On the uniqueness of solutions to the drift-diffusion model of semiconductor devices. (English) Zbl 0801.35133

Gajewski, Herbert

Math. Models Methods Appl. Sci. 4, No. 1, 121-133 (1994).

The paper concerns the following system of semiconductor equations: \[ {\partial u_ i\over \partial t}+ e_ i\text{ div }J_ i+ r_ i= 0\quad (i=n,p),\;e_ n=- 1,\;e_ p= +1,\tag{1} \] \[ - \text{div}(\varepsilon\nabla v_ 0)= D+ u_ p- u_ n\tag{2} \] where \(v_ 0\) is the electrostatic potential, \(u_ i= f_ i(v_ i)\) the carrier density, \(v_ i\) the chemical potential, \(J_ i= J(x,v_ i, z_ i)\), \(z_ i= \nabla w_ i\) the current density, \(w_ i= v_ 0+ e_ i v_ i\) the quasi-Fermi potential, \(\varepsilon\) the dielectric permittivity, \(D\) the density of impurities, \(r_ i\) the recombination/generation rate.
System (1), (2) is completed by mixed boundary conditions and initial conditions on \(u_ n\), \(u_ p\). The author then introduces appropriate structural conditions on \(J_ i\) and various smoothness conditions on \(f_ i\) and \(r_ i\).
The main result of the paper is a uniqueness theorem for bounded weak solutions to (1), (2) which satisfy an additional regularity condition. The proof makes essential use of a kind of distance between two weak solutions which is constructed by an energy functional.

Reviewer: J.Naumann (Berlin)

MSC:

35Q60	PDEs in connection with optics and electromagnetic theory
78A35	Motion of charged particles
35K55	Nonlinear parabolic equations
35K57	Reaction-diffusion equations

Keywords:

semiconductor equations; energy functional; uniqueness