# SIMULATION OF DIFFUSION-DRIFF PROCESSES IN THE ELECTRON-HOLE PLASMA OF THE P-I-N-DIODES ACTIVE REGION UNDER THE CONDITIONS OF A MW PENETRATING IN THE PLASMA BY THE PERTURBATION THEORY METHODS

## DOI:

https://doi.org/10.20535/kpisn.2022.1-2.268688## Keywords:

singularity, asymptotic series, boundary function, electron-hole plasma, p-i-n-diode## Abstract

**Background.** The problem of developing tools for mathematical modeling of the state of electron-hole plasma in the p-i-n structures active region under the influence of an additional factor - a powerful microwave electromagnetic field is considered. The task is important for specialists in the field of microwave electronics, since p-i-n structures are used, in particular, for switching powerful electromagnetic fields and as protective devices for the input paths of radio engineering systems.

**Objective.** It consists in developing a methodology for modeling the electron-hole plasma concentration distribution in the p-i-n- diodes active region taking into account the effect on the dynamics of charge carriers of microwave radiation penetrating into the active region and developing asymptotic methods for solving the corresponding singularly perturbed nonlinear problems.

**Methods.** Achieving the goal is ensured by the use of boundary layer method, complex amplitudes method and classical analytic-numerical methods for solving boundary value problems for systems of ordinary differential equations.

**Results.** A generalized mathematical model of the electron-hole plasma stationary state in the p-i-n diodes active region in the hydrodynamic approximation, which takes into account the effect of microwave radiation on processes in the plasma, is proposed. The model basis is a nonlinear singularly perturbed boundary value problem for the system of electron-hole currents continuity equations and the Poisson. The model boundary value problem is reduced to a recurrent sequence of linear boundary value problems. Solutions of the stated problem are found in the form of asymptotic series containing stationary and non-stationary components. A feature of the proposed mathematical model is that it reflects the effect of detecting an electromagnetic microwave TE-like wave on the charge carrier concentration distribution inhomogeneity in the p-i-n-diode active region.

**Conclusions.** The methodology for modeling the electron-hole plasma stationary state in the p-i-n-diodes active region taking into account the effect on the charge carriers dynamics of microwave radiation penetrating into the active region has been developed.

## References

S. Sze and K. Kwok, Physics of Semiconductor Devices, New York: Wiley-Interscience, 2006, 815 р. doi: https://doi.org/10.1002/0470068329

K. Kwok, Complete Guide to Semiconductor Devices, New York: Wiley-Interscience, 2002, 740 p. URL: https://ieeexplore.ieee.org/book/5271197

E. I. Adirovich et. al., Currents double injection in semiconductors, Moscow: Sov. radio, 1978, 320 p.

V. N. Grydin et. al., Electrodynamics of extremely high frequency structures. Мoscow: Nauka, 2002, 359 p.

B. S. Polsky and J. S. Rimshans, “Numerical simulation of transient processes in 2-D bipolar transistor”, Solid State Electron., vol. 24, pp. 1081–1085, 1981.

V. A. Nikolaeva et. al., “A numerical method for the simulation of two-dimensional semiconductor structures using quasihydrodynamic approach“, Dokl. Akad. Nauk SSSR, no. 298(6), pp. 1367–1371, 1988. URL: http://www.mathnet.ru/links/ffb06774392c934ef68c73f4a11cb67b/dan48205.pdf

A. N. Tikhonov, “Systems of differential equations containing small parameters in the derivatives”, Mat. Sb., vol. 31(73), no. 3, pp. 575–586, 1952. URL: http://www.mathnet.ru/links/7bf5e98e10e302c132f30df4fdad6b9b/sm5548.pdf

M. Vishik and L. A. Lusternik, “Regular degeneration and boundary layer for linear differential equations with small parameter”, Usp. Mat. Nauk., vol.12, no. 5, pp. 3–122, 1957. URL: http://www.mathnet.ru/links/3ad5badbcecb855d7a6ed5e79bf964d1/rm7705.pdf

A. B. Vasil’eva et. al., The Boundary Function Method for Singular Perturbation Problems, Philadelphia: SIAM, 1995. doi: https://doi.org/10.1137/1.9781611970784

D. R. Smith, Singular-Perturbation Theory. An Introduction with Applications, Cambridge: Cambridge Univ. Press, 1985, 520 p.

A. Bomba, “On the approximate solution asymptotic method of one problem of mass transfer during filtration in a porous medium”, Ukr. Math. J., vol. 34, no. 4, pp. 37–40, 1982.

M. P. Belyanin, “On the asymptotic solution of one p-n-junction model”, U.S.S.R. Comput. Math. Math. Phys., vol. 26, no. 2, pp. 306–311, 1986. URL: http://www.mathnet.ru/links/f91b663c991258d0733ea2d02cf7f68a/zvmmf9248.pdf

A. Ya. Bomba and I. P. Moroz, “ Prediction of the charge carriers stationary distribution in the surface-oriented p-i-n structures active region by the perturbation theory methods”, Visnyk KNU seriya «Matematychne modeluvannya. Informatsijni tehnologiji. Avtomatyzovani systemy upravlinnya», iss. 50, pp. 27–36, 2021.

A. Ya. Bomba and I. P. Moroz, “ The numerical-asymptotic method for solving singularly perturbed model problems on the stationary distribution of charge carriers in the active region of p-i-n-diodes“, Bulletin National University of Water and Environmental Engineering. Technical sciences, v. 1(97), pp. 291–306, 2022.

A. Bomba et. al., “Identification of Mass Transfer Distribution Factor and Its Account for Magnetic Filtration Process Modeling“, Journal of Automation and Information Sciences, vol. 45, iss. 4, pp. 16–22, 2013.

A. Ya. Bomba and I. P. Moroz, “The diffusion-drift process with account heating and recombination in the p-i-n diodes active region mathematical modeling by the perturbation theory methods”, Jurnal obchycluval’noi i prykladnoi matematyky, no. 1 (135), pp. 29–35, 2021.

A. B. Vasil’eva and V.G. Stel’makh, “Singularly perturbed systems in the theory of semiconductors”, U.S.S.R. Comput. Math. Math. Phys., vol. 17, no. 2, pp. 339–348, 1977. URL: http://www.mathnet.ru/links/5d86b56f4673e4843abf1810d61ba4e8/zvmmf6023.pdf

A. I. Prokopyev and S. A. Mesheryakov, “Static characteristics of high-barrier Schottky diode under high level injection”, Solid-State Electronics, vol. 43, no. 9, pp. 1747–1753, 1999. doi: https://doi.org/10.1016/S0038-1101(99)00138-0

D. M. Pozar, “Microwave Engineering”, 4th ed. Hoboken, New Jersey: John Wiley & Sons, 2012, 732 p.

## Downloads

## Published

## Issue

## Section

## License

Copyright (c) 2022 Ігор Мороз

This work is licensed under a Creative Commons Attribution 4.0 International License.

Authors who publish with this journal agree to the following terms:

- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under CC BY 4.0 that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work