaN(0001) substrate via plasma-enhanced atomic layer deposition, followed by annealing at 800 °C for 300 s under a flowing N2 atmosphere. The SiO2 layer was then removed using buffered HF solution, and Pt/Al2O3/GaN metal-oxide-semiconductor capacitors were fabricated on the substrate. Positive-bias stress tests revealed that the flat-band voltage shifts were substantially reduced for devices fabricated using this process, probably because of improved interface crystallinity. This method can also be applied to other dielectric/GaN systems.

Radical, ion, and photon’s effects on defect generation at SiO2/Si interface during plasma etching

Defect generation and recovery at the SiO2/Si interface are experimentally studied during SiO2 etching and annealing. The SiO2 etching over the underlaying Si is performed with CF4 plasma, where the remaining SiO2 layer thickness is varied to clarify the effects of radicals, ions, and photons on the defect generation. The defects are generated initially by photon irradiation, and then generated by ion bombardment/penetration, as the remaining layer becomes thin. Around the endpoint, defects are also generated by radical reactions and in-diffusion. The photon-mediated defects are fully recovered by annealing, whereas the ion and radical-mediated defects are not sufficiently recovered; residual defects are created.

Effect of SiO2 Layer Thickness on SiO2/Si3N4 Multilayered Thin Films

Silicon nitride (Si3N4) materials are widely used in the electronics, optoelectronics, and semiconductor industries, with Si3N4 thin films exhibiting high densities, high dielectric constants, good insulation performance, and good thermal and chemical stability. However, direct deposition of Si3N4 thin films on glass can result in considerable tensile stress and cracking. In this study, magnetron sputtering was used to deposit a Si3N4 thin film on glass, and a silicon dioxide (SiO2) thin film was introduced to reduce the Si3N4 binding force and stress. The effect of the SiO2 layer thickness on the SiO2/Si3N4 multilayered thin film was explored. The results indicated that the introduction of the SiO2 layer could improve the weak adhesion characteristics of Si3N4 thin films. Moreover, sputtering the SiO2 layer to 150 nm resulted in the highest hardness and transmittance of the SiO2/Si3N4 multilayered thin films. The findings of this study lay a solid foundation for the application of Si3N4 thin films on glass.

In-situ synthesis of SiC/SiO2 nanowires by catalyst-free thermal evaporation of silicon powder and their photoluminescence properties

Core-shell structured SiC/SiO2 nanowires (SiC/SiO2 NWs) were synthesized by a simple thermal evaporation method without a catalyst. The influence of the growth temperature and holding time on the microstructure and composition of the nanowires were investigated. The results indicate that within the temperature range of 1150 to 1550 °C, the color of SiC/SiO2 nanowires gradually transitions from pure white to off-white with increasing temperature. During this process, the average diameter of SiC/SiO2 increases from ∼74 nm to ∼291 nm, while the surface SiO2 layer thickness decreases from ∼17 nm to ∼5 nm, with nanowires can reach lengths of hundreds of micrometres. Variations in supersaturation of SiO and CO at different temperatures lead to discrepancies in nanowire diameter and SiO2 shell thickness. At 1350 °C, prolonged holding time enhances silicon powder reaction, resulting in increased structural disorder of the obtained SiC/SiO2 nanowires. The nanowire growth follows a vapor-solid (VS) mechanism. Additionally, the study demonstrates that the photoluminescence (PL) characteristics of the nanowires exhibit a blue shift attributable to their nanoscale dimensions, while variations in emission spectra correlate with stacking faults, SiC diameter, and SiO2 thickness. This work may be provide a theoretical basis for preparing SiC/SiO2 nanowires with different diameters and SiO2 shell thicknesses, which is crucial for advancing the development and application of SiC/SiO2 nanoceramics.

Ultrathin Graphite‐Based Full‐Color Electrochromic Devices

AbstractUltrathin graphite‐based materials have revolutionized optoelectronics through their multispectral and energy‐efficient tunability. However, their narrow color range poses a challenge in applications requiring a broad spectrum of colors for effective information delivery or aesthetic enhancement. Here, full‐color tunability is achieved in ultrathin graphite‐based electrochromic devices using Fabry‐Perot (F‐P) nanocavities. By adjusting the optical properties (n, k) of ultrathin graphite through lithium‐ion intercalation/de‐intercalation and the thickness of the dielectronic SiO2 layer, managed to control the optical reflectivity in the visible spectrum. Moreover, this prototype device consumes only 1.59 mW cm−2, just one‐tenth of the commercial organic light‐emitting displays’ energy usage. Furthermore, the pixel size in the Fabry‐Perot nanocavity‐type electrodes can be reduced to 2 µm, under half that of contemporary displays like Micro‐LED. These results pave the way for full‐color displays with minimal energy consumption and inspire extensive research in low‐power photonics.

Low-oxygen silicon preparation using vacuum treatment by oxidation control and moisture removal kinetics investigation

Diamond wire saw silicon powder (DWSSP) is a waste produced during silicon wafer processing for photovoltaic (PV) modules, and it shows significant reutilization potential. However, the presence of moisture leads to silicon oxidation and the formation of a surface SiO2 layer that hinders the recovery of high-purity silicon. In this study, vacuum treatment (VT) was used to prepare low-oxygen silicon by controlling the oxidation and removing moisture from DWSSP waste, thus preventing SiO2 layer formation. The oxygen content in DWSSP decreased by 15.01 %, from 7.26 % to 6.17 %, and the SiO2 layer thickness also decreased by 10.29 %, from 2.04 nm to 1.83 nm after the VT process. Additionally, the VT process lowered the moisture boiling point, facilitating surface evaporation and internal diffusion of moisture. This increased the moisture effective diffusion coefficient from 5.16 × 10−10–6.16 × 10−9 m2 s−1 to 1.21 × 10−9–6.88 × 10−9 m2 s−1 and reduced the apparent activation energy from 32.17 kJ mol−1 to 16.05 kJ mol−1. The efficient moisture removal of VT inhibited the growth of the SiO2 layer to enable the preparation of low-oxygen silicon during silicon recovery.

Modeling and Design Parameter Optimization to Improve the Sensitivity of a Bimorph Polysilicon-Based MEMS Sensor for Helium Detection.

Helium is integral in several industries, including nuclear waste management and semiconductors. Thus, developing a sensing method for detecting helium is essential to ensure the proper operation of such facilities. Several approaches can be used for helium detection, including based on the high thermal conductivity of helium, which is several times higher than air. This work utilizes the high thermal conductivity of helium to design and analyze a bimorph MEMS sensor for helium sensing applications. COMSOL Multiphysics software (version 6.2) is used to carry out this investigation. The sensor is constructed from poly-silicon and SiO2 materials with a trenched cantilever beam configuration. The sensor is electrically heated, and its morphed displacement depends on the surrounding gas's composition, which decreases in the presence of helium. Several factors were investigated to probe their effect on the sensor's sensitivity to helium, including the thickness of the poly-silicon layer, the configuration of the trench, and the thickness and location of SiO2 layer. The simulations showed that the best performance, up to 2 ppm helium detection level, can be achieved with thinner beams and medium trench lengths.

Influence of CoSi Oxidation and Passivation During Silicide Selective Etching on Junction Leakage: Applications to Schottky Diodes Devices

For new analogic microelectronic circuits development based on non-linear devices such as Schottky diodes formed in Si active regions, new Co-silicide integrations are required to reduce junction leakages. To gather targeted device requirements, precise Co silicide/Si interface optimization and a limited silicide formation at the active edges is needed. The selective etching during the “Salicide” process plays a real role in the oxidation and/or passivation of the silicide layer. Here, we propose a systematic study including a very large spectrum of experiments around the main parameters of CoSi selective etching. The main conclusions are (1) diode leakages are directly linked to SiO2 layer thickness formed during the SC1 dispense or by air exposure over the CoSi layer, (2) significant effect of dispense flow on SiO2 formation is measured through X-ray photoelectron spectroscopy and time-of-flight secondary ion mass spectrometry characterizations; (3) optimized diode leakages together with contact resistances are then demonstrated for low SC1 delivery flow and long dispense time; and (4) major changes in final CoSi2 layer morphology and silicide/silicon interface are observed by transmission electron microscopy-energy-dispersive X-ray analyses for different selective etching processes, which are potentially explained by enrichment in Co atoms at CoSi/Si during SiO2 overlayer growth.

O2 and Ar plasma processing over SiO2/Si stack: Effects of processing gas on interface defect generation and recovery

Defect generation and recovery at the interface of a silicon dioxide/silicon (SiO2/Si) stack are studied in oxygen (O2) or argon (Ar) plasma processing and post-annealing. Defect generation is recognized to be dependent on the processing gas and the SiO2 layer thickness. O2 plasma processing shows a strong incident-ion energy dependence, where ion’s implantation, diffusion, and reactions in the SiO2 layer play important roles in defect generation. A similar dependence is observed for Ar plasma processing; however, it also shows the photon effects in defect generation for a thick SiO2 layer. Defect recovery is demonstrated by annealing, where recovery depends on the annealing temperature as well as the amount of defects generated at the interface.

Reactively-sputtered super-hydrophilic ultra-thin titania films deposited at 120 °C

Abstract We investigate super-hydrophilic TiO2 (titania) films for concentrated solar-thermal power applications. Reactive magnetron sputtering has been used to deposit 8 to 12 nm thick titania thin films onto borosilicate microscope glass slides, low-Fe extra-clear architectural glass, or Si(100) wafers with a 500 nm thick thermal SiO2 layer. The effects of deposition temperature and O2 fraction of the O2/Ar working gas were investigated. We demonstrate the importance of the O2 fraction for obtaining optically transparent, super-hydrophilic (contact angle below 1°) thin films. In particular, we show that as the O2 fraction increases, contact angle decreases, obtaining super-hydrophilic titania thin films at deposition temperatures as low as 120 °C. Our work enables to develop low thermal budget cost-efficient industrial synthesis processes, paving the way for commercial applications.

Dual-mode photoluminescent microspheres with silica core, separator layer and envelope for anti-counterfeiting ink

The microsphere is concentrically constructed on the basis of SiO2 nanocore on which Gd2O3:Yb,Er upconversion luminescent layer, SiO2 separator, Gd2O3:Ln@SiO2 (Ln3+ = Tb3+, Eu3+) downconversion luminescent layer, and SiO2 envelope are successively deposited by means of wet chemistry method. Under 980 nm near-infrared laser excitation, the microspheres emit bright upconverted green light due to Gd2O3:Yb,Er luminescent layer. While, under 254 nm portable ultraviolet lamp irradiation, they produce red and green downconversion fluorescence because of Gd2O3:Ln (Ln3+ = Tb3+ or Eu3+) luminescent layer. The silica core fixes the luminescence center, reduces non radiative transitions caused by molecular vibration, enhances fluorescence intensity, and saves rare earth resources. Impressively, the silica separator layer isolates the upconversion luminescence of Er3+ and downconversion of Eu3+, preventing cross relaxation and quenching of Er3+ fluorescence. The silica envelope plays a crucial role in the photoluminescence of the prepared microspheres. The photoluminescence quantum yield of the microspheres initially increases from 90.4 % to 95.9 % and then decreases to 82.9 % as the thickness of SiO2 layer on the ∼3 nm-thick Gd2O3:Ln shell rises from 3 nm to 16 nm. The absorption of photons reaches its maximum when the SiO2 layer is 8 nm thick. This thickness corresponds to the lowest rate constant for non-radiative transitions. The proto inks prepared from the dual-mode microspheres exhibit good performance including good luminescence and high dispersibility for practical anti-counterfeiting applications.

Study on the electrical breakdown failure mode transition of TSV-RDL

This study investigates the electrical breakdown failure modes and mechanisms of TSV-RDL (Through-Silicon Via with Redistribution Layer) structures through experiments and finite element simulation analysis. By employing a Taguchi method-based finite element simulation, the impact of structural parameters on the electrical breakdown failure modes of TSV-RDLs has been explored. The study reveals two electrically induced breakdown failure modes for TSV-RDLs. One case involves failures occurring at the RDL_pad/SiO2 interface, while the other involves failures within the SiO2 layer along the TSV sidewall interface. The design of TSV-RDLs can influence the distribution of electric field strength and subsequently lead to different electrical breakdown failure modes. In the case of double-blind solid TSV-RDLs, double-through solid TSV-RDLs, and double-blind hollow TSV-RDLs, the pad pitch is identified as a crucial parameter that affects their failure modes. As the pad pitch increases, the location of electrical breakdown failure shifts from the RDL_pad/SiO2 interface to the SiO2 layer along the TSV sidewall interface. Conversely, for the double-through hollow TSV-RDL structure, the structural parameters of TSV, including TSV height, TSV pitch, and SiO2 layer thickness, are the principal factors influencing its failure modes. As these parameters decrease, the position of electrical breakdown failure transitions from the RDL_pad/SiO2 interface to the SiO2 layer along the TSV interface.

Effect of SiO2 gate insulator on electrical performance of solution processed SnO2-Based thin film phototransistor

SnO2-based TFTs were fabricated by using the sol-gel spin coating method. SnO2 semiconductor films used as active layers were coated on oxidized Si with different SiO2 thicknesses. AFM was used to analyze the surface morphology. The oxidation states of SnO2 film for Sn3d and O1s were determined by XPS. The parameters demonstrating the electrical performance of a transistor such as mobility (μ), threshold voltage (Vth) and sub-threshold swing (SS) have been determined depending on the thickness of the dielectric layer. The thickness of the dielectric layer played an important role in the device's performance. The highest mobility and Ion/off values were obtained as 7.1 cm2/V.s and 6.98x104 for TFT fabricated with a 150 nm thick SiO2 layer. All the TFTs exhibited photosensing behavior at different light intensities and these results are an indicator that the SnO2 TFT can be effectively utilized in visible photo-detecting device applications.

The effect of SiO2 interlayer on magnetic interactions in multilayer composites ribbons

This article studies the influence of dipole interactions on the magnetic properties. Specifically, FINEMET/SiO2 and FINEMET/SiO2/Fe20Ni80 composite ribbons were prepared by magnetron sputtering. During the manufacturing process, the Fe20Ni80 film with a fixed thickness of 260 nm and the FINEMET ribbon were separated by the SiO2 layer, and the SiO2 layer thickness could be adjusted within a range of 0–150 nm. This configuration enabled the exchange coupling gradually reduced with the SiO2 layer thickness increased, allowing dipole interactions to become the dominant factor. This study not only excludes the influence of the SiO2 in the composite ribbons but analyzes the effect of the exchange couple and dipole interactions for the GMI, respectively. The experimental results show the GMI ratio decreases initially and then increases with the thickness of SiO2, where the optimization of GMI ratio was explored in FINEMET/Fe20Ni80 ribbons. On the other hand, the minimum peak field was shown with the SiO2 thickness of 5 nm. Furthermore, the results also elucidate that both dipole interactions and exchange couple have affected the peak field of GMI, which could be used for the design of multilayer composite ribbons with better performance. Thus, the analysis we showed in this work is thought to be valuable in revealing the mechanism of multilayer composite ribbons for future sensing applications.

From design to realization of high diffraction efficiency Littrow configuration diffraction gratings based on multilayer dielectric mirrors patterned with electron beam lithography

Diffraction efficiency (DE) and laser-induced damage threshold of Chirped pulse amplification (CPA) system diffraction gratings (DGs) are one of the main limiting factors for the anticipated progress of high peak power lasers. Optimization of a high reflectivity dielectric multilayer ZrO2/SiO2 stack mirror with a DG etched in the top SiO2 layer was carried out employing the Rigorous coupled wave analysis method. Two families of solutions possessing DE > 0.99 over the bandwidth of the 1030 nm wavelength ultrashort laser and withstanding different fabrication margins were identified for the Littrow configuration. DGs were produced and verified experimentally. Structures permitting variation in both the depth of the DG and filling factor without loss in DE were obtained for the 690 nm thick SiO2 layer. While the 770 nm thick layer permitted separation of the two parameters requiring either rigorous control of the DG depth or filling factor. DG realization technology was proposed employing electron beam lithography and inductively coupled plasma reactive ion etching. Measurements of the fabricated structures demonstrated 0.91 DE for the first diffraction order over the 60 nm bandwidth and under a 6° mounting error from the Littrow configuration. Spectral angular reflectance DE simulations and respective measurements indicated quantitative agreement. The work provides an integrated overview of high DE periodic structure numerical optimization and manufacturing roadmap which could be applied for the high-peak-power laser CPA system development.

Effects of Thermal-Strain-Induced Atomic Intermixing on the Interfacial and Photoluminescence Properties of InGaAs/AlGaAs Multiple Quantum Wells.

Quantum-well intermixing (QWI) technology is commonly considered as an effective methodology to tune the post-growth bandgap energy of semiconductor composites for electronic applications in diode lasers and photonic integrated devices. However, the specific influencing mechanism of the interfacial strain introduced by the dielectric-layer-modulated multiple quantum well (MQW) structures on the photoluminescence (PL) property and interfacial quality still remains unclear. Therefore, in the present study, different thicknesses of SiO2-layer samples were coated and then annealed under high temperature to introduce interfacial strain and enhance atomic interdiffusion at the barrier-well interfaces. Based on the optical and microstructural experimental test results, it was found that the SiO2 capping thickness played a positive role in driving the blueshift of the PL peak, leading to a widely tunable PL emission for post-growth MQWs. After annealing, the blueshift in the InGaAs/AlGaAs MQW structures was found to increase with increased thickness of the SiO2 layer, and the largest blueshift of 30 eV was obtained in the sample covered with a 600 nm thick SiO2 layer that was annealed at 850 °C for 180 s. Additionally, significant well-width fluctuations were observed at the MQW interface after intermixing, due to the interfacial strain introduced by the thermal mismatch between SiO2 and GaAs, which enhanced the inhomogeneous diffusion rate of interfacial atoms. Thus, it can be demonstrated that the introduction of appropriate interfacial strain in the QWI process is of great significance for the regulation of MQW band structure as well as the control of interfacial quality.

(Digital Presentation) Internal Friction, Mechanical Spectroscopy in SiO2/Si Wafers

Introduction The electrophysical properties of semiconductor devices significantly depend on the perfection of their structure. It is necessary to control the structure of semiconductor wafer-plates disks in the manufacture of large integrated circuits (LIC). Technique was developed and device was made that allows to obtain the information about structural defects in such disks by attenuating elastic oscillations Q-1 to study the defects density nd and the depth hd of their propagation. Defect annealing leads to change in the shape of internal friction (IF) temperature spectrum Q-1(T) [1]. IF method allows to set the spectrum of structural defects on the analysis of positions of maximums IF, on duration of relaxation time τ and on their deposit in attenuation of elastic vibrations [2]. A non-destructive method for the technological control of the structure defects by measuring internal friction (IF) and elastic modulus E after laser radiation was developed. Experimental procedure Ultrasonic (US) pulse-phase method using USMV-LETI, modernized USMV-KNU and computerized “KERN-4” with frequencies f║ ≈ 1 MHz and f┴ ≈ 0,7 MHz, US invariant-polarization method for determining the effective acoustic μil and elastic constants Cijkl were used [3,4,5]. The measured velocity error was equal to ΔV/V = 0.5÷1,5%. The study of influence of structure defects on damping of vibrations in Si/SiO2 wafer-plates by the diameter of D = 100÷60 mm and by the thickness of hSiO2 ≈ 600 nm, hSi ≈ 470 000 nm, allows to estimate the degree of perfection of crystalline structure. Metallography optical supervision of microstructure by means of the microscope ”LOMO MVT”, atomic-force microscopy (AFM) were used. Results and discussion The quasi-longitudinal US velocity V║[001] = 5870 m/s, elastic modulus E001 = ρV║[001] 2 = 80,28 GPa for SiO2/Si from the oscillogram were determined. Temperature dependence of internal friction Q-1(T) in SiO2/Si wafer-plate p-type, doped with B, KDB-7.5(100) diameter D ≈ 76∙10-3 m, thickness hSi ≈ 460∙103 nm with SiO2 layer thickness hSiO2 ≈ 100 nm after X-ray irradiation with dose Dγ ≈ 102 Gy is showed. Conclusions It was found that as the result of the structural defect annealing IF background Q0 -1 significantly decreases during measuring of IF temperature dependence Q-1(T), which indicates the improvement of SiO2/Si crystal structure.The elastic modulus Е, the shear modulus G, Poisson coefficient μ, IF Q-1 are dependent from SiO2/Si wafer-plate anisotropy.The value of IF background Q-1 0 after temperature T, mechanical treatments describes the changes of the elastic stress σi fields in SiO2/Si wafer-plate.The study of vibrations of disk wafer-plate Si/SiO2 at different harmonic frequencies f0, f2 made it possible to develop the technique for determining the structural defects density nD for semiconductor wafers-substrates.The relationship between IF Q-1 value, the logarithmic decrement of ultrasound damping δ and the dislocations density nD was established for disk-shaped semiconductor wafers-substrates. Acknowledgements This work has been supported by Ministry of Education and Science of Ukraine: Grant of the Ministry of Education and Science of Ukraine for perspective development of a scientific direction "Mathematical sciences and natural sciences" at Taras Shevchenko National University of Kyiv.