Qualitative Analysis of a Class of Differential Equations of Heat and Mass Transfer in a Condensed Material
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Journal of Mathematical Sciences, Vol. 250, No. 1, October, 2020
QUALITATIVE ANALYSIS OF A CLASS OF DIFFERENTIAL EQUATIONS OF HEAT AND MASS TRANSFER IN A CONDENSED MATERIAL D. V. Turtin Plekhanov Russian University of Economics, Ivanovo Branch 53, Dzerzhinskii St., Ivanovo 153025, Russia [email protected]
E. V. Seregina Bauman Moscow State Technical University (National Research University), Kaluga Branch 2, Bazhenov St., Kaluga 248000, Russia [email protected]
M. A. Stepovich
∗
Tsiolkovsky Kaluga State University 26, Stepan Razin St., Kaluga 248023, Russia [email protected]
UDC 517.955.2;517.958;537.533.9;51-73;51-74
We propose a mathematical model of three-dimensional diffusion of nonequilibrium minority charge carries caused by a pulsating sharply focused electron beam in a homogeneous semiconductor material and show the correctness of the model. We establish the continuous dependence of the solution on the data and obtain estimates for influence of data errors on the distribution of diffusing impurity. Bibliography: 9 titles.
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Introduction
Mathematical modeling of processes of interaction between kilovolt electrons and a condensed substance, as well as related heat and mass transfer processes are difficult from the mathematical point of view, as well as in view of verifying the proposed models. Thus, it is not easy to determine experimentally the local characteristics of a target irradiated by a sharply focused electron beam in the local signal excitation microvolume because the volume size is small (some micrometer units or less) [1]. Therefore, in the practical electron probe study, it is very important to obtain estimates characterizing physical processes caused by interaction between an electron beam and a target (cf. [2] and the references therein). In fact, no mathematically correct study of ∗
To whom the correspondence should be addressed.
Translated from Problemy Matematicheskogo Analiza 104, 2020, pp. 149-156. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2501-0166
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physical phenomena caused by interaction between electron beams and semiconductors has been carried out. The problem of modeling diffusion for such processes was only considered within the framework of the theory of energy loss by the primary low-energy electrons in a target in the form of a two-dimensional normal (Gaussian) distribution (cf. [3] and the references therein). In the case of kilovolt electrons of average and partly low energies, the quantitative agreement with experiments can be reached by using a mathematical model describing the energy loss by low-angle scattered electrons absorbed by the target separately from the energy loss by highangle reflected electrons left the target. Such an approach to describing the density of energy loss by an electron beam in a solid ρ∗ (x, y, z) (consequently, in the microvolume of generation of minority charge carries) corresponds to the real energy loss by the primary electron beam in a solid and, in the general case, we have x2 + y 2 z − z 2 1, 085(1 − η)
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