Under consideration is a Sturm–Liouville equation with a piecewise entire potential and discontinuity conditions independent of the spectral parameter for the solutions on an unspecified rectifiable curve lying in the complex plane. We study an inverse spectral problem with respect to the ratio of elements of one column or one row of the transfer matrix and give the conditions of uniqueness of a solution. These results are applied to the inverse problem for the Sturm–Liouville equation with piecewise constant complex weight, piecewise entire potential, and discontinuity conditions on a segment.

For the Sturm–Liouville equation of standard form on the complex plane, we study theexistence of potentials with monodromy-quasifree singular points, i.e., singular points such thatsome power of the monodromy matrix $$M$$ is independent of thespectral parameter and equal to $$\pm I$$ , where $$I$$ is the identity matrix. For the matrix $$M$$ and its trace, we state necessary and sufficientconditions for the singular point of the potential to be monodromy-quasifree. Examples ofpotentials with such singular points, including branching points, are given.

A Sturm-Liouville equation with a piecewise entire potential and a non-zero piecewise constant weight function on a curve of an arbitrary shape lying on the complex plane is considered. For such equation, the inverse spectral problem is posed with respect to the ratio of elements of one column or one row of the transfer matrix along the curve. The uniqueness of the solution to the problem is proved with the help of the method of unit transfer matrix using the study of asymptotic solutions of the Sturm-Liouville equation for large values of the absolute value of the spectral parameter. The obtained results allowed to consider inverse problem for a previously unexplored class of Sturm-Liouville equations with three unknown coefficients on a segment of the real axis.

Высокая плотность структурных дефектов является основной проблемой при изготовлении электроники на гетероструктурах «кремний на сапфире» (КНС). Современный метод получения ультратонких структур КНС с помощью твердофазной эпитаксиальной рекристаллизации позволяет значительно снизить дефектность в гетероэпитаксиальном слое КНС. В данной работе ультратонкие (100 нм) слои КНС были получены путем рекристаллизации и утонения субмикронных (300 нм) слоев кремния на сапфире, обладающих различным структурным качеством. Плотность структурных дефектов в слоях КНС оценивалась с помощью рентгеноструктурного анализа и просвечивающей электронной микроскопии. Кривые качания от дифракционного отражения Si(400), полученные в ω-геометрии, продемонстрировали максимальную ширину на полувысоте пика не более 0,19-0,20° для ультратонких слоев КНС толщиной 100 нм. Формирование структурно совершенного субмикронного слоя КНС 300 нм на этапе газофазной эпитаксии обеспечивает существенное уменьшение плотности дислокаций в ультратонком кремнии на сапфире до значений ~1 • 104 см-1. Тестовые n-канальные МОП-транзисторы на ультратонких структурах КНС характеризовались подвижностью носителей в канале 725 см2 Вс-1. The high density of structural defects is the main problem on the way to the production of electronics on silicon-on-sapphire (SOS) heteroepitaxial wafers. The modern method of obtaining ultrathin SOS wafers is solid-phase epitaxial recrystallization which can significantly reduce the density of defects in the SOS heteroepitaxial layers. In the current work, ultrathin (100 nm) SOS layers were obtained by recrystallization and thinning of submicron (300 nm) SOS layers, which have various structural quality. The density of structural defects in the layers was estimated by using XRD and TEM. Full width at half maximum of rocking curves (ω-geometry) was no more than 0.19-0.20° for 100 nm ultra-thin SOS layers. The structural quality of 300 nm submicron SOS layers, which were obtained by CVD, depends on dislocation density in 100 nm ultrathin layers. The dislocation density in ultrathin SOS layers was reduced by ~1 • 104 cm-1 due to the utilization of the submicron SOS with good crystal quality. Test n-channel MOS transistors based on ultra-thin SOS wafers were characterized by electron mobility in the channel 725 cm2 V-1 s-1.

The main technological problem in the manufacture of electronics on silicon-on-sapphire (SOS) structures is the high density of defects in silicon-on-sapphire layers. The modern method of obtaining ultrathin SOS structures using solid-phase epitaxial recrystallization and pyrogenic thinning can significantly reduce the defectiveness in these layers. Nevertheless, the effect of the defectiveness of submicron SOS layers on the structural perfection of ultrathin layers remains unclear. In this work, ultrathin (100 nm) SOS structures have been obtained on submicron (300 nm) SOS structures with different structural quality. The crystallinity of 300 nm layers before the recrystallization process and ultrathin layers has been determined using X-ray diffraction and transmission electron microscopy. It has been found that the lowest values of the full width at half maximum (FWHM) of 0.19°–0.20° have been observed for an ultrathin SOS structure obtained based on the most structurally perfect SOS layers of 300 nm. It has been shown that a more perfect near-surface layer of the basic SOS structure of 300 nm and a double implantation regime make it possible to reduce the density of structural defects in the ultrathin Si layer by an order of magnitude to achieve ~1 × 10 4 cm –1 .

The asymptotics of the transfer matrix of Sturm-Liouville equation with piecewise-entire potential function on a curve in the complex plane is obtained and studied for large absolute values of the spectral parameter.

The inverse problem for the standard Sturm - Liouville equation with a spectral parameter rho and a potential function, piecewise-entire on a rectifiable curve gamma subset of C, on which only the starting point is given, is studied for the first time. A function Q that is bounded on a curve gamma is piecewise-entire on it if gamma can be spited by a finite number of points into parts on which Q coincides with entire functions, different in neighboring parts. The split points, the initial and final points of the curve are called critical points. The problem is to find all the critical points of the curve gamma and the potential on it by the column or row of the transfer matrix (P) over cap along gamma. On the basis of the obtained asymptotics of matrix (P) over cap for vertical bar rho vertical bar -> infinity , it is proved that if at least one of its elements is bounded for for all rho is an element of C, then the curve & nbsp;gamma degenerates to a point after removing all "invisible loops". An "invisible loop" is a loop of the curve gamma (with a given piecewise-entire function) whose knot coincides with two successive critical points. The uniqueness of the solution of the inverse problem for curves without "invisible loops" is proved. On the example of the inverse problem for the equation d/dx (1/r(x) dy/dx) + (q(x) - r(x)lambda(2))y(x) =0 with a piecewise-entire function q(x) and a piecewise constant function r(x) not equal 0 on the segment of the real axis, the usefulness of the results obtained in the article is shown for the study of inverse problems for generalized Sturm - Liouville equations, which can be reduced to the type studied in the article.

The effect of the parameters (energy, dose) of the irradiation of silicon-on-sapphire (SOS) structures with ions Si+ ions on the quality of the silicon-film crystal structure after solid-phase epitaxial recrystallization and annealing is studied. It is shown that the most efficient mechanism of crystal-structure recovery is recrystallization from the silicon surface layer which is a seed.

A method for the unambiguous reconstruction of the spatial profiles of all components (except for χ zzz ) of the quadratic susceptibility complex tensor {ie165-2} (z, ω1 + ω2; ω1, ω2), which is responsible for the sumfrequency generation in a one-dimensionally inhomogeneous plate is proposed and proven for the first time. Such reconstruction is possible if the symmetry of the medium provides the diagonal character of the linear permittivity tensor {ie165-3} (z, ω). The procedure involves the measurement of the complex amplitude of the new wave with the frequency ω1 + ω2 that is reflected from the plate for a certain interval of the angles of incidence of the wave with the frequency ω2. The reflected wave results from the nonlinear interaction of the wave with frequency ω2 and the wave with frequency ω1 that exhibits the normal incidence. A similar approach can be used to determine the profiles of the components of the quadratic susceptibility tensor {ie165-4}(z, ω1 − ω2; ω1, − ω2), which is responsible for the difference-frequency generation.

This study focuses on a one-dimensionally inhomogeneous absorbing plane-parallel plate with dielectric properties characterized by a diagonal tensor of dielectric permittivity. The research has found that based on the reflection and transmission coefficients of p- and s-polarized plane monochromatic waves with frequency omega known for a certain range of incidence angles the coordinate dependence of all the components of the dielectric permittivity tensor of the plate material at frequency omega is determined in a unique manner according to the proposed reconstitution algorithm.

For a one-dimensionally inhomogeneous plate whose linear dielectric properties are also inhomogeneous and are characterized by a diagonal permittivity tensor, it is proved that the spatial profiles of all components of the complex quadratic susceptibility tensor \(\hat \chi ^{(2)} \)(z, 2ω; ω, ω responsible for the second harmonic generation can be uniquely determined, and an appropriate method is proposed. To implement this method, one should measure the complex coefficient of transformation of a fundamental plane wave incident on the plate into a reflected second-harmonic wave in a certain range of angles of incidence. By varying the plane of incidence of the wave and (or) its polarization and measuring the coefficients of transformation into s- and p-polarized waves of double frequency, one can uniquely determine the spatial profiles of all components of the quadratic susceptibility tensor. The method involves the measurement of the intensities of the second-harmonic waves generated under special conditions with the use of two auxiliary reference plates and thus allows one to avoid complicated phase measurements.

A method for the unambiguous reconstruction of the spatial profiles of all components (except for χ zzz ) of the quadratic susceptibility complex tensor {ie165-2} ( z , ω 1 + ω 2 ; ω 1 , ω 2 ), which is responsible for the sumfrequency generation in a one-dimensionally inhomogeneous plate is proposed and proven for the first time. Such reconstruction is possible if the symmetry of the medium provides the diagonal character of the linear permittivity tensor {ie165-3} ( z , ω). The procedure involves the measurement of the complex amplitude of the new wave with the frequency ω 1 + ω 2 that is reflected from the plate for a certain interval of the angles of incidence of the wave with the frequency ω 2 . The reflected wave results from the nonlinear interaction of the wave with frequency ω 2 and the wave with frequency ω 1 that exhibits the normal incidence. A similar approach can be used to determine the profiles of the components of the quadratic susceptibility tensor {ie165-4}( z , ω 1 − ω 2 ; ω 1 , − ω 2 ), which is responsible for the difference-frequency generation.

The effect of energy, dosage, and temperature of irradiation of silicon-on-sapphire structures by Si+ ions, as well as parameters of recrystallization annealing, on crystallinity of silicon film is shown. Implantation conditions and recrystallization annealing conditions are determined.

This paper proves the possibility of and offers a technique for a unique reconstitution of the spatial profiles of various components of the complex second-order susceptibility tensor χ^(2)(z,ω1-ω2;ω1,-ω2) responsible for difference frequency generation in one-dimensionally inhomogeneous plates. To be able to achieve such reconstitution it is necessary to measure the complex amplitude of the reflected difference frequency wave in a certain range of incidence angles of the plane biharmonic wave with frequencies ω1 and ω2 in its spectrum. By altering the incidence plane of the initial biharmonic wave and/or the polarization of its monochromatic components it is possible to uniquely determine the coordinate dependences of more than a half of all second-order susceptibility tensor components. Such reconstitution is successful when the symmetric nature of the medium ensures the diagonal character of the linear dielectric permittivity tensor of the plate. Besides, the dielectric properties of the plate can change arbitrarily along the z axis. The suggested approach includes measuring the intensity of difference frequency waves generated under specific conditions with the use of an auxiliary reference plate to be able to avoid complex phase measurements.

It is proven that the spatial profiles of different components of the complex quadratic susceptibility tensor \(\hat \chi ^{(2)}\) (z,ω 1+ω 2; ω 1, ω 2), which is responsible for the generation of sum-frequency wave in a plate with one-dimensional inhomogeneity, can be reconstructed unambiguously. A reconstruction technique is proposed. To implement it, one has to direct a plane biharmonic wave with monochromatic components at frequencies ω 1 and ω 2 onto a plate and measure (in some range of the angles of incidence) the complex amplitude of the sum-frequency wave reflected from the plate. Changing the plane of incidence of the initial wave and (or) the polarization of its monochromatic components, one can determine the coordinate dependences for more than half of the components of \(\hat \chi ^{(2)}\) (z,ω 1+ω 2; ω 1, ω 2). This reconstruction can be performed if the symmetry of the plate medium provides a diagonal form for its linear permittivity tensor. The technique proposed implies measurement of the intensities of the sum-frequency waves generated under special conditions using an auxiliary reference plate; this approach allows one to do without complex phase measurements.

A method for unambiguously determining the frequency dependence of the relative permittivity of a homogeneous medium has been proposed and implemented from the measured angular dependences of the complex transmission spectra of pulsed terahertz radiation through a plane-parallel plate. It has been shown experimentally that the proposed method makes it possible to reconstruct the dielectric parameters of the plane under investigation, even in the presence of time overlapping of waves rereflected from the medium boundaries.

The present paper is the first to prove that one of the columns of the monodromy matrix and two of the three coefficients (piecewise analytic on the interval [0, 1]) of the equation (f(x)y′)′+(r(x)−λ 2 q(x))y = 0 uniquely determine the third coefficient on this interval provided that the values of the functions f(x) and q(x) lie in the lower (or upper) open complex halfplane and on the positive part of the real axis. This unknown coefficient can be reconstructed by finding the unique zero minimum of a specially constructed functional depending on the solutions of the corresponding Cauchy problem and the given elements of the monodromy matrix.

A model of the potential distribution in voltage terminating structures (VTSs) with floating p + - n junction rings in silicon radiation detectors is proposed. The model is based on experimental current-voltage characteristic of interring gaps, measured for detectors based on high-resistivity silicon with resistivities from 1 to 25 kΩ cm. The physical basis of the model is the injection principle of current flow through VTS interring gaps, which becomes possible at a certain electric field distribution in space charge regions of p + - n junctions of the sensitive contact and rings. It is shown that the injection current flow is a universal operation principle of the VTS with floating rings, which leads to rigid stabilization of potentials of individual rings. As a result, it becomes possible to divide the potential irrespective of the semiconductor material resistivity.