# THE DIFFUSION-DRIFT PROCESS WITH ACCOUNT HEATING AND RECOMBINATION IN THE p-i-n DIODES ACTIVE REGION MATHEMATICAL MODELING BY THE PERTURBATION THEORY METHODS

### Abstract

With prolonged transmission of an electric current through the semiconductor devices, in a particular p-i-n diodes, an electron-hole plasma of their active region is heated. This paper presents the theoretical studies results of the plasma heating effect by the Joule heat release in the p-i-n diode volume and the charge carriers recombination energy release on the plasma concentration distribution in the p-i-n diodes active region. The mathematical model is proposed for predicting the electron-hole plasma stationary concentration distribution and the temperature field in the i-region of the bulk p-i-n diodes in the form of a nonlinear boundary value problem in a given area for the equations system, which consist of the charge carrier current continuity equations, the Poisson and the thermal conductivity. It is shown that the differential equations of the model contain a small parameter in such a way that the Poisson equation is singularly perturbed and the heat conduction equation is regularly perturbed. An approximate solution of the problem posed is obtained in the form of the corresponding asymptotic series in powers of the small parameter. The asymptotic serieses, which describes the behavior of the plasma concentration and potential in the investigated region, containing near-boundary corrections to ensure the fulfillment of the boundary conditions. The terms of these series are found as a result of solving a sequence of boundary value problems, obtained as a result of splitting the original problem, for systems of linear differential equations. The boundary value problem for a nonlinear heat equation is reduced to a sequence of problems for the corresponding linear inhomogeneous equations. The process of refining solutions is iterative. The stabilization of the process is ensured by the existence of negative feedback in the system (as the temperature rises, the mobility of charge carriers decreases).

### References

Sze S., Kwok K. Physics of Semiconductor Devices. New York: Wiley-Interscience, 2006. 815 р.

Adirovich E.I., Karageorgiy-Alkalaev P.M., Leyderman A.Yu. Double injection currents in semiconductors. Moscow: Soviet radio, 1978. 320 p. (in Russian)

Gurtov V.A., Osaulenko R.N., Solid State Physics for Engineers. Moscow: Tekhnosfera, 2012. 560 c. (in Russian)

Tikhonov A. N. Systems of differential equations containing small parameters multiplying some of the derivatives. Mat. Sb., 1952.Vol. 31(73), issue 3, pp. 575-586. (in Russian)

Vishik M.I., Lusternik L.A. Regular degeneration and boundary layer for linear differential equations with small parameter. UMN, 1957. Vol. 12, issue 5 (77), pp. 3-122. (in Russian)

Vasilieva A.B., Butuzov V.F. Asymptotic methods in the theory of singular perturbations. Moscow: Vissh. shkola, 1990. 208 p. (in Russian)

Bomba A.Ya., Prysyagnuk I.M., Prysyagnuk O.V. Methods of perturbation theory for predicting heat and mass transfer processes in porous and microporous media. Rivne: O. Zen, 2017. 291 p. (in Ukrainian)

Bomba A.Ya. On the asymptotic method of approximate solution of one mass transfer problem during filtration in a porous medium. Ukrainian Mathematical Journal, 1982. Vol.34, issue 4, pp.37-40. (in Ukrainian)

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