A new numerical algorithm for a drift-diffusion system
R. Aboulaïch‡, O. Guennoun*, A. Souissi*, J.E. Souza de Cursi¶
‡LERMA, Ecole Mohammadia d’Ingénieurs, Université Mohamed V. Avenue Ibn Sina, B.P. 765 Rabat-Agdal.
*Université Mohamed V, Faculté des Sciences, Département de Mathématiques et InformatiqueBP 1014;Rabat.
¶URA 2213, I.M.R.- I.N.S.A. de Rouen, BP 8, 76131 Mont Saint Aignan Cedex, France.
Abstract : Electrical potentials and free boundary separating the depletion and neutrality regions in a junction field transistor can be computed using a drift-diffusion model for a MESFET. This paper presents a new numerical algorithm of this model using a fixed point method. Numerical results are encouraging and provides a reasonably free boundary.
Keywords: Semiconductors- Free boundary- Regularization- Fixed point-.Finite Difference Method.
I. Introduction
The simple mathematical model for steady state semiconductor analysis is the drift-diffusion model, determined by three equations: Poisson's equation and continuity equations for the electron and hole current densities:
(3) , 1 .
p
(2) , 1 .
t n
(1) ), (
) .(
p p
n n
R q J
t
R q J
N p n q u
=
∇
∂ +
∂
=
∇
∂ −
∂
−
−
=
∇
∇ ε
subject to mixed boundary conditions. u is the electrostatic potential, n is the electron density, N is the doping function, q and ε are respectively the electric charge and the dielectric function of the material, Jn and Jp are respectively the electron and hole current densities.
For sake of simplicity we suppose that the device is unipolar semiconductor or with charge n. The semiconductor is in steady state, uniformly doped.
The net recombination-rate of holes and electrons Rp
and Rn are neglected. The electron mobility, the temperature and the dielectric function are constant.
The transition from the charge neutrality region (denoted in the following by C) to the depletion region (denoted by D) is abrupt (Debye
approximation).
The system {(1), (2), (3)} can be rewritten as:
(5) . in 0 . .J
(4) , in ) (
n =∇ = Ω
∇
Ω
−
=
∆
J N q n
u ε
The current density J is given by:
(6) in
),
( ∇ − ∇ Ω
= D n n u
J µ
µ is the electron mobility and Ω denotes the device domain.
We assume Boltzmann statistics and express the mobile charge carrier density n by
n = Neβ(u-ω) (7) where ω is the quasi-Fermi level for the electrons.
Introducing this variable change n in (4) and (5) we obtain:
) 1 ( ( ) −
=
∆ β −ω
ε e u
qN
u in Ω (8) 0
)
.( ( )∇ =
∇ eβ u−ω ω in Ω (9) This implies that
=0
∆u if u = ω ,
the region where n = N is characterized by u = ω . With these notations, we obtain u and ω as solution of the problem (P) = (P1 ,P2 ) :
(P1) î í ì
Γ
∂ =
∂= Γ
<
=
∆
=
=
∆
N D 1
on u 0
on
D in u
and - u
C in u and 0
ν
ω ζ
ω g
u u
,
2 58 1999 The Moroccan Statistical Physical Society
A new numerical algorithm for a drift-diffusion system 2
(P2 ) î í ì
Γ
∂ =
∂ = Γ
Ω
=
∇
∇ −
N D 2
) (
on 0
on g
in 0 ) .(
ν ωω
ω
ω β u
e
Where,
. on
g
; on 0 g
and on g
; on g
; on 0
d 2
s 2
g 1
D 1
s 1
V g
V V g
Γ
∪ Γ
=
Γ
=
Γ
=
Γ
=
Γ
=
+
− +
The real numbers β, ζ >0 , V- and V+, are given. The boundary ∂Ω of Ω is splitted in two unconnected parts ΓD and ΓN (ΓN = ∂Ω \ ΓD). ν is the outwards unitary normal to Ω .The whole is presented by the following figure:
ΓD = Γs ∪Γd ∪Γg , where Γs , Γd and Γg represent respectively the contact source, the drain and the gate. ΓL = D∩C is the free boundary.
Results concerning the problems (P1) for a given ω and (P2) for a given u can be found in the literature: on the one hand, the questions of existence and uniqueness for (P1) with a given ω have been studied in [12]; on the other hand, the problem (P2) with a given u can be transformed in a classical linear problem by the introduction of the auxiliary variable
φ = e-βω ; then, the questions of the existence and the uniqueness of a solution for a given u follows straightly from the classical Lemma of Lax-Milgram (see, for instance, [4]). Maala proposed in [9] a QVI approach of (P) with existence proof using the Joly- Mosco result [7]. We present here a new approach introducing some multivoque function and some process of regularization. The proof of existence follows from Schauder's theorem which suggests a simple fixed point algorithm for numerical approximation.
II. An equivalent formulation
We shall denote by (•,•)0 and ((•,•))1 the usual scalars products of L2(Ω) and H1(Ω), respectively . The associated norms are noted | • |0 and || • ||1 , respectively. Analogously, the usual norm of L∞(Ω) is noted | • |∞. In the sequel, we consider also
V0 = {v∈ H1(Ω) ; v = 0 on ΓD } , (u , v) 1 = Ω∇u.∇v dx
V = { v∈ H1(Ω) ; v = g1 on ΓD } , W = {v∈ H1(Ω) ; v = g2 on ΓD }
g1 and g2 are the traces of elements of H1(Ω): the components of (u, ω) can be decomposed in a sum of an element of V0 and a given element of H1(Ω) : for instance, u = u0+ u with u0∈V0 and u∈H1(Ω), ω=ω0+ω with, ω0∈V0 and ω∈H1(Ω), u=g1 on ∂Ω and ω=g2 on ∂Ω.
In the following, we shall modify the formulation of problem (P1) in order to eliminate the unknowns C, D and ΓL and we shall apply the Schauder's fixed point theorem .
Let us introduce the multivoque function η defined by :
η(s) = -1 (s<0) ; (10) η(s) = [-1,0] (s=0) ; (11) η(s) = 0 (s>0) (12) Then, we rewrite (P) as follows:
(P1') î í ì
Γ
∂ =
∂= Γ
Ω
−
∈
∆
N D
on u 0
on u u
on ) (
ν
ω ςη u u
(P2')))) î í ì
Γ
∂ =
∂ = Γ
Ω
=
∇
∇ −
N D )
(
on 0
on
in 0 ) .(
ν ω ω ω
ω
ω β u
e
We note (P') = (P'1,P'2). The unknowns of the problem P' are (u,ω). The regions C, D and ΓL are determined from u and ω by using { u = ω in C and u
< ω in D}. Moreover, P' concerns the whole domain
59
Ω: the free boundary is eliminated by the introduction of the multivoque function η , which becomes one of the main difficulties.
We observe that η is the subdifferential of N(s) = s- = (|s| -s)/2 (negative part of s). So, (P'1) corresponds to the Euler-Lagrange equation associated to the strictly convex coercive functional.
J : V0 →IR , J(v0) = ( ).
2 1
0 2
0 +u1 +ςΝ v +u−ω v
So, for a given ω , we show that u0 minimizes J on V0 , and we prove that (P) and (P') are equivalents:
(see [3]).
A. A regularized problem
In order to solve the difficulty introduced by the multivoque function η, we consider a sequence of regularized problems where the η is replaced by a continuous univoque function: let us denote by Nε a regularization of N such that Nε is convex, differentiable; N'ε = ηε is continuous, uniformly bounded such that: |ηε| ≤ α (α independent of ε ) and, for i=1, 2 ,
∃hi(ε) ≥0 such that hi(ε) ε →→0 0 and N(s) - h1(ε) ≤ Nε(s) ≤ N(s) + h2(ε).
The regularized problem reads as :
î í ì
Γ
∂ =
∂ = Γ
Ω
−
=
∆
N D 1 0 on
on u
on ) (
) (
ν
ω ςη
ε ε
ε ε ε ε
ε u
u u u
P
î í ì
Γ
∂ =
∂= Γ
Ω
=
∇
∇ −
N D )
(
2
on 0
on
in 0 ) .(
) (
ν ωω ω
ω
ε ε
ω ε β ε
ε
uε
e P
In the following, we note (Pεεεε) = (P1εεεε
, P2εεεε
) . This problem will be referred in the sequel as being the regularized problem. The unknown is (uε,ωε) and, analogously to the preceding problems, we can write uε = u0ε+ u and ωε = ω0ε+ ω with { u0ε , ω0ε } ⊂ V0
. For a given ω0ε, we can also introduce a map T1ε : V0 →V0 such that u0ε = T1ε(ω0ε) and is a solution of (P1εεεε), we have ω0ε = T2 (u0ε ) a solution of ( P2εεεε) yet.
We can use for example the following function Nε , a regularization of N :
B. Existence result for the regularized problem We establish that the regularized problem (Pεεεε) = (P1εεεε, P2εεεε) has a solution by using Shauder's fixed point theorem. Let us consider F = T1ε • T2 , and introduce the closed bounded convex domain defined by:
{
V0 such that 1 c1}
B= ω∈ ω ≤
Theorem 1
The application F applies B onto B , is continuous and compact. So it has a fixed point in B.
The proof uses the following auxiliary results:
Lemma 1
T1 is well defined in the sense that P1εεεε admits a unique solution.
Proof::
(P1εεεε) corresponds to the Euler-Lagrange's equation associated to the strictly convex coercive functional:
ε: J
).
2 ( ) 1 ( J ,
2 0
0 1 0
0 →IR ε v = v +u +ξNε v +u−ω
V
Existence and uniqueness follows from classical results of optimization (see, [8]).
Lemma 2
There is a constant c2 independent on ε such that .
) ( T :
1 1 2
0 c
V ≤
∈
∀ω ω
Proof: The variational formulation of (P1εεεε) reads as . ) ), (
( ) , (u : V v
;
0 1 0
0
0 V v u v
u ε∈ ∀ ∈ ε =−ξ ηε ε −ω By taking v = u0ε we obtain
0 0 2 / 1 0ε 1 ξαmes( ) u ε
u ≤ Ω
and, from the Poincare's inequality,
• Ω
≤ 1( , , ).
0ε 1 c ξ α
u
Using a maximum's principle, we show :
Let (uε,ωε) ∈V×W be a solution of the regularized problem. Then,
A new numerical algorithm for a drift-diffusion system 4 V-≤ uε≤ V+ ; 0 ≤ωε≤ V+ ; and ï∇ωεï0≤ c
where c is a constant independent of ζ and of the data on the boundary of Ω.
Theorem 2 :
For each ε > 0 , (Pε) has a solution (uε,ωε). Moreover, the sequence {(uε,ωε)}ε > 0 of solutions of (Pε) , is bounded in [H1(Ω)]2∩[L∞(Ω)]2 and any cluster point (u, ω) for the weak topology of this sequence is a solution of the problem (P)'. ♦ See [3] for details.
III. A numerical method
The approach introduced suggests a simple numerical algorithm based on fixed point theorem: a natural way to solve this problem consists in performing fixed point iterations
ω0k
=Fε(ω0k-1
) from given ω00∈B. This corresponds to the resolution of Pk=(P1k
, P2k
),
î í ì
Γ
=
∂
∂
Γ
=
Ω
=
∇
∇ î í ì
Γ
=
∂
∂
Γ
=
Ω
−
=
∆
−
−
N k
D k
) (
2
N k
D k
1
1
on 0 /
on
in 0 ) .(
) (
, on 0 / u
on u
on ) (
) (
ν ω
ω ω
ω ν
ω ξη
ω β
ε
k u
k
k k k
k
k
e k
P
u u u
P
The problem (P2k
) is transformed in a classical linear problem using the auxiliary variable
φ = e-βω and for the numerical resolution we solve:
î í ì
Γ
=
∂
∂
Γ
=
Ω
=
∇
∇
N k
D
on 0 /
on
in 0 ) .(
ν φ
φ φ
φ
β k
k uk
e
After discretization the elements of the matrix of the obtained system, becomes very large when β takes evaluate the exponential to calculate these terms; and the second one, the matrix becomes ill-conditioned for the largest values of (βu). The preconditioned GMRES method developed by J.Abouchabaka and al [1] permits to solve this system for the large values of β.
(P1k
) is a non linear problem and we use the following iterative method for the resolution:
Let u0 a given potential, we compute ul as a solution of:
î í ì
Γ
=
∂
∂
Γ
=
Ω
−
=
∆ − −
N k
l
D k
l
1 1
on 0 /
u
on u
on ) (
ν
ω ξηε
u u
ulk lk k
and uk is a limit of the sequence (ulk) for a given ωk-
1.
For numerical experiments, we use finite difference discretization of Ω =[0,12]×[0,4] with a regular steps h = ∆x and k = ∆y. The source, gate and drain regions are respectively given by segments of {0}×[0,4], [4,8] ×{0} and {12}×[0,4].
We denote Pi,j = (xi,yj) and the free boundary is approximated by the polygonal line: P1P2...PN+1 , with:
Pi = 1/2 (Pij +Pij+1)
such that u(Pij) ≤ ω (Pij) and u(Pij+1) = ω (Pij+1).
The numerical results presented in the following are obtained for ξ = 0.5 , V+ = 0.5.
We represent the error in the potential u, || uk+1 - uk||
L2
(Ω) in figure 1 for ε = 10-2, in figure 2 for ε = 10-8; and the error in the quasi - Fermi potential ω, ||ωk+1- ωk||L2
(ω ) is represented in figure 3 for ε = 10-2 and in figure 4 for ε = 10-8. The approximation of the free boundary is represented in figure 5 for ε = 10-2 and in figure 6 for ε = 10-8. We obtain a comparable behavior of the free boundary obtained by Mâalla [9]
and Nachaoui [12].
IV. Concluding Remarks
We have considered the Laplace-Poisson model for a MESFET transistor and a new equivalent
formulation. The resulting problem contains discontinuous terms which can be treated by
regularization. Schauder's theorem has been applied in order to show the existence of a solution for the resulting problem. The proof shows also that the cluster points of the sequence of solutions of the regularized problem are solutions of the Laplace- Poisson model. This establishes the existence of a solution for such a model and suggests a numerical method. Numerical experiments have been
performed in some simple situations and encouraging results have been obtained.
ACKNOWLEDGMENTS
This work is partially supported by the CNCPRST, AI845/95 and LERMA,EMI.
[1] Abouckabaka J., Aboulaïch R., Souissi A., Numerical approach of a free boundary in the 2 61
junction field effect transistor -MESFET ,
Mathematics and computers in simulation 47 (1998) 531-539.
[2] Aboulaïch R., Guennoun O., Souissi A., Analyse du Modèle de Regier par l'Optimisation de Forme, in Proceedings of the 28th French National Congress on Numerical Analysis} (1996).
[3] Aboulaïch R., Guennoun O., Souissi A, Souza de Cursi, J. E., Analyse d'un système de dérive- diffusion, RI no 25, LERMA, Ecole Mohammadia d'Ingénieurs-Agdal, Rabat (1997)
[4] Dautray, R.; Lions, J. L., Analyse Mathématique et Calcul Numérique pour les Sciences et Techniques, Tome 1, Vol. 2, Masson, Paris (1987).
[5] Ekeland I.and Temam R., Analyse convexe et problèmes variationnels, Vol. 4 , Dunod, Paris (1974).
[6] Jerome J.W. Consistency of semiconductor Modeling: An Existence/Stability Analysis for the stationary Van Roosboeck System, SIAM J. Appl.
Math., 45 (4) pp. 565-590 (1985).
[7] Joly J.L.and Moscou V., C.R. Acad. Sc. Paris t. 279 série A(1974), 499.
[8] Lions J.L., Quelques méthodes de résolution des problèmes aux limites non linéaires , Dunod, Paris (1969).
[9] Mâalla K., Analyse du transistor à effet de champ par une Inégalité Quasi - Variationnelle , Doctoral Thesis, University Pierre et Marie Curie (Paris VI) (1981).
[10] Markowich P.A., The Stationary
Semiconductor Device Equation, Springer, Wien (1986).
[11] Mock M.S., Analysis of Mathematical Models of semiconductor devices, Boole Press, Dublin (1983).
[12] Nachaoui A., Contribution à l'analyse et à l'approximation des modèles dérive -diffusion dans les Semi - Conducteurs, Doctoral Thesis, Dept. of Mathematics, University of Rennes 1 (1991).
[13] Roosnroeck W. Van, Theory of flow of electrons and holes in germanium and other semiconductors, Bell System Tech.J., 29, pp. 560- 607 (1950).
