THE OPTIMIZATION OF THE SHAPE AND SIZE OF THE INJECTION CONTACTS OF THE INTEGRATED P-I-N-STRUCTURES ON THE BASE OF USING THE CONFORMAL MAPPING METHOD

Context. P-i-n-diodes are widely used in a microwave technology to control the electromagnetic field. The field is controlled by the formation of an electron-hole plasma in the region of an intrinsic semiconductor (i-region) under the influence of a control current. The development of control devices on p-i-n-diodes has led to the emergence of integral p-i-n-structures of various types, the characteristics of which (for example, switching speed, switched power level, etc.) exceed the similar characteristics of volume diodes. The properties of p-i-n-structures are determined by a number of processes: the diffusion-drift charge transfer process, the recombination-generation, thermal, injection, and the so on. Obviously, these processes should be taken into account (are displayed) in the mathematical model of the computer-aided design system for control devices of a microwave systems. Integrated process accounting leads to the formulation of complex tasks. One of them is the task of optimizing the shape, geometric dimensions and placement of the injected contacts (an active region). Objective. The goal of the work is the development of a mathematical model and the corresponding software of the process of a microwave waves interaction with electron-hole plasma in an active region of the surface-oriented integral p-i-n-structures with ribbon-type freeform contacts to optimize an active region shape and its geometric dimensions. Method. The main idea of the developed algorithm is to use the conformal mapping method to bring the physical domain of the problem to canonical form, followed by solving internal boundary value problems in this area for the ambipolar diffusion equation and the wave equation using numerical-analytical methods (the finite difference method; partial domains method using projection boundary conditions similar to the Galerkin method). The optimization algorithm is based on a phased solution of the following problems (the shape and geometric dimensions of the active region are specified at each stage): a computational grid of nodes for the physical regions of the problem is being found, in an active region the carriers concentration distribution is being determined and the energy transmitted coefficient in the system under study is being calculated, which is used in the proposed optimization functional. The extreme values of the functional are found by the uniform search method. Results. The proposed mathematical model and the corresponding algorithm for optimizing the shape and geometric dimensions of the active region (i-region) of integrated surface-oriented p-i-n-structures expands the tool base for the design of semiconductor circuits of microwave frequencies (for example, similar to CST MICROWAVE STUDIO). Conclusions. An algorithm has been developed to optimize the shape and geometrical dimensions of the active region of integrated surface-oriented p-i-n-structures with in-depth contacts intended for switching millimeter-wave electromagnetic signals. The universality of the algorithm is ensured by applying the method of conformal transformations of spatial domains. The example of the application of the proposed algorithm to search for the optimal sizes of wedge-shaped (in cross-section) contacts of silicon structures is considered.

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Inverse impedance boundary problem via the conformal mapping method: the case of small impedances

Revue Africaine de la Recherche en Informatique et Mathématiques Appliquées ◽

10.46298/arima.1936 ◽

2010 ◽

Vol Volume 13 - 2010 - Special... ◽

Author(s):

F. Ben Hassen ◽

Y. Boukari ◽

H. Haddar

Keyword(s):

Conformal Mapping ◽

Boundary Curve ◽

Cauchy Data ◽

Mapping Method ◽

Impedance Boundary Condition ◽

Internal Boundary ◽

External Boundary ◽

Impedance Boundary ◽

Conformal Mapping Method ◽

International Audience

International audience Haddar and Kress [9] extended the use of the conformal mapping approach [2, 8] to reconstruct the internal boundary curve Ti of a doubly connected domain from the Cauchy data on the external boundary of a harmonic function satisfying a homogeneous impedance boundary condition on Ti. However, the analysis of this scheme indicates non convergence of the proposed algorithm for small values of the impedance. In this paper, we modify the algorithm proposed in [9] in order to obtain a convergent and stable inversion process for small impedances. We illustrate the performance of the method through some numerical examples that also include the cases of variable impedances. Haddar and Kress [9] ont étendu l’utilisation de la méthode des applications conformes [2, 8] pour construire le bord intérieur Ti d’un domaine doublement connexe à partir de données de Cauchy sur le bord extérieur d’une fonction harmonique qui satisfait une condition homogène d’impédance sur Ti. Cependant, l’analyse de la méthode montre la non-convergence de l’algorithme proposé dans [9] dans le cas des faibles impédances. Dans ce travail, nous proposons des modifications de cet algorithme afin d’obtenir un schéma d’inversion convergent et stable dans ce cas. La méthode est ensuite validée par des exemples numériques qui incluent également le cas d’impédances variables.

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Constructing and analyzing mathematical model of plasma characteristics in the active region of integrated p-i-n-structures by the methods of perturbation theory and conformal mappings

Eastern-European Journal of Enterprise Technologies ◽

10.15587/1729-4061.2021.243097 ◽

2021 ◽

Vol 5 (5 (113)) ◽

pp. 51-61

Author(s):

Andrii Bomba ◽

Igor Moroz ◽

Mykhailo Boichura

Keyword(s):

Mathematical Model ◽

Perturbation Theory ◽

Active Region ◽

Small Parameter ◽

Nonlinear Boundary ◽

Boundary Value ◽

Asymptotic Series ◽

Charge Carriers ◽

Electron Hole ◽

Electron Hole Plasma

The results of mathematical modeling of stationary physical processes in the electron-hole plasma of the active region (i-region) of integral p-i-n-structures are presented. The mathematical model is written in the framework of the hydrodynamic thermal approximation, taking into account the phenomenological data on the effect on the dynamic characteristics of charge carriers of heating of the electron-hole plasma as a result of the release of Joule heat in the volume of the i-th region and the release of recombination energy. The model is based on a nonlinear boundary value problem on a given spatial domain with curvilinear sections of the boundary for the system of equations for the continuity of the current of charge carriers, Poisson, and thermal conductivity. The statement of the problem contains a naturally formed small parameter, which made it possible to use asymptotic methods for its analytical-numerical solution. A model nonlinear boundary value problem with a small parameter is reduced to a sequence of linear boundary value problems by the methods of perturbation theory, and the physical domain of the problem with curvilinear sections of the boundary is reduced to the canonical form by the method of conformal mappings. Stationary distributions of charge carrier concentrations and the corresponding temperature field in the active region of p-i-n-structures are obtained in the form of asymptotic series in powers of a small parameter. The process of refining solutions is iterative, with the alternate fixation of unknown tasks at different stages of the iterative process. The asymptotic series describing the behavior of the plasma concentration and potential in the region under study, in contrast to the classical ones, contain boundary layer corrections. It was found that boundary functions play a key role in describing the electrostatic plasma field. The proposed approach to solving the corresponding nonlinear problem can significantly save computing resources

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