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On the Joint Application of the Matrix Method and the Apparatus of Generalized Powers of Bers for Mathematical Modeling of Heat and Mass Transfer in Semiconductor Materials of Electronic Engineering  

Authors
 Kalmanovich V.V.
 Stepovich M.A.
Date of publication
 2018
DOI
 10.31114/2078-7707-2018-3-194-201

Abstract
 The paper presents some possibilities for the joint use of the matrix method and the method of generalized powers of Bers for mathematical modeling of heat and mass transfer caused by the interaction of charged particles or electromagnetic radiation with the surface of a multilayer semiconductor target. The proposed matrix method is analytical, it is applicable to solve the problems of heat and mass transfer in a homogeneous or multilayered medium with shift, axial or central symmetry for an arbitrary number of layers, including if the parameters of the layers are variable, i.e. the layers are not homogeneous. Such a generality is achieved by using the apparatus of generalized powers of Bers. Simulation is reduced to the sequential multiplication of second-order functional matrices whose components at each point are determined by the physical and geometric parameters of the current layer.
The solution of nonstationary problems of heat and mass transfer is based on the classical Fourier method, and its combination with the proposed matrix method makes it possible to find a solution of the problem relatively easily in the case of a multilayered medium. Some possibilities of this approach are demonstrated by the example of solving a nonstationary homogeneous heat conduction equation (cooling pro-cess) for a three-layer material.
When solving the stationary inhomogeneous problem of heat and mass transfer, it is necessary to know its any particular solution. In cases where the analytic expression of a particular solution is difficult to find or fails, the matrix method can be applied as a numerical method. In this case, the entire material is divided into a large number of thin layers and the right side on each layer is assumed to be equal, for example, to a constant. The possibility of such an approach is demonstrated by the example of solving a stationary inhomogeneous diffusion problem for a homogeneous material with a segmentation into 20 layers.
Keywords
 mathematical modeling, charged particles, electromagnetic radiation, semiconductor, heat and mass transfer phenomena, matrix method, generalized powers of Bers.
Library reference
 Kalmanovich V.V., Stepovich M.A. On the Joint Application of the Matrix Method and the Apparatus of Generalized Powers of Bers for Mathematical Modeling of Heat and Mass Transfer in Semiconductor Materials of Electronic Engineering // Problems of Perspective Micro- and Nanoelectronic Systems Development - 2018. Issue 3. P. 194-201.
URL of paper
 http://www.mes-conference.ru/data/year2018/pdf/D088.pdf

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