|
|
|
[Contents] | [Index] |
Cooperation with: K. Gröger (Humboldt-Universität zu Berlin)
Supported by: DFG: ``Zur Analysis von thermodynamischen Modellen des Stoff-, Ladungs- und Energietransports in heterogenen Halbleitern'' (Analysis of thermodynamic models for the transport of mass, charge and energy in heterogeneous semiconductors)
Description: Models describing the transport of dopants in semiconducting materials via pair diffusion mechanisms are considered at very different modeling levels. Generally, such models contain continuity equations, a Poisson equation, and often nonlocal relations which express invariance properties such as global charge conservation. There exists a hierarchy of such models which is based on physically motivated assumptions concerning the time scales of different kinetic subprocesses (see [1, Chap. 5]). Starting from a basic model where all relevant species (electrons, holes, dopants, defects, and dopant-defect pairs) are balanced by continuity equations, one can finally reach a reduced model where only the total amount of each dopant is balanced. Most of such model equations at the different reduction levels can be written in a unified way as a generalized electro-reaction-diffusion system which we have studied last year. Many other problems in the field of electro-diffusion fit into this general setting, too.
We consider m species Xi where only the first
l ones are assumed to underly drift-diffusion processes.
Let z0 denote the electrostatic potential, let , where
the components zj,
, are
some additional quantities representing the electrochemical potential of species
eliminated by the foregoing model reduction.
The functions
are suitably chosen
reference densities of the special structure
The problem is formulated in terms of the vector u of particle densities ui, of the vector b of chemical activities bi=ui/p0i, and of the vector z. The investigated model equations consist of m continuity equations
![]() |
(1) |
![]() |
(2) |
![]() |
(3) |
![]() |
(4) |
In [4] we proved an existence result for (1)-(3) at a special level in the hierarchy of pair diffusion models. But there we considered only homogeneous structures with smooth boundaries, and all species were assumed to be mobile. In [2] we derived global estimates for solutions of that model but now involving heterostructures and immobile dopants. The existence theory for these model equations is contained in [3].
In the weak formulation of (1)-(3), both
the Poisson equation (2)
and the nonlocal conservation laws
(3) are formulated as a generalized
Poisson equation . For every
fixed u, the operator
is a strongly monotone potential
operator.
By means of the potential of this operator and of the usual chemical ansatz,
the free energy functional
For the existence proof
we carry out two steps of regularization and consider problems
() and (
). The first step
(
) contains regularizations in the volume and boundary reaction
terms of the continuity equations only. By means of energy estimates and
Moser iteration we derive
a priori estimates which are independent of the
regularization level N.
A further regularization in the drift terms of
(
) leads to a
problem (
). The existence of solutions of (
)follows by two fixed-point iterations
(Banach's fixed-point theorem for the equations of the immobile species,
Schauder's fixed-point theorem for the equations of the mobile species).
Moreover, we verify estimates for the solutions of (
)which are independent of the regularization level M which finally
leads to the
existence result.
The presented results concerning the generalized electro-reaction-diffusion system (1)-(3) and their proofs can be found in [1, Chap. 8].
References:
|
|
|
[Contents] | [Index] |