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Uniform asymptotic error estimates for semiconductor device and electrochemistry equations
Nonbnear Analysis, Theory. Printed in Great Brirain.
Merhods
& Apphcorions.
Vol. 14. No. 2. pp. 123-139.
1990.
0362-546X190 S3.00+ .OO D 1990 Pngamon Press plc
UNIFORM ASYMPTOTIC ERROR ESTIMATES FOR SEMICONDUCTOR DEVICE AND ELECTROCHEMISTRY EQUATIONS
Arizona
FATIHAAL-AU* State University,Departmentof Mathematics,Tempe, AZ 85287,U.S.A. (Received 1 July
1988;received for publication 20 February 1989)
Key words and phrases:Semiconductordevice equations, electrochemistry equations, singular perturbation, asymptotic expansions.
THE PAPERis organized as follows. Section 1 sets up the problem, our purpose and results and the existing results in the literature. Section 2 gives the notations used in the paper. Sections 3 and 4 set up the problem in the case of a symmetric piecewise constant junction and recall the associated asymptotic expansion. In Section 5 our method is introduced and uniform asymptotic error estimates for a symmetric piecewise constant junction are derived. Section 6 is an application of our technique for the case of a biological membrane with a constant concentration of fixed charges. Uniform asymptotic error estimates are also proved to hold in that case. A brief summary and conclusion are given. In the appendix we give the proofs of the lemmas which are used in Sections 5 and 6. 1.
INTRODUCTION
The one dimensional semiconductor device equations form a nonlinear system of three second order differential equations subject to Dirichlet boundary conditions. It is already known (see [6-81) that after an appropriate scaling the problem can be reformulated as a singularly perturbed one, the singular perturbation parameter &being the minimal normed Debye length of the device. We assume for the following that the generation-recombination term is equal to 0 and that the electron (respectively hole) diffusion and mobility coefficients are constants. Then the sciled one dimensional semiconductor device equations (see [7] for more details about the scaling) in the device domain Q = ]a, b[ are given by: EW” e = a2e+e, E - b2e%E (P,)
- N
(e%.&)’ = 0
Poisson’s equation electron continuity equation
L(e-GE) = 0 subject to the boundary conditions (BC):
hole continuity equation
(u, = e+A
(BC. 1)
v, = eUA
(BC.2)
(W i
vE = In
N + (N2 + 4a4)“’ + t( A 2d2 >
* Present address: UniversitC Bordeaux 1, Departement de MathCmatiques, 35 1 Cours de la Liberation, 33405 Talence, France. 123
F. Aumu
124
where: 0 the dependent variables are wE(electrostatic potential), u, and V, (Slotboom variables) 0 N is the normed doping profile. It has a jump discontinuity across the junction P = (x0] (where x0 belongs to &2)and it is continuous in the two disjoint subdomains f2, = ]a,~,[ and a2 = 1x0,6] 0 u,, is the normed applied voltage. We set U, = max(u,(a), I&)) and u_ = min(uA(a), u”(b)). The applied potential difference (u, - u_) is denoted by urnax. 0 16 is the singular perturbation parameter, it is also the characteristic normed Debye length of the device. For realistic diodes E is a small parameter l a2 is the scaled intrinsic carrier concentration. Since the Slotboom variables uc, V, have no physical meaning (but are very practical for mathematical analysis) it is useful to recall here the definitions of the scaled concentrations and of the scaled current densities. The scaled electron (respectively hole) concentration is given by: n,
= B2e%dc
(respectively pE = 62e%,).
The scaled current densities are given by: J n,c = d2e+cu’ L
and
J
PA
=
-d2e-$cv’
C-
In [6-81 a formal asymptotic expansion (for E small) has been associated to the solutions of the singularly perturbed system and it has been proved that these solutions exhibit interior layers at the points across which the doping profile has jump discontinuities. In this paper we consider the formal asymptotic expansion defined in the above cited references and prove that the first term of this expansion is close to the solutions of the singularly perturbed system by means of uniform asymptotic error estimates of order .sp(p > 0) for the difference between these solutions and the first term of the expansion. There already exist some results of this type in the literature. In [6] Markowich has proved uniform asymptotic error estimate for the equilibrium case (i.e. zero applied potential, in that case the system is reduced to the nonlinear Poisson’s equation). A similar result has been derived in [8] for a one dimensionalp-n junction provided that the applied potential difference is shfficiently small (an alternative proof of this result is also given in [3]). The only result providing uniform asymptotic error estimates which allows large voltages has been performed in (91 for a one dimensional symmetric piecewise constant bipolar junction. The way to obtain these results is based on a method developed in [l 1] which uses the implicit function theorem and requires the existence of an isolated solution for the reduced problem (obtained from the singularly perturbed system by setting E equal to 0). The main difficulty in applying this method for semiconductor device equations comes from the fact that there is no general proof of the existence of isolated solutions for the reduced problem. This paper introduces another alternative based on variational techniques. Its important features are that it provides uniform asymptotic error estimates without any restriction on the applied voltages and does not require the existence of isolated solutions for the reduced problem. In this paper, we first apply this technique to a one dimensional symmetric piecewise constant bipolar junction. We obtain uniform asymptotic error estimates of order E”~ without any restrictions on the applied voltages or on E. This result should be compared to the one previously mentioned, obtained in [9]. In this last paper the authors have proved for the same type of devices, uniform asymptotic error estimates of order fi, but E has to be sufficiently
Semiconductor device and electrochemistry
equations
12s
small, this smallness depending on 6 and u,,, (the applied potential difference). Therefore for u mmlarge (which is the interesting case in practice) E has to be smaller than a certain value. The limitation of such a result is that in practice E is in fact a fixed quantity and u,,, is the quantity which varies. Besides this we proved that any sufficiently smooth solution of the singularly perturbed system is well approximated by the first term of the expansion, as opposed to the method used in [9] which proves only that there exists a solution of the perturbed system which is well approximated by the first term of the expansion. The equations governing the electrical behavior of a biological membrane are the same as the semiconductor device equations. The equivalent of the doping profile is called the concentration of fixed charges. Other than in the semiconductor case this concentration is a smooth function of the spatial variable. Layers occur at the boundary where in general the Dirichlet conditions do not satisfy the electroneutrality relation (obtained by setting the second term in the Poisson’s equation equal to 0). The same type of asymptotic analysis as for the semiconductor case has been performed in [5]. In this last paper the authors have proved that the asymptotic error converges to 0 as E goes to 0 but they have not obtained any estimates. In our paper we obtain uniform asymptotic error estimates provided that the concentration of fixed charges is constant. Even under this restriction there is no proof that there exists an isolated solution of the corresponding problem (in the case of a symmetric piecewise constant bipolar junction it was possible to prove such a result because of the symmetry properties of the device), therefore the method used in (91 cannot be applied. 2. DEFINITIONS
AND
NOTATIONS
We now give definitions of function spaces and norms which will be used in the sequel. We denote by Lq(Q) the space of q-integrable real valued functions defined on S2 with the norm:
Ilfilo,,,~ =
({nh91q by”
(denotedalso by IlfliLq).
*L”(R) is the space of bounded functions on R with the norm: WesetII l10.2.R = II IIO.R
IlflL.n =
sup IfWl XER
We denote the Lz scalar product by:
(denoted also by Ilfllr_). n
C”(Q) stands for the space of all functions defined on R which together with their derivatives of order up to m are continuous in S2. We denote by Hm(S2) the completion of In E c”(n),
Il4lm.n< +a1
with respect to the norm 1) Ilm,R, where:
We denote by H&2) the space lu E H’(Q), u/S2 = 0). We set ]~],,a = Ildu/dxllo,u for u E H’(O). In all the sequel we will denote by c a generic nonnegative constant independent of E.
126
F. Auam 3. SYMMETRIC
PIECEWISE
CONSTANT
BIPOLAR
JUNCTION
In this section we give the definition of a symmetric piecewise constant bipolar junction. As it has been proved in [9] the performance of such devices is described by a two point boundary value problem (using the fact that for such devices the full problem (P,) has symmetry properties). We recall here the formulation of this two point boundary value problem. We set:
62 = l-1, I[. A symmetric piecewise constant bipolar junction satisfies: ox,=0
(point where the doping profile has a jump discontinuity)
0 N(x) = i
for 05 x 5 1 for -1 lx50
_;
0 UA(_I) = -1(,4(l). This last restriction is not really a constraint since the electrostatic potential vE is defined up to an arbitrary additive constant. The only serious restriction comes from the fact that N is assumed to be odd. Using this last assumption Markowich and Schmeiser in [9] have replaced the initial problem (P,) by the following two point boundary value problem: &Vi = 62e*v4 E - B2e+vE - 1 e%4: = c,
forOIx5
1
e-+zui = -c, (SP,)
c; = 0
IL,(O)=
w,(O) = 0, y/,(l) = ln
1 + (1 + 4CJ4)“2 + u/I(l) 2d2 >
(
= esUA(‘), v,(l) = euAo).
u,(O), u,(l)
Given a solution (w,, ut, u,) of (SP,) we extend it to Cl (keeping the same notations for the extended functions) by setting: V,(-x)
= - V,(X)
05x51
u,(-x)
= %(X)
05x51
Q(-x)
= u,(x)
OlXll.
It is easy to check that these new functions now defined on all Sz define a solution of the original problem (P,). In the next two sections we shall only consider solutions of (P,) defined by this way. 4. ASYMPTOTIC
EXPANSION
FOR
BIPOLAR
SYMMETRIC
PIECEWISE
CONSTANT
JUNCTIONS
As we have mentioned in the introduction the semiconductor device equations form a singularly perturbed system, the normed Debye length being the singular perturbation parameter. In several papers Markowich, Ringhofer et al. (see [6-81) have introduced a formal
Semiconductor device and electrochemistry
127
equations
asymptotic expansion associated to the solutions of the singularly perturbed problem (P*). In the case of a symmetric piecewise constant bipolar junction, the different terms of this expansion have symmetric properties. In [9] the authors have used these properties to simplify the asymptotic analysis. In particular they have proved that boundary layer terms occur at x = 0 for the solutions of (SP,) (instead of internal layer terms for (P,)). We recall briefly here the procedure used in their article and give the equations for the reduced and boundary layer problems. Following standard techniques of asymptotic analysis, the authors of [9] look at the expansions of the form:
where r(x) = x/v% and where for a given number (Y> 0, M is a C” function satisfying: M(x) =
1
for 0 s x 5 ‘y and 0 5 M(x) I 1 for 2 I x I (Y 2 2
0
forcrsxzz
1.
The functions v/;, ui, Vi (solutions of reduced problems) are independent of E. The functions marked with *are defined on [0, +a[ and decay exponentially to 0 together with their derivatives as 7 goes to +a0 (boundary layer terms). 7 is the fast independent variable, x is the slow one. We will only consider here the first term of these expansions (for which one we drop the index 0). (w, u, V) is a solution of what is called the reduced problem (it is expected to be a “good approximation” to solutions of (SP,) away from the layer area). (I,?, li, D) are solutions of boundary layer problems (they are corrections to reduced solutions inside the layer area). In [9] it has been proved that the reduced problem is given by: 0 = a2eGu - a*e-+u - 1
.
e&u’ = c, e-+u = -c, c’ = 0 (RP)
u(0) = v(O), u(1) = e-“A”), u(1) = e”A”’ y/(O) = -In(a),
w(1) = In
1 + (1 + 464)“* 26*
(
>
+ UAl)
where b is a number depending on 6 and ~~(1) (see [9] for more details on the dependence). In [9] the authors also give the boundary layer problem corresponding to the electrostatic potential, that is: 2
= g2e+:Cot+@u(()) - ~2e-+(o)-~~(())
-
1
for
7 >
0
They have also proved that: ti = fi = 0. We recall here the asymptotic expansions of n,, pE as they have been performed in (6-91. The reduced electron (respectively hole) concentration is
F. ALABAU
128
given by: n = 6’e*u (respectivelyp = #e-$‘o). The reduced current densities are: J,, = d2eGu’, 6’e-ICv’. The corresponding layer terms for respectively n and p are:
Jp = -
ii(~)
5. UNIFORM
=
82e+!‘co),(0)(e~cti- 1) and
ASYMPTOTIC
ERROR
j(7)
=
ESTIMATES
62e-~(0’v(0)(e-G”’ - 1). FOR
SYMMETRIC
JUNCTIONS
In this section we first introduce some notations which simplify the asymptotic error analysis. We, then, give intermediate results which are needed in order to prove the main theorem of this section, namely that for a symmetric piecewise constant bipolar junction the solutions of the singularly perturbed problem (SP,) are well represented by the first term of the asymptotic expansion with no restriction on the applied voltages or on the singular perturbation parameter. As it has been said in the introduction this result should be compared to the one obtained in [9]. In this last paper the authors have obtained a better order of convergence for the uniform asymptotic error estimates but they have to restrict the size of the singular perturbation parameter which has to be sufficiently small. This restriction, as a consequence, leads to a restriction on the applied voltages (see [9] for more details), which is not convenient on a practical point of view since physicists are interested in applying higher voltages to a fixed device (this means that the voltage varies while E is kept fixed). In order to prove these estimates the authors have used the results of [ 1l] which involve the strong form of the equation and a version of the implicit function theorem. We develop here a method based on the weak form of the equation without using any implicit function theorem. The consequence is that our proof does not require that the reduced problem admits isolated solutions as it is the case in 191. Let us introduce now some notation which will be used in the sequel to this paper. We set: co,(x) = V(X) + S,(x)
aw
=
(4
= (4
where
S(7(X))MX)
7(x)
=
P,)(X),
SW
= (n + P)(X)
- P,)(X),
D(x)
= (n - P)(X).
+
We also associate to p,ethe solutions ti,, & of the two corresponding means that li, and t’, satisfy:
2 dz
continuity equations, this
(e%;)’ = 0 in 52 and (e-%3;)’ =0 in R + (BC. 1) 1 l + (BC.2) We now set: S; = fi, + fi,
where A, = 6*e%, and 8, = 8*eMpcfie & = ri, - p,
S(T) = 6
2e~‘0,+~‘T+4())
+
~2e-w3’-wv(())
for T > 0
b(7) = @eN3’+&7+40) _ ~Ze4L’O’-S’c7440) 9,(x) =
@(7(X)),
&(x)
=
9(7(x)),
b,(x)
for
7 >
0
= B(T(X)).
The main result of this section holds in theorem 5.1. In order to prove this theorem, we need to prove several lemmas. Since these proofs are very technical they are deferred to the appendix. We will also constantly use the two following facts which have been proved respectively in [6]
Semiconductor
device and electrochemistry
129
equations
(theorem 2.1) and in [5] (theorem 3), without always recalling them: v/~, u,, u, are uniformly bounded independently of E (see 161) v/+-~ is bounded in H’(0) independently of E (see [5]). LEMMA5.1. We consider a symmetric piecewise constant bipolar junction, then the following estimates are satisfied: (5-L)
112% - ull,.* + II& - &.,l3 = c& II& -
41,fl + Ilfie- 4.n = cP4
(5.2)
IISE- ~IIP(R) + IIQ - alr~~n, 5 4 II& - ~IIOJI+ IID, - Dllo,* 5 cP4.
(5.3) (5.4)
The proof is deferred to the appendix. LEMMA5.2. Assume the hypothesis of lemma 5.1, then:
aYf(x)l 5 c
VXE [O, l]
(5.5)
am)l
v x E [O, 11.
(5.6)
5c
The proof is also deferred to the appendix. LEMMA5.3. Let the hypothesis of lemma 5.1 hold. Then: ID, - Qll0.R 5
4ku/, - C4c)“l/0,R + cfi.
(5.7)
The proof of this lemma is also deferred to the appendix. .
LEMMA5.4. Assume the hypothesis of lemma 5.1, then: (SC - Q(x)
= k,(x) + (WC- K)(X)
in Sz
(5.8)
where k, satisfies
-1’41jDE - dclJO,* + c&1’4 IlhlL,* 5 CC
(5.9)
kllo,n I CllD,- DEllo,n+ cb5. The proof is deferred to the appendix. The next lemma gives the weak form of the twice differentiated
(5.10)
Poisson’s equation.
LEMMA 5.5. Assume the hypothesis of lemma 5.1. Then for 8, in H2(Q), such that 0,/X2 = 0 the following equality is satisfied:
1
& t(v/c - &“e:dx s0
+
c ,O
S,(Jy, - ql,)‘e;dx
= -
l(S, - ~&9:egx s0
+ 6, + T,
(5.11)
F. ALABAIJ
130
where T, and 6, satisfy: Ir,l =
c~“411~~llo,*
(5.12)
I T,I 5
cwclllP,n
(5.13)
(&I d cfi
if 0, 5 ry, - qr
(5.14)
Id s 44ll,,n.
(5.15)
The proof is deferred to the appendix. LEMMA5.6. Let the hypothesis of lemma 5.1 hold. Then the following estimates hold:
mw, - cpJIIo,o =c lla - &llo.n5 4 Ilk, Ilq-I5
(5.16) (5.17)
c.P4
(5.18)
llhllo,n= cfi
(5.19)
where k, has been defined in lemma 5.4. The previous lemmas allow us to prove the main result of this section, namely theorem 5.1. 5.1. We consider a symmetric piecewise constant bipolar junction, then the following estimates hold: THEOREM
IIK - 4%llce,R + II%- n - fiMll,,* + IIPE-P I&
- m41ce,*
5 CE”4
(5.20) (5.21)
- J,I + /.I,,, - Jpl 5 cE”4.
Proof. By using symmetry we work in &, (w,, u,, u,) being a solution of (SP,) extended to h by symmetry as it has been defined in Section 3. We introduce w, as the solution of the following problem. ” = wc - q,c)zp+l in Rz = IO, 11 WC I w,(O) = o,(l) = 0. From Markowich [6] we know that vt and qE are continuous in Q,, therefore w, is a C2 function. Since we have o,(O) = w,(l) = 0 there exists x* in R2 such that w:(x*) = 0. Thus:
Il4ll,.n, 5 IIWE - dl?~+1.n*. We now use (5.11) with c9,= w, and obtain: -
sk4 -
$Pcp:bldx -
E
0
-
ql,)“w;
dx = - 6, -
r,
where & and T, satisfy (5.15) and (5.13). Clearly: SEw: - $,cp; = (I!?,- $)(w, - co,)’ + S&E - v7,)’ + (S, - s;)Vd
(5.22)
Semiconductor device and electrochemistry
equations
131
holds. Using (5.8) in this last equality, we derive: S,& -
s&7;=
tv, -
9,)(w, - 9,)’ +
~,tv,- 9,)’ + (WC- 9,)9I + k,(ly, -
9J + kl9i (5.23)
where kr satisfies (5.9) and (5.10). Using the fact that (w,, uE, u,) is a solution of (SP,) it is easy to check that: for x in slz. S;(x) = d,(x)co:(x) Therefore: 9;(x) = 3; - (D, - 1)9j. Using this last relation in (5.23), we derive: SE9; - $9;
= [+(I& - 90,)z + Q&
- 9Jl’ + [k,(Wc - 9J’ + kI9: - (PC - 9,)(&
- 1)9J*
Therefore: 1
-
I(S,w: - $9;)w:
dX =
s0
l(w, - 9,)2p+2(t(W, - 9,) + SC)d.X+
s0
R,w; dx
(5.24)
s0
where: R, = (w, - co,)@, - 1)9: - kr(~e - 9,)’ - k,9:.
(5.25)
By using (5.8) once again we obtain: +(9, - 9,) + 9, = &S, + SE) - $c, . Using this last relation in (5.24) and then using (5.24) in (5.22), we obtain: + r(s, + Q(w, r .,O
- 9J 2p+2dx + (2P + 1)e ll(w, - 9J12(w, - UltJZPb s0 1
1
R&dx
=s0
k,(Wc - 9,)2p+2 dx - T, - r?&.
+ f
(5.26)
s0
We need now to estimate the righthand side of this last expression, we proceed as follows:
Using (5.4) and (5.18) in this last equality, we obtain: (5.27) Using now the fact that the function (5, + &) is bounded away from 0 by a nonnegative number independent of E (see Markowich [6], theorem 2.1) and the inequalities (5.27), (5.18), (5.13) and (5.15) in the relation (5.26), we finally obtain: 11~~- 9~ll0,2p+~,a~5 ccl’4 where c is a constant independent of p. (see Adams (11). Therefore we obtain: As P --) +a IIV’c- ~Io,z/J+z.R~ + IIWE- 9r114z I)IJI~- 9EII_,RZ I CE”~. Also we have:
llnc - n - AMI/CO.R> 5 11%-
fisllre,R2 + II% - n - fima,nZ.
132
F. ALABAU
Since: n, - ii, = 62e”c(u, - tl,) + g2(eGc,- eVc)& it is easy to check that: iln, - tiJ+*
5 c&1’4.
Since: ii, - n - fiM = 62ev$i c - u) + a2(e+Gu- e”O’u(0))(eGc - 1) in [0, (~121we obtain: II& - n - iiMll,,a2 d cd?. Analogous proof holds of (p, - p - fiM). This proves (5.20). (5.21) is easy to deduce using (5.20). 6. ELECTROCHEMISTRY
EQUATIONS
We also apply our technique to the case of biological membranes, for which the equations are the same as for the semiconductor device ones except that the equivalent of the doping profile (called the concentration of fixed charges) is a smooth function of the spatial variable and that layers occur at the boundary where the electroneutrality condition is not satisfied. In [5] Henry and Louro have performed the same type of asymptotic analysis as in [6]. We will consider here this asymptotic expansion and we will see that under minor modifications, the method used in Section 5 for symmetric piecewise constant bipolar junctions is still valid for the biological membranes. We will keep in this section the notation previously introduced. We first give the new boundary conditions associated to electrochemistry equations. The equations are given by (P,) (see Section 1) in S2 = 10, 11. The boundary conditions are different: forx = 0 (UE, v,, w,) = (n0, uo* wo) (EBC) forx= 1. L(UE,t)E, w,) = (u,, v1, w,) We set: d2 = 1 (without loss of generality). For the sake of simplicity, we will assume that boundary layers occur only at x = 0, that means, we assume:
t
e”Ouo-
e-$Ou
e+:'u,
e-+iv
-
0 1
-
N(O)
#
0
-
N(1)
=
0.
The reduced problem is then given by: e3, - e-*y - N = 0 (e&u’)’ = 0 (e-“v’)’ = 0 i + boundary conditions (REBC) where (REBC) is given by: (u, ZJ)= (uo, u,) at x = 0
(u, u) = (ur, u,) at x = 1 I,Y= w1 at x = 1 and w(O) = In
N(0) + (N2(0) + 4UoUo)“2
2uo
Semiconductor
device and electrochemistry
133
equations
The layer problem is given by:
d2@
!b(o)+~~o_
z=e
e
+co)-quo - N(0)
for T > 0
i
k!w-x =
!llo
l//(O), @(+oo) = 0.
-
We will assume in the sequel of this section that: N=
1.
(6.1)
It is easy to check that the lemmas 5.1, 5.2, 5.3 are still valid. The lemma (5.6) requires some modifications, in order to prove the main result of this section (theorem 6.1). LEMMA6.1. We assume (6.1) and that uo, vo, ui, u1 are nonnegative. estimates hold: fll(YG - (P~)“IlO.R 5 c
Then the following (6.2) (6.3)
ID, - ~cll0.R 5 cfi. The proof is deferred to the appendix. This enables us to prove the following theorem. THEOREM6. I. Assume the hypothesis of lemma 6.1. Then the following estimates hold:
llv, - dl.m,n + II& - n - fi~ll,,*
+ lb, - p - fwco,n 5 cF4
(6.4)
I.& - J,I + I.$, - .$I S c&1’4* Proof.
(6.5)
Let us introduce two functions B and o in H’(Q) such that:
e/as
= 0,
w/an
= 0.
We will specify later on these functions. From the relation: (S, - &)” = (D,w; - d,(p:)‘, and the fact that D is constant we derive: -
Iq, - s’,)&lx i ,O
= k,(8) -
l(w, - &)De’, dX i ,O
(6.6)
‘~4 - 8,)p;
(6.7)
where: k,(B) = “l(DC - D)(w,
- h)‘e’
I
dx +
1
-0
8’ dw.
CO
We now introduce 0, , e2 as solutions of:
e; = ~p;d e,/as2 = 0.
e; = (w, - pc)w eliat = 0
We use the technique already developed for the symmetric bipolar junction (see lemma 5.5) with 8, = o and obtain: 1
c ‘(WC- P,)‘fw’Vdr = - i’(SC - s~,)(~, - (DE)Iotti s ..O -0 s0 -
k i ,O
-
Qp;o’
dx
+
c?,
+
r,
~c,cv,
-
(DJ’O’
dx
(6.8)
F. ALABAU
134
where 6, and T, satisfy:
(6.9) (6.10) Using (6.6) with 0 = 0, and f3 = ~9,we derive: E
s
‘(w, - ppc)nd dw = k,(e,) 0
s 1
+ k,(e,)
-
0
1
my/,-
%)(ly, - vd’~ dx
s 1
$,
q&;w’dx
+ c?&+ T,
(6.11)
0
0
where k,(B,) and k,(&) satisfy (6.7) with respectively 8 = 8,, 8 = 0,. Since: D& (D, - D)q$ - &,, where J??~is a constant bounded with respect to E, we derive:
= s,’ -
1
1 (v, -
c~,Kb"
+
WJ%
-
D(w,
-
s0
co,Y)l~ - E (w, - co,Y’~“~ s0
=
-k,(e,)
-
k,(e,) -
'(WC -
q@o’(Dc
- 0)~:
(6.12)
- c& - T,.
s0
IJJipg (5.4) and (6.3) in (6.7), we obtain: ; _,* + IJ8;ll,,n I cIIw’L,~. Therefore: lkml
jk,(e)l 5 ce1’4110’llm,n. It is easy to check that:
= cc~%~~ll~,~
fori=
(6.13)
1,2.
Using now (5.4), (6.3), the fact that (w, - V,)(O) = 0 and that (w, - q~,)is bounded in H’(Q) independently of E (see [S]) we obtain that: (iy, -
cp,)o’(D,
- D)&dx
(6.14)
5 CE~‘~~‘/~,,S-P
0
We set:
- Q%)2p+‘(x)with
f(x) = m(w, x I,
g(x) =
dt
p EN
where B,(x) = EC - D(v,
- (D,)‘(X)
OZ5
* f(f)e”“’
F(x) =
0
It.
W)
Let us now define w as: x
o(x)
= Cl
eeg(') dt + i ./O
* ebg(‘)F(f) i ,0
where: c, =
-5; eeg(‘)F(t) dt
j:, e-B”’ dr
.
Then w satisfies: (6.15) (6.16)
Semiconductor device and electrochemistry
equations
135
Since: ][&,][,l 5 c, it is easy to check that: (6.17)
Therefore: Ilo’lL.n 5 41wc- CP~II?$+~,~.Using
(6.15), (6.13), (6.14), (6.9), (6.10) in (6.12)
we derive:
s 1
0
s’(w, c
-
~p*2
dx + QP + I)&
illir - cp,)‘12W- t%)“~ s0
1 =
-_E
(w,
-
q~,)“$w’
s 0
dx
+
R,
E
where R, satisfies: JR,/ I ct~'"~~o'~~~,~. Using (6.2) and the fact that & is bounded in L’(Q) independently of E we obtain: IIv/c - e&Z~+2,R I CE”~ where c is independent of p. Thus: IIWr- 9Aa,* 5 C&1’4follows (see Adams [I]). In the same way as in theorem 5.1 we derive (6.4) and (6.5).
SUMMARY
AND CONCLUSIONS
We consider the asymptotic analysis developed by P. A. Markowich et (11. for the scaled semiconductor device equations. Since a small parameter is multiplying the highest order derivative and because of the jump discontinuity of the doping profile, a layer occurs at the junction in the electrostatic potential and in the electron and hole concentrations. A formal asymptotic expansion was defined by Markowich et al. The purpose of this paper is to justify this formal expansion by obtaining uniform asymptotic error estimates for the difference between the solutions of the singularly perturbed system and the first term of the asymptotic expansion. There already exist some results of this type in the literature, they principally use a result obtained by Weiss and Schmeiser [ll] which requires the existence of an isolated solution for the reduced problem. The other constraint of this method is that it requires some restrictions on the applied voltages (see [6], [S]) or on the type of devices and on the singular perturbation parameter (and consequently on the applied voltages, see [9]). These restrictions and their consequences are given in more detail in the introduction of this paper. We introduce in this article, another method based on variational techniques which provides uniform asymptotic error estimates without any restriction on the applied voltages or on the singular perturbation parameter, and without requiring the existence of an isolated solution for the reduced problem. We give two applications of this method: the case of piecewise constant symmetric bipolar junction (the results of (91 are extended here in the sense that we obtain uniform asymptotic error estimates without any restriction on the singular perturbation parameter, as it is the case in [9]), and the case of a biological membrane with a concentration of fixed charges which is constant. In both cases we derive uniform asymptotic error estimates without any restriction on the datas. Acknowledgemenfs-The
author is grateful to J. Henry and C. A. Ringhofer for many discussions.
F. ALABAU
136
REFERENCES I. ADAMSR. A., Sobolev Spaces. Academic Press, New York (1975).
2. ALABAUF., Analyse asymptotique des equations semiconducteurs, Rapport de Recherche INRIA, No. 547 (1986). 3. ALABAUF., Analyse asymptotique et simulation numerique des equations de base des semiconducteurs, Doctorat de I’llniversitt Paris VI (1987). 4. FIFE P. C., Semilinear elliptic boundary value problems with small parameters, Arch. Rot. Mech. Anal. 52, 205-232 (1973). 5. HENRYJ. & LOUROB., Singular perturbation theory applied to the electrochemistry equations in case of electroneutrality, Nonlineul Analysis 13, 787-801 (1989). P. A., A singular perturbation analysis of the fundamental semiconductor device equations, Sium 1. 6. MARKOWICH Appl. Math. 44 (October 1984). 7. MARKOWKHP. A., The Basic Semiconductor Device Equarions. Springer, New York (1986). P. A., RINGHOFER C. A., SELBERHERR S. & LANGERE., An asymptotic analysis of single junction 8. MARKOWICH semiconductor devices, MRC-Technical Summary Report No. 2527 (1983). C., Asymptotic representation of solutions of the basic semiconductor device 9. MARKOWICHP. A. & SCHMEISER equations, MRC-Technical Summary Report No. 2772 (1984). of Mathemarical Models of Semiconducror Devices. Boole Press, Dublin (1983). 10. Mocx M. S., An&is C., Asymptotic analysis of singular singularly perturbed boundary value problems, SIAM II. WEISSR. & SCHMEISER 1. Math. Anal. 17, 560-579 (1986). APPENDIX Proofs of the lemmas of Section 5
Proof. By the change of variable r = x/v;, it is easy to check that: l]cp,- I&~,~, c &and Ilot - e&n I CE”~. We set now: W(X) = (ti, - U)(X) and f(x) = -(ecr - e”)u’(x-). Then w satisfies: (e’%‘) = f’ in R and ~(-1) = w(l) = 0. Integrating both sides of this last equality, we derive: w’(x) = f(x)e+“’ - c,e-or(” where:
c, = I!, e-s?(s) ds j!, e-“r”’ ds . Therefore w(x) = ~,f(r)e-Gc”’ dr - c,s”_, e-9r”’ dt. Using the fact that (Pi is bounded independently of E (see Markowich [a], theorem 4.1, theorem 4.2) we obtain: J/w]],,, zs ~l/~]l~+~, and ]Iw’]]~,~5 ~llfl&,~. Using the estimates on (pz- w in respectively L’(Q) and L’(R), we derive: (/wI],,n I ~\‘Eand ]lw’]lo,n -E CE”~. Analogous results hold for i$ - t’. (5.3), (5.4) are obvious using (5.1) and (5.2). . LEMMA5.2 Proof. From the hypothesis v,/Ri and (o/R, are in C’(Q) for i = I,2 (see Markowich 161, theorem 2.2). Since: (f< - q,)(O) = (w, - (p,)(l) = 0 it implies that there exists x* in Cl2 such that: w;(x*) = &(x*). Therefore: velw;(x’)l 5 c. From Poisson’s equation, we derive:
E(1I;W;= S; - w; in Q,
(since S; = DCI&).
We integrate this last relation from .t-’ to x and derive: iv;,
= ; v;2(~‘) + S,(x) - S,(P)
- (v,(x) - V,(X’)).
We use the fact that S, and v, are bounded uniformly independently of E. This concludes the proof. LEMMA5.3 Proof.
From Poisson’s equation and the layer equation, we derive: (D, - D) - (d, - D(0’))
= &(vz - 9,)” + EV” + cz in R,
Semiconductor
device and electrochemistry
equations
137
where:IIJill,.n, errc.?’ v q > 0 (using the exponential decay of I$ and its derivatives as T -+ + 0~). Therefore: ]I@, - D) (6, - o(O’))ll,.,,
5
+ CE. We now analyse the term: yr = fiZ - D - d, + D(O+). We find: 4I(VI,- dDc)‘llo,o yr = fj2[e”(tie - u) - e-*c(Oc*- u)] + d2[e”u(eCM - 1) - e*(‘),(0)(e9c - l)] _ ~2[e-&v(e-J.'M - 1) _ e-+(")@)(e-G* - ])I.
In [0, a/2] we have: M = 1. Therefore: e”u(e ItrM - 1) - etio’u(0)(eaL - 1) = [e’u - e*“‘u(0)](e3c - 1). Using (5.1) for the first term at the right hand side and by the change of variable T = x/G for the two last terms, we derive: Ilvrllo.nl5 cfi. Since D, - 6, = (0, - D) - (d, - D(0’)) - y, we derive (5.7). LEMMA
5.4
Proof.
Since: S; = D, (II;and &’ = 4 w;, we derive: (S, - S,)‘(r) = (0, - l)(w, - cp,)’ + (Q - &)col + (w, - (P,)‘.
We integrate this equation from 1 to x and use the fact that: (w, - (pJ(l) = (S, - $)(I) = 0. Therefore, we derive: (S, - S,)(x) = k,(x) + (u/, - (P,)(X), where: k,(x) = E,(x) + E,(x) + with: P,(x) = “‘(0, - d,)(r)(w, iI
- o,)‘(r) dr +
L,(x)
x (Q - &)(r)(w’ + @&f’)(r) dr
sI
Using (5.4) we easily obtain (5.9) (we also use the fact that vr - (p, is bounded independently of E in H’(Q) and that D = 1). Using Cauchy-Schwarz inequality and Fubini’s theorem, we derive: lI~2llo.n, 5 IV, - co,l,,“,ll\i’~~, - 1)1/O,“, 5 cvz. By the same way, we derive:
This concludes the proof. LEMMA
5.5
Proof. We differentiate Poisson’s equation twice in R, and evaluate: E((v, - (p,)@‘,0,). Using the fact that w, satisfies Poisson’s equation and the @ is the solution of the layer equation, we derive (5.11) with: Gr = &[w”(o)e:(o) - w”(l)e;(l)] where:
F. ALNMU
138
and where: lBzl za ce41~0;l10,,V q > 0 (using the exponential decay of 9 and its derivatives at +m). The terms T,, , T2 are easily estimated using respectively the exponential decay of @and d$/dr at +a, and the inequality (5.4). Using (5.1) we derive: T ~ c~“‘118;110.111
r.3
I ~~l~:ll~.o,~
Since N/S& = 1, w is in Cm(R2) (in fact IJ can be explicitly evaluated, see 191for more details). Therefore:
and using the results of Iemma 5.2 it is easy to check that 6, satisfies (5.14) and (5.15). This concludes the proof.
Proof
We use (5.11) with 0, = (w, - (o<)and replace in (5.11) (S, - s,) by the expression given in (5.8) we derive:
4lcw, - bPt)“llo.a,+
I c ,O
S,lCw,- (p,)‘V dx = -
1 ~AY)(P:(x)(w,- 9*)‘(x)dy -
s0
1
(ulr - (P,)&(u/, - cp,)‘dy + T, + d, c .O
where 7, and 6, satisfy (5.12) and (5.14). For the first term on the right hand side we use (5.9) and obtain:
Using (5.7), we derive: 1 k,Co:(Cu,- ~0,)‘~
s c + cvzil(ul, - +‘r)“llo,n>.
-0 IC Since: \(v, - o,)(x)1 5 c& (using the fact that (vu, - (o,)(O) = 0 and that V~ - o, is bounded independently of E in H’(Q)) we derive: “I (w, - cp,)Vp;(V,- 4%)’dr 5 cll~‘;(p;(~~llo.o,= c. I\LO Therefore we obtain: elltu/, - (PrYllO.R* 5 c + G(w,
- (D,)llllo.*2
(5.16) is then easily derived from this last equality. (5.17). (5.18), (5.19) are obvious using (5.7), (5.9), (5.10) and (5.16). This concludes the proof. Proof of rhe lemmas of Section 6
LEMMA6.1 Proof
We use the techniques developed in lemma 5.5 with 6, = vu, - CO,.We derive: “1 -1 S,l(w, - @%)‘I2 b = - \ c.9, - SM(Vu, - $3,)’d.Xi 6, + 7, & ?‘,(w. - rp,f”lLd.X+ I I -0 .O .O
where: ItiC 5 cv? and /7,1 s c?‘~~w, - (P,I,.~. We just need to estimate the first term on the right hand side of the previous equation. We proceed as follows: (S, - S;)&(w, -
(4,)‘dx
=
c
.O
i
I
,O
(SC-
d@
%)z
0
Since: (s, - Q(X) = j$j, - $)‘(I) dr we derive: I(& - &)‘(x)l or v’III(S, - S,)llo,, . Therefore:
EIkW, - cp,,“II;.” 5 c + CIIG,- Se)‘llo.“~
x x
(W, - 4%)’du.
Semiconductor device and electrochemistry equations
139
Since: (S, - s,)‘(x) = D,I,u; - 6,cp; + E, - ,!$ where E, and l?Care constants bounded independently of E. We derive: ID, - si,)‘ll,.,, 5 c + 5 IID, - &1t,., . Using the results of lemma 5.3, namely the inequality (5.7) (which is still valid in case of electrochemistry equations) we obtain: This concludes the proof.
Merhods
& Apphcorions.
Vol. 14. No. 2. pp. 123-139.
1990.
0362-546X190 S3.00+ .OO D 1990 Pngamon Press plc
UNIFORM ASYMPTOTIC ERROR ESTIMATES FOR SEMICONDUCTOR DEVICE AND ELECTROCHEMISTRY EQUATIONS
Arizona
FATIHAAL-AU* State University,Departmentof Mathematics,Tempe, AZ 85287,U.S.A. (Received 1 July
1988;received for publication 20 February 1989)
Key words and phrases:Semiconductordevice equations, electrochemistry equations, singular perturbation, asymptotic expansions.
THE PAPERis organized as follows. Section 1 sets up the problem, our purpose and results and the existing results in the literature. Section 2 gives the notations used in the paper. Sections 3 and 4 set up the problem in the case of a symmetric piecewise constant junction and recall the associated asymptotic expansion. In Section 5 our method is introduced and uniform asymptotic error estimates for a symmetric piecewise constant junction are derived. Section 6 is an application of our technique for the case of a biological membrane with a constant concentration of fixed charges. Uniform asymptotic error estimates are also proved to hold in that case. A brief summary and conclusion are given. In the appendix we give the proofs of the lemmas which are used in Sections 5 and 6. 1.
INTRODUCTION
The one dimensional semiconductor device equations form a nonlinear system of three second order differential equations subject to Dirichlet boundary conditions. It is already known (see [6-81) that after an appropriate scaling the problem can be reformulated as a singularly perturbed one, the singular perturbation parameter &being the minimal normed Debye length of the device. We assume for the following that the generation-recombination term is equal to 0 and that the electron (respectively hole) diffusion and mobility coefficients are constants. Then the sciled one dimensional semiconductor device equations (see [7] for more details about the scaling) in the device domain Q = ]a, b[ are given by: EW” e = a2e+e, E - b2e%E (P,)
- N
(e%.&)’ = 0
Poisson’s equation electron continuity equation
L(e-GE) = 0 subject to the boundary conditions (BC):
hole continuity equation
(u, = e+A
(BC. 1)
v, = eUA
(BC.2)
(W i
vE = In
N + (N2 + 4a4)“’ + t( A 2d2 >
* Present address: UniversitC Bordeaux 1, Departement de MathCmatiques, 35 1 Cours de la Liberation, 33405 Talence, France. 123
F. Aumu
124
where: 0 the dependent variables are wE(electrostatic potential), u, and V, (Slotboom variables) 0 N is the normed doping profile. It has a jump discontinuity across the junction P = (x0] (where x0 belongs to &2)and it is continuous in the two disjoint subdomains f2, = ]a,~,[ and a2 = 1x0,6] 0 u,, is the normed applied voltage. We set U, = max(u,(a), I&)) and u_ = min(uA(a), u”(b)). The applied potential difference (u, - u_) is denoted by urnax. 0 16 is the singular perturbation parameter, it is also the characteristic normed Debye length of the device. For realistic diodes E is a small parameter l a2 is the scaled intrinsic carrier concentration. Since the Slotboom variables uc, V, have no physical meaning (but are very practical for mathematical analysis) it is useful to recall here the definitions of the scaled concentrations and of the scaled current densities. The scaled electron (respectively hole) concentration is given by: n,
= B2e%dc
(respectively pE = 62e%,).
The scaled current densities are given by: J n,c = d2e+cu’ L
and
J
PA
=
-d2e-$cv’
C-
In [6-81 a formal asymptotic expansion (for E small) has been associated to the solutions of the singularly perturbed system and it has been proved that these solutions exhibit interior layers at the points across which the doping profile has jump discontinuities. In this paper we consider the formal asymptotic expansion defined in the above cited references and prove that the first term of this expansion is close to the solutions of the singularly perturbed system by means of uniform asymptotic error estimates of order .sp(p > 0) for the difference between these solutions and the first term of the expansion. There already exist some results of this type in the literature. In [6] Markowich has proved uniform asymptotic error estimate for the equilibrium case (i.e. zero applied potential, in that case the system is reduced to the nonlinear Poisson’s equation). A similar result has been derived in [8] for a one dimensionalp-n junction provided that the applied potential difference is shfficiently small (an alternative proof of this result is also given in [3]). The only result providing uniform asymptotic error estimates which allows large voltages has been performed in (91 for a one dimensional symmetric piecewise constant bipolar junction. The way to obtain these results is based on a method developed in [l 1] which uses the implicit function theorem and requires the existence of an isolated solution for the reduced problem (obtained from the singularly perturbed system by setting E equal to 0). The main difficulty in applying this method for semiconductor device equations comes from the fact that there is no general proof of the existence of isolated solutions for the reduced problem. This paper introduces another alternative based on variational techniques. Its important features are that it provides uniform asymptotic error estimates without any restriction on the applied voltages and does not require the existence of isolated solutions for the reduced problem. In this paper, we first apply this technique to a one dimensional symmetric piecewise constant bipolar junction. We obtain uniform asymptotic error estimates of order E”~ without any restrictions on the applied voltages or on E. This result should be compared to the one previously mentioned, obtained in [9]. In this last paper the authors have proved for the same type of devices, uniform asymptotic error estimates of order fi, but E has to be sufficiently
Semiconductor device and electrochemistry
equations
12s
small, this smallness depending on 6 and u,,, (the applied potential difference). Therefore for u mmlarge (which is the interesting case in practice) E has to be smaller than a certain value. The limitation of such a result is that in practice E is in fact a fixed quantity and u,,, is the quantity which varies. Besides this we proved that any sufficiently smooth solution of the singularly perturbed system is well approximated by the first term of the expansion, as opposed to the method used in [9] which proves only that there exists a solution of the perturbed system which is well approximated by the first term of the expansion. The equations governing the electrical behavior of a biological membrane are the same as the semiconductor device equations. The equivalent of the doping profile is called the concentration of fixed charges. Other than in the semiconductor case this concentration is a smooth function of the spatial variable. Layers occur at the boundary where in general the Dirichlet conditions do not satisfy the electroneutrality relation (obtained by setting the second term in the Poisson’s equation equal to 0). The same type of asymptotic analysis as for the semiconductor case has been performed in [5]. In this last paper the authors have proved that the asymptotic error converges to 0 as E goes to 0 but they have not obtained any estimates. In our paper we obtain uniform asymptotic error estimates provided that the concentration of fixed charges is constant. Even under this restriction there is no proof that there exists an isolated solution of the corresponding problem (in the case of a symmetric piecewise constant bipolar junction it was possible to prove such a result because of the symmetry properties of the device), therefore the method used in (91 cannot be applied. 2. DEFINITIONS
AND
NOTATIONS
We now give definitions of function spaces and norms which will be used in the sequel. We denote by Lq(Q) the space of q-integrable real valued functions defined on S2 with the norm:
Ilfilo,,,~ =
({nh91q by”
(denotedalso by IlfliLq).
*L”(R) is the space of bounded functions on R with the norm: WesetII l10.2.R = II IIO.R
IlflL.n =
sup IfWl XER
We denote the Lz scalar product by:
(denoted also by Ilfllr_). n
C”(Q) stands for the space of all functions defined on R which together with their derivatives of order up to m are continuous in S2. We denote by Hm(S2) the completion of In E c”(n),
Il4lm.n< +a1
with respect to the norm 1) Ilm,R, where:
We denote by H&2) the space lu E H’(Q), u/S2 = 0). We set ]~],,a = Ildu/dxllo,u for u E H’(O). In all the sequel we will denote by c a generic nonnegative constant independent of E.
126
F. Auam 3. SYMMETRIC
PIECEWISE
CONSTANT
BIPOLAR
JUNCTION
In this section we give the definition of a symmetric piecewise constant bipolar junction. As it has been proved in [9] the performance of such devices is described by a two point boundary value problem (using the fact that for such devices the full problem (P,) has symmetry properties). We recall here the formulation of this two point boundary value problem. We set:
62 = l-1, I[. A symmetric piecewise constant bipolar junction satisfies: ox,=0
(point where the doping profile has a jump discontinuity)
0 N(x) = i
for 05 x 5 1 for -1 lx50
_;
0 UA(_I) = -1(,4(l). This last restriction is not really a constraint since the electrostatic potential vE is defined up to an arbitrary additive constant. The only serious restriction comes from the fact that N is assumed to be odd. Using this last assumption Markowich and Schmeiser in [9] have replaced the initial problem (P,) by the following two point boundary value problem: &Vi = 62e*v4 E - B2e+vE - 1 e%4: = c,
forOIx5
1
e-+zui = -c, (SP,)
c; = 0
IL,(O)=
w,(O) = 0, y/,(l) = ln
1 + (1 + 4CJ4)“2 + u/I(l) 2d2 >
(
= esUA(‘), v,(l) = euAo).
u,(O), u,(l)
Given a solution (w,, ut, u,) of (SP,) we extend it to Cl (keeping the same notations for the extended functions) by setting: V,(-x)
= - V,(X)
05x51
u,(-x)
= %(X)
05x51
Q(-x)
= u,(x)
OlXll.
It is easy to check that these new functions now defined on all Sz define a solution of the original problem (P,). In the next two sections we shall only consider solutions of (P,) defined by this way. 4. ASYMPTOTIC
EXPANSION
FOR
BIPOLAR
SYMMETRIC
PIECEWISE
CONSTANT
JUNCTIONS
As we have mentioned in the introduction the semiconductor device equations form a singularly perturbed system, the normed Debye length being the singular perturbation parameter. In several papers Markowich, Ringhofer et al. (see [6-81) have introduced a formal
Semiconductor device and electrochemistry
127
equations
asymptotic expansion associated to the solutions of the singularly perturbed problem (P*). In the case of a symmetric piecewise constant bipolar junction, the different terms of this expansion have symmetric properties. In [9] the authors have used these properties to simplify the asymptotic analysis. In particular they have proved that boundary layer terms occur at x = 0 for the solutions of (SP,) (instead of internal layer terms for (P,)). We recall briefly here the procedure used in their article and give the equations for the reduced and boundary layer problems. Following standard techniques of asymptotic analysis, the authors of [9] look at the expansions of the form:
where r(x) = x/v% and where for a given number (Y> 0, M is a C” function satisfying: M(x) =
1
for 0 s x 5 ‘y and 0 5 M(x) I 1 for 2 I x I (Y 2 2
0
forcrsxzz
1.
The functions v/;, ui, Vi (solutions of reduced problems) are independent of E. The functions marked with *are defined on [0, +a[ and decay exponentially to 0 together with their derivatives as 7 goes to +a0 (boundary layer terms). 7 is the fast independent variable, x is the slow one. We will only consider here the first term of these expansions (for which one we drop the index 0). (w, u, V) is a solution of what is called the reduced problem (it is expected to be a “good approximation” to solutions of (SP,) away from the layer area). (I,?, li, D) are solutions of boundary layer problems (they are corrections to reduced solutions inside the layer area). In [9] it has been proved that the reduced problem is given by: 0 = a2eGu - a*e-+u - 1
.
e&u’ = c, e-+u = -c, c’ = 0 (RP)
u(0) = v(O), u(1) = e-“A”), u(1) = e”A”’ y/(O) = -In(a),
w(1) = In
1 + (1 + 464)“* 26*
(
>
+ UAl)
where b is a number depending on 6 and ~~(1) (see [9] for more details on the dependence). In [9] the authors also give the boundary layer problem corresponding to the electrostatic potential, that is: 2
= g2e+:Cot+@u(()) - ~2e-+(o)-~~(())
-
1
for
7 >
0
They have also proved that: ti = fi = 0. We recall here the asymptotic expansions of n,, pE as they have been performed in (6-91. The reduced electron (respectively hole) concentration is
F. ALABAU
128
given by: n = 6’e*u (respectivelyp = #e-$‘o). The reduced current densities are: J,, = d2eGu’, 6’e-ICv’. The corresponding layer terms for respectively n and p are:
Jp = -
ii(~)
5. UNIFORM
=
82e+!‘co),(0)(e~cti- 1) and
ASYMPTOTIC
ERROR
j(7)
=
ESTIMATES
62e-~(0’v(0)(e-G”’ - 1). FOR
SYMMETRIC
JUNCTIONS
In this section we first introduce some notations which simplify the asymptotic error analysis. We, then, give intermediate results which are needed in order to prove the main theorem of this section, namely that for a symmetric piecewise constant bipolar junction the solutions of the singularly perturbed problem (SP,) are well represented by the first term of the asymptotic expansion with no restriction on the applied voltages or on the singular perturbation parameter. As it has been said in the introduction this result should be compared to the one obtained in [9]. In this last paper the authors have obtained a better order of convergence for the uniform asymptotic error estimates but they have to restrict the size of the singular perturbation parameter which has to be sufficiently small. This restriction, as a consequence, leads to a restriction on the applied voltages (see [9] for more details), which is not convenient on a practical point of view since physicists are interested in applying higher voltages to a fixed device (this means that the voltage varies while E is kept fixed). In order to prove these estimates the authors have used the results of [ 1l] which involve the strong form of the equation and a version of the implicit function theorem. We develop here a method based on the weak form of the equation without using any implicit function theorem. The consequence is that our proof does not require that the reduced problem admits isolated solutions as it is the case in 191. Let us introduce now some notation which will be used in the sequel to this paper. We set: co,(x) = V(X) + S,(x)
aw
=
(4
= (4
where
S(7(X))MX)
7(x)
=
P,)(X),
SW
= (n + P)(X)
- P,)(X),
D(x)
= (n - P)(X).
+
We also associate to p,ethe solutions ti,, & of the two corresponding means that li, and t’, satisfy:
2 dz
continuity equations, this
(e%;)’ = 0 in 52 and (e-%3;)’ =0 in R + (BC. 1) 1 l + (BC.2) We now set: S; = fi, + fi,
where A, = 6*e%, and 8, = 8*eMpcfie & = ri, - p,
S(T) = 6
2e~‘0,+~‘T+4())
+
~2e-w3’-wv(())
for T > 0
b(7) = @eN3’+&7+40) _ ~Ze4L’O’-S’c7440) 9,(x) =
@(7(X)),
&(x)
=
9(7(x)),
b,(x)
for
7 >
0
= B(T(X)).
The main result of this section holds in theorem 5.1. In order to prove this theorem, we need to prove several lemmas. Since these proofs are very technical they are deferred to the appendix. We will also constantly use the two following facts which have been proved respectively in [6]
Semiconductor
device and electrochemistry
129
equations
(theorem 2.1) and in [5] (theorem 3), without always recalling them: v/~, u,, u, are uniformly bounded independently of E (see 161) v/+-~ is bounded in H’(0) independently of E (see [5]). LEMMA5.1. We consider a symmetric piecewise constant bipolar junction, then the following estimates are satisfied: (5-L)
112% - ull,.* + II& - &.,l3 = c& II& -
41,fl + Ilfie- 4.n = cP4
(5.2)
IISE- ~IIP(R) + IIQ - alr~~n, 5 4 II& - ~IIOJI+ IID, - Dllo,* 5 cP4.
(5.3) (5.4)
The proof is deferred to the appendix. LEMMA5.2. Assume the hypothesis of lemma 5.1, then:
aYf(x)l 5 c
VXE [O, l]
(5.5)
am)l
v x E [O, 11.
(5.6)
5c
The proof is also deferred to the appendix. LEMMA5.3. Let the hypothesis of lemma 5.1 hold. Then: ID, - Qll0.R 5
4ku/, - C4c)“l/0,R + cfi.
(5.7)
The proof of this lemma is also deferred to the appendix. .
LEMMA5.4. Assume the hypothesis of lemma 5.1, then: (SC - Q(x)
= k,(x) + (WC- K)(X)
in Sz
(5.8)
where k, satisfies
-1’41jDE - dclJO,* + c&1’4 IlhlL,* 5 CC
(5.9)
kllo,n I CllD,- DEllo,n+ cb5. The proof is deferred to the appendix. The next lemma gives the weak form of the twice differentiated
(5.10)
Poisson’s equation.
LEMMA 5.5. Assume the hypothesis of lemma 5.1. Then for 8, in H2(Q), such that 0,/X2 = 0 the following equality is satisfied:
1
& t(v/c - &“e:dx s0
+
c ,O
S,(Jy, - ql,)‘e;dx
= -
l(S, - ~&9:egx s0
+ 6, + T,
(5.11)
F. ALABAIJ
130
where T, and 6, satisfy: Ir,l =
c~“411~~llo,*
(5.12)
I T,I 5
cwclllP,n
(5.13)
(&I d cfi
if 0, 5 ry, - qr
(5.14)
Id s 44ll,,n.
(5.15)
The proof is deferred to the appendix. LEMMA5.6. Let the hypothesis of lemma 5.1 hold. Then the following estimates hold:
mw, - cpJIIo,o =c lla - &llo.n5 4 Ilk, Ilq-I5
(5.16) (5.17)
c.P4
(5.18)
llhllo,n= cfi
(5.19)
where k, has been defined in lemma 5.4. The previous lemmas allow us to prove the main result of this section, namely theorem 5.1. 5.1. We consider a symmetric piecewise constant bipolar junction, then the following estimates hold: THEOREM
IIK - 4%llce,R + II%- n - fiMll,,* + IIPE-P I&
- m41ce,*
5 CE”4
(5.20) (5.21)
- J,I + /.I,,, - Jpl 5 cE”4.
Proof. By using symmetry we work in &, (w,, u,, u,) being a solution of (SP,) extended to h by symmetry as it has been defined in Section 3. We introduce w, as the solution of the following problem. ” = wc - q,c)zp+l in Rz = IO, 11 WC I w,(O) = o,(l) = 0. From Markowich [6] we know that vt and qE are continuous in Q,, therefore w, is a C2 function. Since we have o,(O) = w,(l) = 0 there exists x* in R2 such that w:(x*) = 0. Thus:
Il4ll,.n, 5 IIWE - dl?~+1.n*. We now use (5.11) with c9,= w, and obtain: -
sk4 -
$Pcp:bldx -
E
0
-
ql,)“w;
dx = - 6, -
r,
where & and T, satisfy (5.15) and (5.13). Clearly: SEw: - $,cp; = (I!?,- $)(w, - co,)’ + S&E - v7,)’ + (S, - s;)Vd
(5.22)
Semiconductor device and electrochemistry
equations
131
holds. Using (5.8) in this last equality, we derive: S,& -
s&7;=
tv, -
9,)(w, - 9,)’ +
~,tv,- 9,)’ + (WC- 9,)9I + k,(ly, -
9J + kl9i (5.23)
where kr satisfies (5.9) and (5.10). Using the fact that (w,, uE, u,) is a solution of (SP,) it is easy to check that: for x in slz. S;(x) = d,(x)co:(x) Therefore: 9;(x) = 3; - (D, - 1)9j. Using this last relation in (5.23), we derive: SE9; - $9;
= [+(I& - 90,)z + Q&
- 9Jl’ + [k,(Wc - 9J’ + kI9: - (PC - 9,)(&
- 1)9J*
Therefore: 1
-
I(S,w: - $9;)w:
dX =
s0
l(w, - 9,)2p+2(t(W, - 9,) + SC)d.X+
s0
R,w; dx
(5.24)
s0
where: R, = (w, - co,)@, - 1)9: - kr(~e - 9,)’ - k,9:.
(5.25)
By using (5.8) once again we obtain: +(9, - 9,) + 9, = &S, + SE) - $c, . Using this last relation in (5.24) and then using (5.24) in (5.22), we obtain: + r(s, + Q(w, r .,O
- 9J 2p+2dx + (2P + 1)e ll(w, - 9J12(w, - UltJZPb s0 1
1
R&dx
=s0
k,(Wc - 9,)2p+2 dx - T, - r?&.
+ f
(5.26)
s0
We need now to estimate the righthand side of this last expression, we proceed as follows:
Using (5.4) and (5.18) in this last equality, we obtain: (5.27) Using now the fact that the function (5, + &) is bounded away from 0 by a nonnegative number independent of E (see Markowich [6], theorem 2.1) and the inequalities (5.27), (5.18), (5.13) and (5.15) in the relation (5.26), we finally obtain: 11~~- 9~ll0,2p+~,a~5 ccl’4 where c is a constant independent of p. (see Adams (11). Therefore we obtain: As P --) +a IIV’c- ~Io,z/J+z.R~ + IIWE- 9r114z I)IJI~- 9EII_,RZ I CE”~. Also we have:
llnc - n - AMI/CO.R> 5 11%-
fisllre,R2 + II% - n - fima,nZ.
132
F. ALABAU
Since: n, - ii, = 62e”c(u, - tl,) + g2(eGc,- eVc)& it is easy to check that: iln, - tiJ+*
5 c&1’4.
Since: ii, - n - fiM = 62ev$i c - u) + a2(e+Gu- e”O’u(0))(eGc - 1) in [0, (~121we obtain: II& - n - iiMll,,a2 d cd?. Analogous proof holds of (p, - p - fiM). This proves (5.20). (5.21) is easy to deduce using (5.20). 6. ELECTROCHEMISTRY
EQUATIONS
We also apply our technique to the case of biological membranes, for which the equations are the same as for the semiconductor device ones except that the equivalent of the doping profile (called the concentration of fixed charges) is a smooth function of the spatial variable and that layers occur at the boundary where the electroneutrality condition is not satisfied. In [5] Henry and Louro have performed the same type of asymptotic analysis as in [6]. We will consider here this asymptotic expansion and we will see that under minor modifications, the method used in Section 5 for symmetric piecewise constant bipolar junctions is still valid for the biological membranes. We will keep in this section the notation previously introduced. We first give the new boundary conditions associated to electrochemistry equations. The equations are given by (P,) (see Section 1) in S2 = 10, 11. The boundary conditions are different: forx = 0 (UE, v,, w,) = (n0, uo* wo) (EBC) forx= 1. L(UE,t)E, w,) = (u,, v1, w,) We set: d2 = 1 (without loss of generality). For the sake of simplicity, we will assume that boundary layers occur only at x = 0, that means, we assume:
t
e”Ouo-
e-$Ou
e+:'u,
e-+iv
-
0 1
-
N(O)
#
0
-
N(1)
=
0.
The reduced problem is then given by: e3, - e-*y - N = 0 (e&u’)’ = 0 (e-“v’)’ = 0 i + boundary conditions (REBC) where (REBC) is given by: (u, ZJ)= (uo, u,) at x = 0
(u, u) = (ur, u,) at x = 1 I,Y= w1 at x = 1 and w(O) = In
N(0) + (N2(0) + 4UoUo)“2
2uo
Semiconductor
device and electrochemistry
133
equations
The layer problem is given by:
d2@
!b(o)+~~o_
z=e
e
+co)-quo - N(0)
for T > 0
i
k!w-x =
!llo
l//(O), @(+oo) = 0.
-
We will assume in the sequel of this section that: N=
1.
(6.1)
It is easy to check that the lemmas 5.1, 5.2, 5.3 are still valid. The lemma (5.6) requires some modifications, in order to prove the main result of this section (theorem 6.1). LEMMA6.1. We assume (6.1) and that uo, vo, ui, u1 are nonnegative. estimates hold: fll(YG - (P~)“IlO.R 5 c
Then the following (6.2) (6.3)
ID, - ~cll0.R 5 cfi. The proof is deferred to the appendix. This enables us to prove the following theorem. THEOREM6. I. Assume the hypothesis of lemma 6.1. Then the following estimates hold:
llv, - dl.m,n + II& - n - fi~ll,,*
+ lb, - p - fwco,n 5 cF4
(6.4)
I.& - J,I + I.$, - .$I S c&1’4* Proof.
(6.5)
Let us introduce two functions B and o in H’(Q) such that:
e/as
= 0,
w/an
= 0.
We will specify later on these functions. From the relation: (S, - &)” = (D,w; - d,(p:)‘, and the fact that D is constant we derive: -
Iq, - s’,)&lx i ,O
= k,(8) -
l(w, - &)De’, dX i ,O
(6.6)
‘~4 - 8,)p;
(6.7)
where: k,(B) = “l(DC - D)(w,
- h)‘e’
I
dx +
1
-0
8’ dw.
CO
We now introduce 0, , e2 as solutions of:
e; = ~p;d e,/as2 = 0.
e; = (w, - pc)w eliat = 0
We use the technique already developed for the symmetric bipolar junction (see lemma 5.5) with 8, = o and obtain: 1
c ‘(WC- P,)‘fw’Vdr = - i’(SC - s~,)(~, - (DE)Iotti s ..O -0 s0 -
k i ,O
-
Qp;o’
dx
+
c?,
+
r,
~c,cv,
-
(DJ’O’
dx
(6.8)
F. ALABAU
134
where 6, and T, satisfy:
(6.9) (6.10) Using (6.6) with 0 = 0, and f3 = ~9,we derive: E
s
‘(w, - ppc)nd dw = k,(e,) 0
s 1
+ k,(e,)
-
0
1
my/,-
%)(ly, - vd’~ dx
s 1
$,
q&;w’dx
+ c?&+ T,
(6.11)
0
0
where k,(B,) and k,(&) satisfy (6.7) with respectively 8 = 8,, 8 = 0,. Since: D& (D, - D)q$ - &,, where J??~is a constant bounded with respect to E, we derive:
= s,’ -
1
1 (v, -
c~,Kb"
+
WJ%
-
D(w,
-
s0
co,Y)l~ - E (w, - co,Y’~“~ s0
=
-k,(e,)
-
k,(e,) -
'(WC -
q@o’(Dc
- 0)~:
(6.12)
- c& - T,.
s0
IJJipg (5.4) and (6.3) in (6.7), we obtain: ; _,* + IJ8;ll,,n I cIIw’L,~. Therefore: lkml
jk,(e)l 5 ce1’4110’llm,n. It is easy to check that:
= cc~%~~ll~,~
fori=
(6.13)
1,2.
Using now (5.4), (6.3), the fact that (w, - V,)(O) = 0 and that (w, - q~,)is bounded in H’(Q) independently of E (see [S]) we obtain that: (iy, -
cp,)o’(D,
- D)&dx
(6.14)
5 CE~‘~~‘/~,,S-P
0
We set:
- Q%)2p+‘(x)with
f(x) = m(w, x I,
g(x) =
dt
p EN
where B,(x) = EC - D(v,
- (D,)‘(X)
OZ5
* f(f)e”“’
F(x) =
0
It.
W)
Let us now define w as: x
o(x)
= Cl
eeg(') dt + i ./O
* ebg(‘)F(f) i ,0
where: c, =
-5; eeg(‘)F(t) dt
j:, e-B”’ dr
.
Then w satisfies: (6.15) (6.16)
Semiconductor device and electrochemistry
equations
135
Since: ][&,][,l 5 c, it is easy to check that: (6.17)
Therefore: Ilo’lL.n 5 41wc- CP~II?$+~,~.Using
(6.15), (6.13), (6.14), (6.9), (6.10) in (6.12)
we derive:
s 1
0
s’(w, c
-
~p*2
dx + QP + I)&
illir - cp,)‘12W- t%)“~ s0
1 =
-_E
(w,
-
q~,)“$w’
s 0
dx
+
R,
E
where R, satisfies: JR,/ I ct~'"~~o'~~~,~. Using (6.2) and the fact that & is bounded in L’(Q) independently of E we obtain: IIv/c - e&Z~+2,R I CE”~ where c is independent of p. Thus: IIWr- 9Aa,* 5 C&1’4follows (see Adams [I]). In the same way as in theorem 5.1 we derive (6.4) and (6.5).
SUMMARY
AND CONCLUSIONS
We consider the asymptotic analysis developed by P. A. Markowich et (11. for the scaled semiconductor device equations. Since a small parameter is multiplying the highest order derivative and because of the jump discontinuity of the doping profile, a layer occurs at the junction in the electrostatic potential and in the electron and hole concentrations. A formal asymptotic expansion was defined by Markowich et al. The purpose of this paper is to justify this formal expansion by obtaining uniform asymptotic error estimates for the difference between the solutions of the singularly perturbed system and the first term of the asymptotic expansion. There already exist some results of this type in the literature, they principally use a result obtained by Weiss and Schmeiser [ll] which requires the existence of an isolated solution for the reduced problem. The other constraint of this method is that it requires some restrictions on the applied voltages (see [6], [S]) or on the type of devices and on the singular perturbation parameter (and consequently on the applied voltages, see [9]). These restrictions and their consequences are given in more detail in the introduction of this paper. We introduce in this article, another method based on variational techniques which provides uniform asymptotic error estimates without any restriction on the applied voltages or on the singular perturbation parameter, and without requiring the existence of an isolated solution for the reduced problem. We give two applications of this method: the case of piecewise constant symmetric bipolar junction (the results of (91 are extended here in the sense that we obtain uniform asymptotic error estimates without any restriction on the singular perturbation parameter, as it is the case in [9]), and the case of a biological membrane with a concentration of fixed charges which is constant. In both cases we derive uniform asymptotic error estimates without any restriction on the datas. Acknowledgemenfs-The
author is grateful to J. Henry and C. A. Ringhofer for many discussions.
F. ALABAU
136
REFERENCES I. ADAMSR. A., Sobolev Spaces. Academic Press, New York (1975).
2. ALABAUF., Analyse asymptotique des equations semiconducteurs, Rapport de Recherche INRIA, No. 547 (1986). 3. ALABAUF., Analyse asymptotique et simulation numerique des equations de base des semiconducteurs, Doctorat de I’llniversitt Paris VI (1987). 4. FIFE P. C., Semilinear elliptic boundary value problems with small parameters, Arch. Rot. Mech. Anal. 52, 205-232 (1973). 5. HENRYJ. & LOUROB., Singular perturbation theory applied to the electrochemistry equations in case of electroneutrality, Nonlineul Analysis 13, 787-801 (1989). P. A., A singular perturbation analysis of the fundamental semiconductor device equations, Sium 1. 6. MARKOWICH Appl. Math. 44 (October 1984). 7. MARKOWKHP. A., The Basic Semiconductor Device Equarions. Springer, New York (1986). P. A., RINGHOFER C. A., SELBERHERR S. & LANGERE., An asymptotic analysis of single junction 8. MARKOWICH semiconductor devices, MRC-Technical Summary Report No. 2527 (1983). C., Asymptotic representation of solutions of the basic semiconductor device 9. MARKOWICHP. A. & SCHMEISER equations, MRC-Technical Summary Report No. 2772 (1984). of Mathemarical Models of Semiconducror Devices. Boole Press, Dublin (1983). 10. Mocx M. S., An&is C., Asymptotic analysis of singular singularly perturbed boundary value problems, SIAM II. WEISSR. & SCHMEISER 1. Math. Anal. 17, 560-579 (1986). APPENDIX Proofs of the lemmas of Section 5
Proof. By the change of variable r = x/v;, it is easy to check that: l]cp,- I&~,~, c &and Ilot - e&n I CE”~. We set now: W(X) = (ti, - U)(X) and f(x) = -(ecr - e”)u’(x-). Then w satisfies: (e’%‘) = f’ in R and ~(-1) = w(l) = 0. Integrating both sides of this last equality, we derive: w’(x) = f(x)e+“’ - c,e-or(” where:
c, = I!, e-s?(s) ds j!, e-“r”’ ds . Therefore w(x) = ~,f(r)e-Gc”’ dr - c,s”_, e-9r”’ dt. Using the fact that (Pi is bounded independently of E (see Markowich [a], theorem 4.1, theorem 4.2) we obtain: J/w]],,, zs ~l/~]l~+~, and ]Iw’]]~,~5 ~llfl&,~. Using the estimates on (pz- w in respectively L’(Q) and L’(R), we derive: (/wI],,n I ~\‘Eand ]lw’]lo,n -E CE”~. Analogous results hold for i$ - t’. (5.3), (5.4) are obvious using (5.1) and (5.2). . LEMMA5.2 Proof. From the hypothesis v,/Ri and (o/R, are in C’(Q) for i = I,2 (see Markowich 161, theorem 2.2). Since: (f< - q,)(O) = (w, - (p,)(l) = 0 it implies that there exists x* in Cl2 such that: w;(x*) = &(x*). Therefore: velw;(x’)l 5 c. From Poisson’s equation, we derive:
E(1I;W;= S; - w; in Q,
(since S; = DCI&).
We integrate this last relation from .t-’ to x and derive: iv;,
= ; v;2(~‘) + S,(x) - S,(P)
- (v,(x) - V,(X’)).
We use the fact that S, and v, are bounded uniformly independently of E. This concludes the proof. LEMMA5.3 Proof.
From Poisson’s equation and the layer equation, we derive: (D, - D) - (d, - D(0’))
= &(vz - 9,)” + EV” + cz in R,
Semiconductor
device and electrochemistry
equations
137
where:IIJill,.n, errc.?’ v q > 0 (using the exponential decay of I$ and its derivatives as T -+ + 0~). Therefore: ]I@, - D) (6, - o(O’))ll,.,,
5
+ CE. We now analyse the term: yr = fiZ - D - d, + D(O+). We find: 4I(VI,- dDc)‘llo,o yr = fj2[e”(tie - u) - e-*c(Oc*- u)] + d2[e”u(eCM - 1) - e*(‘),(0)(e9c - l)] _ ~2[e-&v(e-J.'M - 1) _ e-+(")@)(e-G* - ])I.
In [0, a/2] we have: M = 1. Therefore: e”u(e ItrM - 1) - etio’u(0)(eaL - 1) = [e’u - e*“‘u(0)](e3c - 1). Using (5.1) for the first term at the right hand side and by the change of variable T = x/G for the two last terms, we derive: Ilvrllo.nl5 cfi. Since D, - 6, = (0, - D) - (d, - D(0’)) - y, we derive (5.7). LEMMA
5.4
Proof.
Since: S; = D, (II;and &’ = 4 w;, we derive: (S, - S,)‘(r) = (0, - l)(w, - cp,)’ + (Q - &)col + (w, - (P,)‘.
We integrate this equation from 1 to x and use the fact that: (w, - (pJ(l) = (S, - $)(I) = 0. Therefore, we derive: (S, - S,)(x) = k,(x) + (u/, - (P,)(X), where: k,(x) = E,(x) + E,(x) + with: P,(x) = “‘(0, - d,)(r)(w, iI
- o,)‘(r) dr +
L,(x)
x (Q - &)(r)(w’ + @&f’)(r) dr
sI
Using (5.4) we easily obtain (5.9) (we also use the fact that vr - (p, is bounded independently of E in H’(Q) and that D = 1). Using Cauchy-Schwarz inequality and Fubini’s theorem, we derive: lI~2llo.n, 5 IV, - co,l,,“,ll\i’~~, - 1)1/O,“, 5 cvz. By the same way, we derive:
This concludes the proof. LEMMA
5.5
Proof. We differentiate Poisson’s equation twice in R, and evaluate: E((v, - (p,)@‘,0,). Using the fact that w, satisfies Poisson’s equation and the @ is the solution of the layer equation, we derive (5.11) with: Gr = &[w”(o)e:(o) - w”(l)e;(l)] where:
F. ALNMU
138
and where: lBzl za ce41~0;l10,,V q > 0 (using the exponential decay of 9 and its derivatives at +m). The terms T,, , T2 are easily estimated using respectively the exponential decay of @and d$/dr at +a, and the inequality (5.4). Using (5.1) we derive: T ~ c~“‘118;110.111
r.3
I ~~l~:ll~.o,~
Since N/S& = 1, w is in Cm(R2) (in fact IJ can be explicitly evaluated, see 191for more details). Therefore:
and using the results of Iemma 5.2 it is easy to check that 6, satisfies (5.14) and (5.15). This concludes the proof.
Proof
We use (5.11) with 0, = (w, - (o<)and replace in (5.11) (S, - s,) by the expression given in (5.8) we derive:
4lcw, - bPt)“llo.a,+
I c ,O
S,lCw,- (p,)‘V dx = -
1 ~AY)(P:(x)(w,- 9*)‘(x)dy -
s0
1
(ulr - (P,)&(u/, - cp,)‘dy + T, + d, c .O
where 7, and 6, satisfy (5.12) and (5.14). For the first term on the right hand side we use (5.9) and obtain:
Using (5.7), we derive: 1 k,Co:(Cu,- ~0,)‘~
s c + cvzil(ul, - +‘r)“llo,n>.
-0 IC Since: \(v, - o,)(x)1 5 c& (using the fact that (vu, - (o,)(O) = 0 and that V~ - o, is bounded independently of E in H’(Q)) we derive: “I (w, - cp,)Vp;(V,- 4%)’dr 5 cll~‘;(p;(~~llo.o,= c. I\LO Therefore we obtain: elltu/, - (PrYllO.R* 5 c + G(w,
- (D,)llllo.*2
(5.16) is then easily derived from this last equality. (5.17). (5.18), (5.19) are obvious using (5.7), (5.9), (5.10) and (5.16). This concludes the proof. Proof of rhe lemmas of Section 6
LEMMA6.1 Proof
We use the techniques developed in lemma 5.5 with 6, = vu, - CO,.We derive: “1 -1 S,l(w, - @%)‘I2 b = - \ c.9, - SM(Vu, - $3,)’d.Xi 6, + 7, & ?‘,(w. - rp,f”lLd.X+ I I -0 .O .O
where: ItiC 5 cv? and /7,1 s c?‘~~w, - (P,I,.~. We just need to estimate the first term on the right hand side of the previous equation. We proceed as follows: (S, - S;)&(w, -
(4,)‘dx
=
c
.O
i
I
,O
(SC-
d@
%)z
0
Since: (s, - Q(X) = j$j, - $)‘(I) dr we derive: I(& - &)‘(x)l or v’III(S, - S,)llo,, . Therefore:
EIkW, - cp,,“II;.” 5 c + CIIG,- Se)‘llo.“~
x x
(W, - 4%)’du.
Semiconductor device and electrochemistry equations
139
Since: (S, - s,)‘(x) = D,I,u; - 6,cp; + E, - ,!$ where E, and l?Care constants bounded independently of E. We derive: ID, - si,)‘ll,.,, 5 c + 5 IID, - &1t,., . Using the results of lemma 5.3, namely the inequality (5.7) (which is still valid in case of electrochemistry equations) we obtain: This concludes the proof.