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Existence of Weak Solutions of the Drift Diffusion Model Coupled with Maxwell's Equations
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
204, 655]676 Ž1996.
0460
Existence of Weak Solutions of the Drift Diffusion Model Coupled with Maxwell’s Equations F. Jochmann Fachbereich Mathematik, Technische Uni¨ ersitaet Berlin, Strasse des 17, Juni 136, 10623 Berlin, Germany Submitted by Colin Rogers Received December 13, 1994
1. INTRODUCTION The aim of this paper is to prove existence of weak solutions of the drift diffusion model for semiconductors involving Maxwell’s equations, which is governed by the equations j1 s yD 1 Ž x, r 1 , r 2 . =r 1 q m 1 r 1 E
Ž 1.1.
j 2 s yD 2 Ž x, r 1 , r 2 . =r 2 y m 2 r 2 E
Ž 1.2.
t r t s y= ? j k y R Ž x, r 1 , r 2 . ,
k g 1, 2 4
Ž 1.3.
t E s curl H q j 2 y j1
Ž 1.4.
t H s ycurl E
Ž 1.5.
= ? E s r 1 y r 2 q C,
= ? H s 0.
Ž 1.6.
r 1 , r 2 denote the densities of j1 , j 2 denote the current densities of the holes and electrons, respectively. The self-consistent electromagnetic field ŽE, H. obeys Maxwell’s equations Ž1.4., Ž1.5., and Ž1.6.. The unknown functions r 1 , r 2 , E, H depend on Ž t, x . g w0, `., where t, x denote the time and space variable resp. V ; R 3 is a bounded Lipschitz-domain with V s GD j GN , where GD , GN are disjoint subsets of V. GD represents the perfectly conducting Ohmic contacts and GN represents the insulating boundary of the semiconductor device. The mobilities m 1 , m 2 of the holes and electrons resp. are assumed to be positive constants. The diffusion coefficients D 1 , D 2 and the recombination generation rate R are functions of the densities r 1 , r 2 and the space variable x. C is a bounded function of x, which describes the doping profile of the device. 655 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
656
F. JOCHMANN
Mathematical analysis of the drift diffusion model for semiconductors has been presented in w2, 3, 5, 8]10x. But in all these references Maxwell’s equations Ž1.4. ] Ž1.6. are replaced by Poisson’s equation for an electrostatic field E s y=V with yDV s r 1 y r 2 y C, which simplifies the mathematical analysis, since the electric field depends continuously on the total charge density with respect to the L2 Ž V . topology in this case. However, at very high frequencies, i.e., if the wavelength of the electromagnetic field has the same order than the size of the device, we have to replace Poisson’s equation by Maxwell’s equations Ž1.4., Ž1.5.. Since we do not have an electrostatic potential for the electric field we have to find new physically reasonable boundary conditions for the electromagnetic field. On the perfectly conducting Ohmic contacts GD the tangential component of the electric field must vanish. The tangential component of the magnetic field is prescribed on the perfectly insulating boundary. The system Ž1.1. ] Ž1.6. is supplemented by the initial boundary conditions
rsU ª
n ? jk s 0
on GD
on GN for k g 1, 2 4
ª
nnEs0
on GD
ª
n?Es0
on GN
ª
n n H sª n n HG
on GN
r Ž 0, x . s r 0 Ž x . E Ž 0, x . s E 0 Ž x . ,
H Ž 0, x . s H 0 Ž x . .
Ž 1.7. Ž 1.8. Ž 1.9. Ž 1.10. Ž 1.11. Ž 1.12. Ž 1.13.
Obviously, E 0 , H 0 , r 0 , and H G must be consistent with Maxwell’s equations Ž1.4. ] Ž1.6. and the initial-boundary conditions, that means = ? H0 s 0
Ž 1.14.
= ? E 0 s r 1, 0 y r 2 , 0 q C
Ž 1.15.
ª
n ? E0 s 0
on GN .
Ž 1.16.
Moreover, by Ž1.8., Ž1.10., Ž1.11., and Ž1.4. we have to require ª
n ? Ž curl H G . s 0
on GN .
Ž 1.17.
657
THE DRIFT DIFFUSION MODEL
Now, if ŽE, H. is a solution of Maxwell’s equations Ž1.4., Ž1.5., and the initial boundary conditions Ž1.9., Ž1.11., and Ž1.13., the equations Ž1.6. and the boundary condition Ž1.10. are automatically fulfilled for all times t G 0. Therefore, problem Ž1.1. ] Ž1.6., Ž1.7. ] Ž1.13. can be reformulated as j1 s yD 1 Ž x, r 1 , r 2 . =r 1 q m 1 r 1 E
Ž 1.18.
j 2 s yD 2 Ž x, r 1 , r 2 . =r 2 y m 2 r 2 E
Ž 1.19.
t r t s y= ? j k y R Ž x, r 1 , r 2 . ,
k g 1, 2 4
Ž 1.20.
t E s curl h q j 2 y j1 y j 0
Ž 1.21.
t h s ycurl E y t H G
Ž 1.22.
rsU ª
n ? jk s 0
on GD
Ž 1.23.
on GN for k g 1, 2 4
Ž 1.24.
ª
nnEs0
on GD
Ž 1.25.
ª
on GN
Ž 1.26.
nnhs0
r Ž 0, x . s r 0 Ž x . E Ž 0, x . s E 0 Ž x . ,
Ž 1.27.
h Ž 0, x . s h 0 Ž x . ,
Ž 1.28.
where h [ H y H G , j 0 [ ycurl H G . 2. NOTATION, ASSUMPTIONS, AND RESULTS Let V ; R 3 be a bounded Lipschitz-domain with V s GD j GN , where GD is closed and GD l GN s B. The diffusion coefficients D 1 , D 2 : V = w0, `. 2 ª R are uniformly positive and bounded, i.e., there exist d ) 0 and d g Ž0, `., such that
d F D k Ž x, y . F d
2
for all x g V , y g 0, ` . , k g 1, 2 4 . Ž 2.29.
For fixed y g w0, `. 2 the functions D k Ž?, y ., RŽ?, y . are measurable and R, D k are assumed to be locally Lipschitz-continuous with respect to r , i.e., for M g Ž0, `. there exists L M g Ž0, `., such that for all x g V, y, z g w0, M x 2 , D k Ž x, y . y D k Ž x, z . F L M < y y z < and R Ž x, y . y R Ž x, z . F L M < y y z < .
Ž 2.30.
658
F. JOCHMANN
Moreover, we assume that for all x g V and u g w0, `. 2 , yR Ž x, u . F K 0 Ž 1 q u1 q u 2 .
Ž 2.31.
with some K 0 g Ž0, `. independent of x, u and that R Ž x, u . F 0
if u1 s 0 or u 2 s 0.
Ž 2.32.
In particular, these assumptions on R are fulfilled in the case R Ž x, u1 , u 2 . s r Ž x, u . Ž u1 u 2 y 1 . , where r is a nonnegative and Lipschitz-continuous function w7x. In the sequel we use the following function spaces: Let Y be the closure of C0`ŽR 3 _ GD . in H 1 Ž V ., where H 1 Ž V . is the usual first order Sobolev space of L2 type and C0`ŽR 3 _ GD . denotes the space of all infinitely differentiable functions with compact support contained in R 3 _ GD . Thus, all w g Y obey w < GD s 0 weakly. Let W0 be the space of all w g L2 Ž V, R 3 . with curl w g L2 Ž V, R 3 . in the sense of distributions. By WE we denote the space of all w g W0 , such that the tangential component of w vanishes on GD in a generalized sense, i.e.,
HV Ž w ? curl c y c ? curl w. dw s 0
Ž 2.33.
for all c g C`ŽR 3, R 3 . whose support does not intersect GN . WH denotes the set of all u g W0 , such that
HV Ž w ? curl u y u ? curl w. dx s 0
Ž 2.34.
for all w g WE , which is a weak formulation of the boundary condition u nª n s 0 on GN . The assumptions on he initial boundary of the densities are U g H 1 Ž V . l L` Ž V . ,
r 0, k G 0,
r 0 g L` Ž V . Uk G 0.
Ž 2.35. Ž 2.36.
Moreover C g L` Ž V . ,
E 0 , H 0 g L2 Ž V .
Ž 2.37.
and 2 2 2 H G g Wl 1, o c Ž R, L Ž V . . l L l o c Ž R, W0 . .
Ž 2.38.
659
THE DRIFT DIFFUSION MODEL
We assume that Ž1.14. is fulfilled in the sense of distributions and that Ž1.15., Ž1.16. are satisfied weakly in the sense that
HVE =w dx s HV Ž r
y
0
0, 1
y r 0, 2 q C . w dx
Ž 2.39.
for all w g Y. A weak formulation for Ž1.17. is given by
HV Žcurl H
G
Ž t . . ? =w dx s 0
Ž 2.40.
for all w g Y, t g R. Now, we state the main result: THEOREM 1. There exists a global weak solution Ž r , E, H. to problem Ž1.18. ] Ž1.28., such that Ž r y U . g L2 ŽŽ0, T ., Y . l L`ŽŽ0, T ., L`Ž V .. l C Žw0, T x, L2 Ž V .. with t r g L2 ŽŽ0, T ., Y *. for all T g Ž0, `., and ŽE, H. g C Žw0, `., L2 Ž V .., where Ž1.18. ] Ž1.20. and the boundary condition Ž1.24. are fulfilled in the sense that `
H0 HV r
k
s
Ž t . t w Ž t . dx dt `
H0 HV
k
D k Ž x, r Ž t . . =r k Ž t . q Ž y1 . m k r k Ž t . E Ž t . =w Ž t . dx dt
`
H0 HV R Ž x, r Ž t . . w Ž t . dx dt
q
for all w g C0`ŽŽ0, `., Y .. The weak formulation of the Maxwell system Ž1.21., Ž1.22. supplemented by the boundary conditions Ž1.25., Ž1.26. is gi¨ en by d dt
HVEŽ t . ? c dx s HVh Ž t . ? curl c dx q HV Žj Ž t . y j Ž t . y j Ž t . . ? c dx 2
1
0
for all c g WE , and d dt
HVh Ž t . ? c dx s yHVEŽ t . ? curl c dx q HV Ž H t
G
Ž t . . ? v dx
for all c g WH . 3. GLOBAL EXISTENCE OF SOLUTIONS The proof of existence of a weak solution to Ž1.18. ] Ž1.28. consists of the following steps: Firstly, we introduce a sequence of regularization opera-
660
F. JOCHMANN
tors R n : L2 Ž V . ª C`Ž V ., such that R n u ª u for n ª ` in L2 Ž V . strongly for all u g L2 Ž V .. Then the regularized system
t r kŽ n , M . s = ? D k x, Ž R n r Ž n , M . .
ž
k
q
/ =r
Žn, M . k
Ž 3.41.
q Ž y1 . m k min M, Ž R n r kŽ n , M . .
½
y R x, min M, Ž R n r 1Ž n , M . .
½
ž
q
q
5E
Žn, M .
5 , min ½ M, Ž R
n
r 2Ž n , M . .
q
5/,
k g 1, 2 4
t E Ž n , M . s curl hŽ n , M . y jŽ n , M . y j 0
Ž 3.42.
t hŽ n , M . s ycurl E Ž n , M . y t H G ,
Ž 3.43.
where jŽ n , M . s yD 1 x, Ž R n r Ž n , M . .
ž
q
q D 2 x, Ž R n r Ž n , M .
ž
/ =Ž R r . . / =Ž R r . Žn, M . 1
n
q
Žn, M . 2
n
q m 1 min M, Ž R n r 1Ž n , M . .
½
q
Ž 3.44.
5 qm
2
min M, Ž R n r 2Ž n , M . .
½
q
5
=E Ž n , M .
r Žn, M . s U
on GD
ª
Ž 3.45.
n D k x, Ž R n r Ž n , M . .
ž
k
q
/ =r
Žn, M . k
q Ž y1 . m k min M, Ž R n r kŽ n , M . .
½
q
5E
Žn, M .
s0
Ž 3.46.
on GN for k g 1, 24 . ª
n n EŽ n , M . s 0
on GD
Ž 3.47.
ª
on GN
Ž 3.48.
n n hŽ n , M . s 0
r Ž n , M . Ž 0, x . s r 0 Ž x . E
Žn, M .
Ž 0, x . s E 0 Ž x . ,
Ž 3.49. h
Žn, M .
Ž 0, x . s h 0 Ž x .
Ž 3.50.
is solved by using the contraction mapping principle in a suitable subset of C Žw0, T x. L2 Ž V ... Next, we prove a priori estimates for r Ž n, M . y U in L2 ŽŽ0, T ., Y ., t r Ž n, M . in L2 ŽŽ0, T ., Y *. and for ŽE Ž n, M ., hŽ n, M . . in C Žw0, T x, L2 Ž V .., which are independent on n.
661
THE DRIFT DIFFUSION MODEL
Thus, we obtain a subsequence Ž r Ž n m , M ., E Ž n m , M ., hŽ n m , M . . m g N , which converges weakly to some Ž r Ž M ., E Ž M ., hŽ M . . with r Ž M . y U g L2 ŽŽ0, T ., Y ., t r Ž M . g L2 ŽŽ0, T ., Y *. and ŽE Ž M ., hŽ M . . g L`ŽŽ0, T ., L2 Ž V ... By a compactness argument the convergence r Ž n m , M . ª r Ž M . for m ª ` is strong in L2 ŽŽ0, T ., L2 Ž V .., so that we can conclude that Ž r Ž M ., E Ž M ., hŽ M . . solves the system
t r kŽ M . s = ? D k Ž x, Ž r Ž M . .
q
q
. =r kŽ M . q Ž y1. k mk min ½ M, Ž r kŽ M . . 5 E Ž 3.51.
y R x, min M, Ž r 1Ž M . .
½
ž
q
5 , min ½ M, Ž r
ŽM. 2
.
q
5/, k g 1, 2 4
t E
y j0
Ž 3.52.
t hŽ M . s ycurl E Ž M . y t H G ,
Ž 3.53.
ŽM.
s curl h
ŽM.
yj
ŽM.
where jŽ M . s yD 1 Ž x, Ž r Ž M . .
q
q
. =r 1Ž M . q D2 Ž x, Ž r Ž M . . . =r 2Ž M .
q m 1 min M, Ž r 1Ž M . .
½
rŽM. s U
q
5 qm
2
min M, Ž r 2Ž M . .
½
q
Ž 3.54.
5E
on GD
ª
n ? D k Ž x, Ž r Ž M . .
q
ŽM.
Ž 3.55. q
. =r kŽ M . q Ž y1. k mk min ½ M, Ž r kŽ M . . 5 EŽ M .
s0
Ž 3.56. on GN for k g 1, 24 , ª
n n EŽ M . s 0
on GD
Ž 3.57.
ª
on GN
Ž 3.58.
n n hŽ M . s 0
r Ž M . Ž 0, x . s r 0 Ž x . E Ž M . Ž 0, x . s E 0 Ž x . ,
Ž 3.59. hŽ M . Ž 0, x . s h 0 Ž x . .
Ž 3.60.
Finally, we prove the positivity of r kŽ M . and L` estimates for r Ž M ., which are independent of M by applying suitable testing functions to Ž3.51.. Then it follows immediately that Ž r Ž M ., E Ž M ., hŽ M . . is a solution of Ž1.18. ] Ž1.28. if we choose M large enough. Now, we start with some
662
F. JOCHMANN
preliminaries concerning the Maxwell system Ž1.21., Ž1.22. supplemented by the initial boundary conditions Ž1.25., Ž1.26., Ž1.28.. We define an unbounded operator B in L2 Ž V, R 6 . by Bu s Ž curl Ž u 4 , u 5 , u 6 . y curl Ž u 1 , u 2 , u 3 . . with domain DŽ B . s WE = WH , where WE , WH are the spaces defined in Section 2. Since C0`Ž V, R 3 . ; WE l WH , B is densely defined. LEMMA 1. Ži. B is skew self-adjoint in L2 Ž V, R 6 ., i.e., B* s yB. Žii. Let w g Y. Then Ž =w , 0, 0, 0. g ker B. Proof. From the definition of WE , WH it follows immediately that B is skew-symmetric, i.e., B ; yB*. Now, let u g DŽ B*.. For w g WE we have w g Žw, 0, 0, 0. g DŽ B . and hence
HV Ž u
4,
u 5 , u 6 . ? curl w dx s y² u, Bw : s y² B*u, w : sy
Hv Ž g , g 1
2,
g 3 . ? w . dx
with g s B*u. Thus Ž u 4 , u 5 , u 6 . g WH with curlŽ u 4 , u 5 , u 6 . s yŽ g 1 , g 2 , g 3 .. Now let w g C`ŽR 3 , R 3 . with GN l supp w s B. Then w g WH and hence w s Ž0, 0, 0, w. g DŽ B .. Thus
HV Ž u , u 1
2,
u 3 . ? curl w dx s ² u, Bw : s ² B*u, w : s
HV Ž g
4,
g 5 , g 6 . ? w dx.
Therefore Ž u1 , u 2 , u 3 . g WE with curlŽ u1 , u 2 , u 3 . s Ž g 4 , g 5 , g 6 .. Thus, we have shown u g DŽ B . and Bu s yg s yB*u. In order to prove Žii., let w g Y and c g C`ŽR 3, R 3 . such that the support of c does not intersect GN . We choose a sequence wn , n g N in C0`ŽR 3 _ GD . with 5 wn y w 5 H 1 Ž V . ª 0 for n ª `. Since =wn n c s 0 on a neighbourhood of V, we obtain lim H Ž =w . ? curl c dx HV Ž =w . ? curl c dx s nª` V n
s lim
nª`
HV divŽ Ž =w . n c . dx s 0.
Hence =w g WE and the assertion follows.
n
663
THE DRIFT DIFFUSION MODEL
Next, the weak formulation of the Maxwell system Ž1.21., Ž1.22., Ž1.25., Ž1.26., and Ž1.28. is given. Let j g L2 ŽŽ0, T ., L2 Ž V, R 3 ... ŽE, h. g C Žw0, T x, L2 Ž V, R 6 .. is called a weak solution of
t E s curl h y j 0 y j
Ž 3.61.
t h s ycurl E y t H G
Ž 3.62.
ª
n n E s 0 on GD , EŽ 0. s E 0 ,
ª
n n h s 0 on GN
Ž 3.63.
h Ž 0. s h 0
Ž 3.64.
if d dt
HVEŽ t . ? c dx s HVh Ž t . ? curl c dx y HV Žj Ž t . q j Ž t . . ? c dx 0
Ž 3.65.
for all c g WE , and d dt
HVh Ž t . ? c dx s yHVEŽ t . ? curl c dx y HV Ž H t
G
Ž t . . ? c dx Ž 3.66.
for all c g WH , and EŽ 0. s E 0 ,
h Ž 0. s h 0 .
Ž 3.67.
Note that Ž3.65., Ž3.66. include the boundary conditions Ž3.63. in a generalized sense. By setting uŽ t . s ŽEŽ t ., hŽ t .., f j Ž t . s yŽjŽ t . q j 0 Ž t ., t HŽ t .., Ž3.65. ] Ž3.67. can be reformulated as d dt
² u Ž t . , w: s y² u Ž t . , Bw: q² f j Ž t . , w: s² u Ž t . , B*w: q² f j Ž t . , w: u Ž 0. s Ž E 0 , h 0 .
for all w g DŽ B ., i.e., u g C Žw0, T x, L2 Ž V, R 6 .. is a mild solution of the L2 Ž V, R 6 . valued evolution equation d dt
u Ž t . s Bu Ž t . q f j Ž t . ,
u Ž 0. s Ž E 0 , h 0 . .
664
F. JOCHMANN
This Cauchy problem has a unique weak solution u g C Žw0, T x, L2 Ž V, R 6 .. given by
u Ž t . s exp Ž tB . Ž E 0 , h 0 . q
t
H0 exp Ž Ž t y s . B . f Ž s . ds, j
Ž 3.68.
where expŽ tB ., t g R is the C0 group of unitary operators in L2 Ž V . generated by B w1, 10x. Equation Ž3.68. yields the energy estimate 1 d 2 dt
uŽ t .
2 L2 Ž V .
s² f j Ž t . , u Ž t .:
Ž 3.69.
HVj Ž t . ? EŽ t . dx y HV Žj Ž t . ? EŽ t . q Ž H Ž t . . ? h Ž t . . dx.
sy
t
0
Next, it is shown that the solution ŽE, h. of the Maxwell system Ž3.21. ] Ž3.24. obeys =E s r , ª n ? E s 0 on GN weakly. LEMMA 2.
Let r g C ŽŽ0, T ., L2 Ž V .., j g L2 ŽŽ0, T ., L2 Ž V .. with y
HVE
0
? =w dx s
HVr Ž 0. w dx
and d
HVj Ž t . =w dx s dt HVr Ž t . w dx for all w g Y, i.e., = ? j s y t r and ª n ? j s 0 on GN . Then
HVEŽ t . =w dx s HVr Ž t . w dx
y for all w g Y.
665
THE DRIFT DIFFUSION MODEL
Proof. Let w g Y. By Lemma 1Žii. we have c s Ž =w , 0, 0, 0. g ker B. Thus, we obtain from Ž3.68.
HVEŽ t . =w dx s² u Ž t . , c: s² Ž E 0 , h 0 . , exp Ž ytB . c: q s² Ž E 0 , h 0 . , c: q
t
H0² f Ž s . , exp Ž ysB . c: ds j
t
H0² f Ž s . , c: ds j
t
s
HVE =w dx y H0 HV Žj Ž s . q j Ž s . . ? =w dx ds
s
HVE =w dx y H0 HVj Ž s . ? =w dx ds
s
HVE =w dx y HV
0
0
t
0
0
r Ž t . y r Ž 0 . w dx
HVr Ž t . w dx.
sy
Now, we prove existence and uniqueness for the regularized system Ž3.41. ] Ž3.50.. The regularization operars R n are defined as follows: Since V is a Lipschitz-domain, we find open sets U1 , . . . , UN ; R 3, k 1 , . . . , k N g R 3 , such that V ; D Njs1Uj and x q t k j g V for x g V l Uj , t g Ž0, 1.. Now, let G1 , . . . , GN g C`ŽR 3 . with supp Gj ; Uj and Ý Njs1Gj Ž x . s 1 for all x g V. Since V l supp Gj is a compact subset of the open set x g R 3 : x q Ž1rn.k j g V 4 , n g N, there exists some dn g Ž0, 1rN . with x y y q Ž1rn.k j g V for all x g V l supp Gj , n g N, y g R 3 with < y < F dn . Choose v g C0`ŽR 3 . with HR 3 v dx s 1, supp v ; B1Ž0.. Now, we define for u g L2 Ž V ., n g N, x g V, N
Ž R n u. Ž x . s
Ý Gj Ž x . H
u xq
Bd nŽ0 .
js1
ž
1 n
k j y y v jŽ n. Ž y . dy,
/
2Ž . Ž . where v jŽ n. Ž y . s dy3 n v yrd n . For u g L V we have
N
Rn ª
Ý xj u s u js1
for n ª `
Ž 3.70.
666
F. JOCHMANN
in L2 Ž V . strongly. From Young’s inequality we obtain constants k 0 , k n g Ž0,’`., where k n depends only on n g N, such that 5 R n u 5 C 1 Ž V . F k n 5 u 5 L2 Ž V .
Ž 3.71.
5 R n u 5 L2 Ž V . F k 0 5 u 5 L2 Ž V .
Ž 3.72.
and
for all u g L2 Ž V .. Now, let u g H 1 Ž V .. Then we have by the definition of R n N
=Ž R n u . Ž x . s
Ý Ž =Gj Ž x . . H
Bd nŽ0 .
js1
1
u xq
ž
n
k j y y v jŽ n. Ž y . dy
/
N
q
u Ž z . =v jŽ n. x q
Ý Gj Ž x . H
ž
Bd nŽ xq Ž1rn .k j .
js1
1 n
k j y z dz
/
N
s
Ý Ž =Gj Ž x . . H
u Ž x q Ž 1rn . k j y y . v jŽ n. Ž y . dy
Bd nŽ0 .
js1 N
q
Ý Gj Ž x . H
Bd nŽ xq Ž1rn .k j .
js1
1
Ž =u Ž z . . v jŽ n. x q k j y z dz
ž
n
/
since Bd nŽ x q Ž 1rn . k j . ; V for x g V l supp Gj . Therefore, Young’s inequality yields the estimate 5 R n u 5 2H 1 Ž V . F k 0 5 u 5 2H 1 Ž V .
Ž 3.73.
for all u g H 1 Ž V .. Next, problem Ž3.41. ] Ž3.50. will be solved by using a fixed point argument. Let r g C Žw0, T x, L2 Ž V, R 2 .. and ŽE, h. g C Žw0, T x, L2 Ž V, R 6 .. the unique weak solution of
t E s curl h y j 0 y j r
Ž 3.74.
t h s ycurl E y t H G
Ž 3.75.
©
n n E s 0 on GD , EŽ 0. s E 0 ,
©
n n h s 0 on GN
h Ž 0. s h 0
Ž 3.76. Ž 3.77.
where j r s yD 1 Ž x, Ž R n r . q
q
q
. =R n r 1 q D 2 Ž x, Ž R n r . . =R n r 2 q q m 1 min M, Ž R n r 1 . 4 q m 2 min M, Ž R n r 2 . 4
E. Ž 3.78.
667
THE DRIFT DIFFUSION MODEL
Now, we define AŽ n, M .r s r g C Žw0, T x, L2 Ž V, R 2 .., where r is the unique solution of the linear parabolic problem
t r k s y= ? D k Ž x, Ž R n r .
q
q . =r k q Ž y1. k mk min M, Ž R n r k . 4 E
y R Ž x, min M, Ž R n r 1 . ª
n ? D k Ž x, Ž R n r .
q
q
4 , min M, Ž R n r 2 . q 4 .
Ž 3.79.
q . =r k q Ž y1. k mk min M, Ž R n r k . 4 E
s 0 Ž 3.80.
on GN for k g 1, 24
rŽ t. y U g Y
Ž 3.81.
r Ž 0. s r 0 .
Ž 3.82.
By assumption D k Ž x, Ž R n r .q. is uniformly positive. It follows from the standard theory of monotone operators in the space L2 ŽŽ0, T ., Y ., that Ž3.79. ] Ž3.82. has a unique solution r g C Žw0, T x, L2 Ž V .. with r y U g L2 ŽŽ0, T ., Y . and t r g L2 ŽŽ0, T ., Y *.. Now, it is clear that r g C Žw0, T x, L2 Ž V .. is a solution to the regularized problem Ž3.41. ] Ž3.50. iff r s AŽ n, M .r . This fixed point problem is solved by using the contraction mapping principle. For this purpose we need the estimates 1 d 2 dt
rŽ t. y U
2 L2 Ž V .
2
sy
ÝH
ks1 V
D k Ž x, Ž R n r .
q
. =r k ? =Ž r k y U .
k
q Ž y1 . m k min M, Ž R n r k . y
HVR Ž x, min M, Ž R
n
r1 .
q
q
4 E ? =Ž r k y U .
dx
4 , min M, Ž R n r 2 . q 4 .
= Ž r 1 q r 2 y U1 y U2 . dx F yd =r Ž t . Fy
d 2
rŽ t.
2 L2 Ž V .
q k 2 , M Ž1 q EŽ t .
L2 Ž V .
2 H 1Ž V .
q k 3 , M 1 q EŽ t .
2 L2 Ž V .
ž
.
rŽ t.
H 1Ž V .
q rŽ t.
2 L2 Ž V .
/.
Ž 3.83.
668
F. JOCHMANN
The energy estimate Ž3.69. yields d dt
2 L2 Ž V .
EŽ t .
q hŽ t .
2 L2 Ž V .
F k 4, M Ž 1 q = Ž R n r Ž t . . Fy
d 4
=Ž R n r Ž t . .
2 L2 Ž V .
L2 Ž V .
.
2 L2 Ž V .
EŽ t .
q hŽ t .
q k 5, M 1 q E Ž t .
ž
2 L2 Ž V .
1r2 2 L2 Ž V .
q hŽ t .
2 L2 Ž V .
/,
Ž 3.84. where k 0 is the constant occurring in Ž3.73., and k 2, M , . . . , k 5, M are independent of r and n. By Ž3.83. and Ž3.84. there exists a constant k 1, M depending only on M, such that d dt
rŽ t. y U F
d 4 k0
2 L2 Ž V .
q EŽ t .
=Ž R n r Ž t . .
2 L2 Ž V .
q k 1, M 1 q E Ž t .
2 L2 Ž V .
y
2 L2 Ž V .
d 2
q hŽ t .
rŽ t.
q hŽ t .
2 L2 Ž V .
2 H 1Ž V .
2 L2 Ž V .
q rŽ t. y U
2 L2 Ž V .
.
Ž 3.85. According to Ž3.71., we find constants L n, M , k n, M, 3 g Ž0, `., such that AŽ n, M .r g Sn, M for r g Sn, M , where Sn, M is the set of all r g C Žw0, T x, L2 Ž V .. with
rŽ t. y U
2 L2 Ž V .
F k n , M , 3 exp L n , M t.
Ž 3.86.
By Ž3.85., Ž3.86. there exists a constant k n, M , 4 , such that for all r g S n, M
Ž AŽ n , M .r . y U
L2 ŽŽ 0, T . , Y .
q 5E 5 CŽŽ0 , T ., L2 Ž V .. q 5h 5 CŽŽ0 , T ., L2 Ž V .. F k n , M , 4 ,
Ž 3.87. where ŽE, h. g C Žw0, T x, L2 Ž V .. is the solution of the Maxwell system Ž3.74. ] Ž3.77.. Next, it is shown that AŽ n, M . is a contraction in C Žw 0, T x , L 2 Ž V .. w ith re spe ct to th e n orm A r A s sup t g w0, T x 5 r Ž t .5 L2 Ž V . exp Ž l n, M t . with some l n, M g Ž0, `. depending only on n, M. Let r , r g Sn, M , r s AŽ n, M .r , r s AŽ n, M .r . Then we obtain from
669
THE DRIFT DIFFUSION MODEL
Ž3.87. by using r Ž t . y r Ž t . g Y, 1 d
rŽ t. y rŽ t.
2 dt
2 L1 Ž V .
2
sy
ÝH
ks1 V
D k Ž x, Ž R n r .
q
. =rk y Dk Ž x, Ž R n r . q . =rk
k
q
k
q
q Ž y1 . m k min M, Ž R n r k . q Ž y1 . m k min M, Ž R n r k . y R Ž x, min M, Ž R n r 1 .
q
F yd r Ž t . y r Ž t .
4E
? = r k y r k dx
4 , min M, Ž R n r 2 . q 4 .
y R x, min M, Ž R n r 1 .
ž
4E
q
4 , min M, Ž R n r 2 . q 4 /
2 H 1Ž V .
qk n , M , 5 D k Ž x, R n Ž r Ž t . . . y D k Ž x, R n Ž r Ž t . . . rŽ t. y rŽ t. qk n , M , 5
`
H 1Ž V .
rŽ t. y rŽ t.
rŽ t. y rŽ t.
H 1Ž V .
rŽ t. y rŽ t.
L2 Ž V .
F kn, M , 6
rk y rk
L2 Ž V .
q EŽ t . y EŽ t .
q kn, M , 5 r Ž t . y r Ž t .
rŽ t. y rŽ t.
2 L2 Ž V .
L2 Ž V .
L2 Ž V .
q EŽ t . y EŽ t .
2 L2 Ž V .
.
Ž 3.88. Here ŽE, h., ŽE, h. are the solutions of Maxwell system Ž3.74. ] Ž3.77. corresponding to r , r , respectively. From Ž3.68., Ž3.87., and Ž3,78. we obtain EŽ t . y EŽ t . F
t
H0
L2 Ž V .
jr Ž s. y jr Ž s.
F kn, M , 8
t
H0
L2 Ž V .
ds
r Ž s. y r Ž s.
L2 Ž V .
q EŽ s . y EŽ s .
L2 Ž V .
ds.
Hence, there exists some k n, M , 9 g Ž0, `. depending only on n, M, such that EŽ t . y EŽ t .
2 L2 Ž V .
F kn, M , 9
t
H0
r Ž s. y r Ž s.
2 L2 Ž V .
ds.
Ž 3.89.
670
F. JOCHMANN
From Ž3.88., Ž3.89., we obtain gn, M g Ž0, 1. and l n, M g Ž0, `., which depend only on n, M, such that A AŽ n , M .r y AŽ n , M .r A F gn , M A r y r A , for all r , r g Sn, M , where the norm A ? A in C Žw0, T x, L2 Ž V .. is given by A r A s sup t g w0, T x 5 r Ž t .5 L2 Ž V . exp l n, M t. Hence, AŽ n, M . has a unique fixed point r Ž n, M . g Sn, M , i.e., the regularized problem Ž3.41. ] Ž3.50. has a unique solution r Ž n, M . g C Žw0, T x, L2 Ž V .. with r Ž n, M . y U g L2 ŽŽ0, T ., Y ., and t r Ž n, M . g L2 ŽŽ0, T ., Y *.. Next, we prove some estimates for r Ž n, M ., which are independent of n in order to pass to the limit n ª `. From Ž3.73. and Ž3.85. we obtain d
r Žn, M .Ž t . y U
dt
Fy
d 4
2 L2 Ž V .
r Žn, M .Ž t . .
2 L2 Ž V .
q EŽ n , M . Ž t .
q hŽ n , M . Ž t .
2 L2 Ž V .
2 H 1Ž V .
q k 1, M 1 q E Ž n , M . Ž t .
2 L2 Ž V .
q hŽ n , M . Ž t .
2 L2 Ž V .
q rŽn, M .Ž t . .
2 L2 Ž V .
.
Hence, there exists a constant k 6, M g Ž0, `., such that
r Žn, M . y U
L2 Ž0, T . , Y
q hŽ n , M . Ž t .
q EŽ n , M . Ž t .
L`ŽŽ 0, T . , L2 Ž V ..
Ž 3.90.
L`ŽŽ0 , T . , L2 Ž V ..
F k 6, M and by Ž3.41.
t r Ž n , M .
L2 ŽŽ0, T . , Y * .
F k 6, M .
Ž 3.91.
By the reflexivity of L2 ŽŽ0, T ., Y . there exists a subsequence Ž r Ž n m , M . . m g N and Ž r Ž M . , E Ž M . , h Ž M . . with r Ž M . y U g L2 ŽŽ0, T ., Y ., t r Ž M . g L2 ŽŽ0, T ., Y *., and ŽE Ž M ., hŽ M . . g L`ŽŽ0, T ., L2 Ž V .., such that
r Ž nm , M . y r Ž M . ª 0
for m ª `
Ž 3.92.
2 ŽŽ
in L
0, T ., Y . weakly,
r Ž n m , M . ª r Ž M .
for m ª `
Ž 3.93.
2 ŽŽ
in L
0, T ., Y *. weakly,
Ž E Ž n m , M . , hŽ n m , M . . ª Ž E Ž M . hŽ M . . `ŽŽ
in L 0, T ., L V .. weak *. 2Ž
for m ª `
Ž 3.94.
671
THE DRIFT DIFFUSION MODEL
By Ž3.90., Ž3.91., and a standard compactness result, see w4, 6x, the sequence Ž r Ž n m , M . . m g N is precompact to L2 ŽŽ0, T ., L2 Ž V .. s L2 ŽŽ0, T . = V .. Hence
r Ž nm , M . ª r Ž M .
for m ª `
Ž 3.95.
in L2 ŽŽ0, T ., L2 Ž V .. strongly. Relation Ž3.70. and the dominated convergence theorem yield R n mŽ r Ž n m , M . . y r Ž M .
L2 ŽŽ 0, T . , L2 Ž V ..
F R n mŽ r Ž n m , M . y r Ž M . . q ª0
ž
T
H0
L2 ŽŽ 0, T . , L2 Ž V .. 1r2
2
R n mŽ r Ž M . Ž t . . y r Ž M . Ž t .
L V. 2Ž
dt
/
for m ª `
Ž 3.96.
and therefore R n mŽ r Ž n m , M . Ž s . . Ž x . ª r Ž M . Ž s, t .
for m ª `
Ž 3.97.
a.e. in Ž0, T . = V after extracting a further subsequence. Moreover, from Ž3.94. and Ž3.97. we obtain min M, Ž R n m r kŽ n m , M . .
½
q
5E
Ž nm , M .
ª min M, Ž r kŽ M . .
½
q
5E
ŽM.
for m ª `
Ž 3.98.
in L2 ŽŽ0, T ., L2 Ž V .. weakly. Hence, we have jŽ n m , M . ª jŽ M .
for m ª `
Ž 3.99.
in L2 ŽŽ0, T ., L2 Ž V .. weakly, where jŽ n , M . s yD 1 x, Ž R n r Ž n , M . .
ž
q
q D 2 x, Ž R n r Ž n , M .
ž
/ =R r . / =R r
Žn, M . 1
n
q
n
q m 1 min M, Ž R n r 1Ž n , M . .
½
q
qm 2 min M, Ž R n r 2Ž n , M . .
½
Žn, M . 2
5 q
5E
Žn, M .
672
F. JOCHMANN
and jŽ M . s yD 1 Ž x, Ž r Ž M . .
q
q
. =r 1Ž M . q D2 Ž x, Ž r Ž M . . . =r 2Ž M .
q m 1 min M, Ž r 1Ž M . .
½
q
5 qm
2
min M, Ž r 2Ž M . .
½
q
5E
ŽM.
.
From Ž E Ž n m , M . Ž t . , h Ž n m , M . Ž t .. s exp Ž tB .Ž E 0 , h 0 . q H0t exp ŽŽ t y s . B .f jŽ n, M . Ž s . ds and Ž3.99. we obtain t
Ž EŽ n m , M . , hŽ n m , M . . ª exp Ž tB . Ž E 0 , h 0 . q H exp Ž Ž t y s . B . f jŽ M . Ž s . ds 0
for m ª ` in L V . weakly for all t g w0, T x. According to Ž3.94. we get 2Ž
t
Ž EŽ M . , hŽ M . . s exp Ž tB . Ž E 0 , h 0 . q H exp Ž Ž t y s . B . f jŽ M . Ž s . ds, 0
i.e., ŽE , h g C Žw0, T x, L V .. is the weak solution of the Maxwell system Ž3.52., Ž3.53., Ž3.57., Ž3.58., and Ž3.60.. From Ž3.41. ] Ž3.50., Ž3.92. ] Ž3.94., and Ž3.97. it follows easily that for all w g C0`ŽŽ0, T ., Y ., ŽM.
T
H0 HV r
Ž M ..
ŽM. k
s
2Ž
t w dx dt T
H0 HV
D k Ž x, Ž r Ž M . . k
q
. =r kŽ M .
q Ž y1 . m k min M, Ž r kŽ M . .
½
q
q
5E
ŽM.
=w dx dt q
T H0 HV R ž x, min ½ M, Ž r . 5 , min ½ M, Ž r . 5 / w dx dt
q
ŽM. 1
ŽM. 2
Moreover, we have
rŽM .Ž t. y U g Y for all t g Ž0, T ., and
r Ž M . Ž 0. s r 0 . Hence, Ž r Ž M ., E Ž M ., hŽ M . . is a solution of the system Ž3.51. ] Ž3.60.. Next, the positivity of r Ž M . is shown. For this purpose we choose some h g C`ŽR. with h9 G 0, h0 F 0, hŽ t . s 0 for t G 0 and hŽ t . - 0 for t - 0. Since U G 0 and r Ž M . Ž t . y U g Y, it follows easily by the definition of Y
673
THE DRIFT DIFFUSION MODEL
that hŽ r Ž M . Ž t .. g Y. Hence, we obtain from Ž3.51. d dt
HVh Ž r
ŽM. k
Ž t . . dx
HVh0 Ž r .
sy
ŽM. k
D k Ž x, Ž r Ž M . .
q
k
. =r kŽ M .
q Ž y1 . m k min M, Ž r kŽ M . .
½
ŽM. q 1
q
5E
ŽM.
q
=r kŽ M . Dx
HVR ž x, min ½ M, Ž r . ˙ 5 , min ½ M, Ž r . 5 / h9 Ž r . dx
y
ŽM. 2
q
q
ŽM. k
HVR ž x, min ½ M, Ž r . 5 , min ½ M, Ž r . 5 / h9 Ž r . Dx G 0
Gy
ŽM. 1
ŽM. 2
ŽM. k
since RŽ x, u. F 0 if u1 s 0 or u 2 s 0 by the assumptions on R. Hence, we have HV hŽ r kŽ M . Ž t .. dx G 0 and this implies r kŽ M . Ž t . G 0 in V a.e. Finally, we prove a L` estimate for r Ž M ., which is independent of M. Therefore, if we choose M larger than this bound, Ž r Ž M ., E Ž M ., hŽ M . . is a solution to the original problem Ž3.58. ] Ž3.68.. Assume that h g C`ŽR. such that h, h9 are bounded, h9 G 0, and hŽ t . s 0 for t F max5 U 5 ` , 5 r 0 5 `4 . With g Ž t . s H0t hŽ s . ds and HM Ž t . s
t
H0 min M, s4 h9 Ž s . ds F th Ž t .
we obtain from Ž3.51. and hŽ r kŽ M . Ž t .. g Y, d dt
HV
ŽM. ŽM. my1 Ž t . . q my1 Ž t . . dx 1 g Ž r1 2 g Ž r2
2
sy
Ý H my1 k ž
ks1 V
D k Ž x, r Ž M . . =r kŽ M . k
q Ž y1 . m i min M, r kŽ M . 4 E Ž M . =h Ž r kŽ M . . qR Ž x, min M, r 1Ž M . 4 , min M, r 2Ž M . 4 . h Ž r kŽ M . . dx
/
F
HV min M, r 4 =h Ž r . ŽM. 1
ŽM. 1
674
F. JOCHMANN
ymin M, r 2Ž M . 4 =h Ž r 2Ž M . . E Ž M . dx q K0 F
HV Ž1 q r
ŽM. 1
q r 2Ž M . . h Ž r 1Ž M . . q h Ž r 2Ž M . . dx
HM Ž r 1Ž M . . y HM Ž r Ž M . . E Ž M . dx
HV=
HV Ž1 q r . h Ž r . q Ž1 q r . h Ž r .
qk
ŽM. 1
ŽM. 1
ŽM. 2
ŽM. 2
dx, Ž 3.100.
since Ž u1 q u 2 .w hŽ u1 . q hŽ u 2 .x F 4w u1 hŽ u1 . q u 2 hŽ u 2 .x for u g w0, `. 2 , because h is nonnegative and nondecreasing. By setting r Ž t . s r 1Ž M . y r 2Ž M . q C we obtain from Ž3.51. ] Ž3.54., Ž3.60., Ž2.39., and Lemma 2,
HV=
HM Ž r 1Ž M . . y HM Ž r 2Ž M . . E Ž M . dx sy
HM Ž r 1Ž M . . y HM Ž r 2Ž M . .
Fy
HM Ž r 1Ž M . . y HM Ž r 2Ž M . . C dx
F 5 C 5`
Ý H r kŽ M . h Ž r kŽ M . . dx
HV HV
Ž r 1Ž M . y r 2Ž M . q C . dx
2 ks1 V
and hence by using Ž3.100. d
2
H gŽ r . q gŽ r . dt V ŽM. 1
ŽM. 2
dx F k 0, 1
Ý H Ž 1 q r kŽ M . . h Ž r kŽ M . . dx
ks1 V
Ž 3.101. with some k 0, 1 g Ž0, `. independent of M. In order to prove a Ln estimate for r Ž M . Ž t ., we introduce the functions hn , N Ž u. s nŽ N y A. hn , N Ž u. s nŽ u y A. hn , N Ž u. s 0 gn , N Ž u. s
ny 1
ny 1
if u - A u
H0 h
n, N
Ž s . ds,
if u ) N f u g w A, N x
675
THE DRIFT DIFFUSION MODEL
where A s max5 r 0 5 ` , 5 U 5 `4 and N ) A. By approximation with smooth functions Ž3.101. can be applied to h n, N and g n, N . A simple calculation yields uh n , N Ž u . F ng n , N Ž u . q Ah n , N Ž u .
Ž 3.102.
for u g w A, `.. Since h n, N Ž u. F n for u F A q 1 and h n, N Ž u. F ng n, N Ž u. for u ) A q 1 we have
HVh Ž r n, N
ŽM. k
Ž t . . dx F n < V < q H
r kŽ M . Ž t .GAq1 4
F n< V < q n
HVg Ž r n, N
h n , N Ž r kŽ M . Ž t . . dx
ŽM. k
Ž t . . dx.
Ž 3.103.
From Ž3.101. ] Ž3.103. we obtain the Gronwall-type estimate d dt
HV g Ž r . q g Ž r . ŽM. 1
HV g Ž r . q g Ž r .
dx F k 2 n q k 2 n
ŽM. 2
ŽM. 1
ŽM. 2
dx
Ž 3.104. and hence
HV g Ž r
ŽM. 1
Ž t . . q g Ž r 2Ž M . Ž t . . dx F 2 k 2 Tn exp 2 k 2 tn.
Ž 3.105.
Since g n, N Ž u. ª ŽŽ u y A.q. n for n ª `, we obtain from the monotone convergence theorem and Ž3.46.,
r kŽ M . Ž t . g Ln Ž V . and
Ž r kŽ M . Ž t . y A .
q Ln Ž V .
F Ž 2 k 2 Tn .
1rn
exp 2 k 2 T .
Now, letting n ª ` yields
r kŽ M . Ž t . g L` Ž V . and
Ž r kŽ M . Ž t . y A .
q L`Ž V .
F exp 2 k 2 T .
Hence, Ž r Ž M ., E Ž M ., hŽ M . . is a solution of the system Ž3.58. ] Ž3.68., if M ) A q exp 2 k 2 T.
676
F. JOCHMANN
REFERENCES 1. J. M. Ball, Strongly continuous semigroups, weak solutions and the ¨ ariation of constant formula, Proc. Amer. Math. Soc. 63 Ž1977., 370]373. 2. H. Gajewski, On existence, uniqueness and asymptotic behavior of solutions of the basic equations for carrier transport in semiconductors, Z. Angew. Math. Mech. 65 Ž1985., 101]108. 3. H. Gajewski and K. Groger, On the basic equations for carrier transport in semiconductors, J. Math. Anal. Appl. 113 Ž1986., 12]35. 4. H. Gajewski, K. Groger, and K. Zacharias, ‘‘Nichtlineare Operatorengleichungen und Operatordifferentialgleichungen,’’ Akademie Verlag, Berlin, 1974. 5. A. Juengel, On existence and uniqueness of transient solutions of a degenerate nonlinear drift diffusion model for semiconductors, submitted for publication. 6. J. L. Lions, ‘‘Quelques methods de resolution des problems aux limites non lineaires,’’ DunodrGauthier]Villars, Paris, 1969. 7. P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, ‘‘Semiconductor Equations,’’ Springer-Verlag, New YorkrBerlin, 1990. 8. M. S. Mock, An initial value problem from semiconductor device theory, SIAM J. Math. Anal. 5 Ž1974., 597]612. 9. M. S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices, Math. Anal. Appl. 49 Ž1975., 215]225. 10. Pazy, ‘‘Semigroups of Linear Operators and Applications to Partial Differential Equations,’’ Springer-Verlag, New York, 1983. 11. T. I. Seidman and G. M. Troianiello, Time dependent solutions of a nonlinear system arising in semiconductor theory, Nonlinear Anal. 9 Ž1985., 1137]1157. 12. V. Roosbroeck, Theory of flow of electrons and holes in Germanium and other semiconductors, Bell System Tech. J. 29 Ž1950., 560]607.
204, 655]676 Ž1996.
0460
Existence of Weak Solutions of the Drift Diffusion Model Coupled with Maxwell’s Equations F. Jochmann Fachbereich Mathematik, Technische Uni¨ ersitaet Berlin, Strasse des 17, Juni 136, 10623 Berlin, Germany Submitted by Colin Rogers Received December 13, 1994
1. INTRODUCTION The aim of this paper is to prove existence of weak solutions of the drift diffusion model for semiconductors involving Maxwell’s equations, which is governed by the equations j1 s yD 1 Ž x, r 1 , r 2 . =r 1 q m 1 r 1 E
Ž 1.1.
j 2 s yD 2 Ž x, r 1 , r 2 . =r 2 y m 2 r 2 E
Ž 1.2.
t r t s y= ? j k y R Ž x, r 1 , r 2 . ,
k g 1, 2 4
Ž 1.3.
t E s curl H q j 2 y j1
Ž 1.4.
t H s ycurl E
Ž 1.5.
= ? E s r 1 y r 2 q C,
= ? H s 0.
Ž 1.6.
r 1 , r 2 denote the densities of j1 , j 2 denote the current densities of the holes and electrons, respectively. The self-consistent electromagnetic field ŽE, H. obeys Maxwell’s equations Ž1.4., Ž1.5., and Ž1.6.. The unknown functions r 1 , r 2 , E, H depend on Ž t, x . g w0, `., where t, x denote the time and space variable resp. V ; R 3 is a bounded Lipschitz-domain with V s GD j GN , where GD , GN are disjoint subsets of V. GD represents the perfectly conducting Ohmic contacts and GN represents the insulating boundary of the semiconductor device. The mobilities m 1 , m 2 of the holes and electrons resp. are assumed to be positive constants. The diffusion coefficients D 1 , D 2 and the recombination generation rate R are functions of the densities r 1 , r 2 and the space variable x. C is a bounded function of x, which describes the doping profile of the device. 655 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
656
F. JOCHMANN
Mathematical analysis of the drift diffusion model for semiconductors has been presented in w2, 3, 5, 8]10x. But in all these references Maxwell’s equations Ž1.4. ] Ž1.6. are replaced by Poisson’s equation for an electrostatic field E s y=V with yDV s r 1 y r 2 y C, which simplifies the mathematical analysis, since the electric field depends continuously on the total charge density with respect to the L2 Ž V . topology in this case. However, at very high frequencies, i.e., if the wavelength of the electromagnetic field has the same order than the size of the device, we have to replace Poisson’s equation by Maxwell’s equations Ž1.4., Ž1.5.. Since we do not have an electrostatic potential for the electric field we have to find new physically reasonable boundary conditions for the electromagnetic field. On the perfectly conducting Ohmic contacts GD the tangential component of the electric field must vanish. The tangential component of the magnetic field is prescribed on the perfectly insulating boundary. The system Ž1.1. ] Ž1.6. is supplemented by the initial boundary conditions
rsU ª
n ? jk s 0
on GD
on GN for k g 1, 2 4
ª
nnEs0
on GD
ª
n?Es0
on GN
ª
n n H sª n n HG
on GN
r Ž 0, x . s r 0 Ž x . E Ž 0, x . s E 0 Ž x . ,
H Ž 0, x . s H 0 Ž x . .
Ž 1.7. Ž 1.8. Ž 1.9. Ž 1.10. Ž 1.11. Ž 1.12. Ž 1.13.
Obviously, E 0 , H 0 , r 0 , and H G must be consistent with Maxwell’s equations Ž1.4. ] Ž1.6. and the initial-boundary conditions, that means = ? H0 s 0
Ž 1.14.
= ? E 0 s r 1, 0 y r 2 , 0 q C
Ž 1.15.
ª
n ? E0 s 0
on GN .
Ž 1.16.
Moreover, by Ž1.8., Ž1.10., Ž1.11., and Ž1.4. we have to require ª
n ? Ž curl H G . s 0
on GN .
Ž 1.17.
657
THE DRIFT DIFFUSION MODEL
Now, if ŽE, H. is a solution of Maxwell’s equations Ž1.4., Ž1.5., and the initial boundary conditions Ž1.9., Ž1.11., and Ž1.13., the equations Ž1.6. and the boundary condition Ž1.10. are automatically fulfilled for all times t G 0. Therefore, problem Ž1.1. ] Ž1.6., Ž1.7. ] Ž1.13. can be reformulated as j1 s yD 1 Ž x, r 1 , r 2 . =r 1 q m 1 r 1 E
Ž 1.18.
j 2 s yD 2 Ž x, r 1 , r 2 . =r 2 y m 2 r 2 E
Ž 1.19.
t r t s y= ? j k y R Ž x, r 1 , r 2 . ,
k g 1, 2 4
Ž 1.20.
t E s curl h q j 2 y j1 y j 0
Ž 1.21.
t h s ycurl E y t H G
Ž 1.22.
rsU ª
n ? jk s 0
on GD
Ž 1.23.
on GN for k g 1, 2 4
Ž 1.24.
ª
nnEs0
on GD
Ž 1.25.
ª
on GN
Ž 1.26.
nnhs0
r Ž 0, x . s r 0 Ž x . E Ž 0, x . s E 0 Ž x . ,
Ž 1.27.
h Ž 0, x . s h 0 Ž x . ,
Ž 1.28.
where h [ H y H G , j 0 [ ycurl H G . 2. NOTATION, ASSUMPTIONS, AND RESULTS Let V ; R 3 be a bounded Lipschitz-domain with V s GD j GN , where GD is closed and GD l GN s B. The diffusion coefficients D 1 , D 2 : V = w0, `. 2 ª R are uniformly positive and bounded, i.e., there exist d ) 0 and d g Ž0, `., such that
d F D k Ž x, y . F d
2
for all x g V , y g 0, ` . , k g 1, 2 4 . Ž 2.29.
For fixed y g w0, `. 2 the functions D k Ž?, y ., RŽ?, y . are measurable and R, D k are assumed to be locally Lipschitz-continuous with respect to r , i.e., for M g Ž0, `. there exists L M g Ž0, `., such that for all x g V, y, z g w0, M x 2 , D k Ž x, y . y D k Ž x, z . F L M < y y z < and R Ž x, y . y R Ž x, z . F L M < y y z < .
Ž 2.30.
658
F. JOCHMANN
Moreover, we assume that for all x g V and u g w0, `. 2 , yR Ž x, u . F K 0 Ž 1 q u1 q u 2 .
Ž 2.31.
with some K 0 g Ž0, `. independent of x, u and that R Ž x, u . F 0
if u1 s 0 or u 2 s 0.
Ž 2.32.
In particular, these assumptions on R are fulfilled in the case R Ž x, u1 , u 2 . s r Ž x, u . Ž u1 u 2 y 1 . , where r is a nonnegative and Lipschitz-continuous function w7x. In the sequel we use the following function spaces: Let Y be the closure of C0`ŽR 3 _ GD . in H 1 Ž V ., where H 1 Ž V . is the usual first order Sobolev space of L2 type and C0`ŽR 3 _ GD . denotes the space of all infinitely differentiable functions with compact support contained in R 3 _ GD . Thus, all w g Y obey w < GD s 0 weakly. Let W0 be the space of all w g L2 Ž V, R 3 . with curl w g L2 Ž V, R 3 . in the sense of distributions. By WE we denote the space of all w g W0 , such that the tangential component of w vanishes on GD in a generalized sense, i.e.,
HV Ž w ? curl c y c ? curl w. dw s 0
Ž 2.33.
for all c g C`ŽR 3, R 3 . whose support does not intersect GN . WH denotes the set of all u g W0 , such that
HV Ž w ? curl u y u ? curl w. dx s 0
Ž 2.34.
for all w g WE , which is a weak formulation of the boundary condition u nª n s 0 on GN . The assumptions on he initial boundary of the densities are U g H 1 Ž V . l L` Ž V . ,
r 0, k G 0,
r 0 g L` Ž V . Uk G 0.
Ž 2.35. Ž 2.36.
Moreover C g L` Ž V . ,
E 0 , H 0 g L2 Ž V .
Ž 2.37.
and 2 2 2 H G g Wl 1, o c Ž R, L Ž V . . l L l o c Ž R, W0 . .
Ž 2.38.
659
THE DRIFT DIFFUSION MODEL
We assume that Ž1.14. is fulfilled in the sense of distributions and that Ž1.15., Ž1.16. are satisfied weakly in the sense that
HVE =w dx s HV Ž r
y
0
0, 1
y r 0, 2 q C . w dx
Ž 2.39.
for all w g Y. A weak formulation for Ž1.17. is given by
HV Žcurl H
G
Ž t . . ? =w dx s 0
Ž 2.40.
for all w g Y, t g R. Now, we state the main result: THEOREM 1. There exists a global weak solution Ž r , E, H. to problem Ž1.18. ] Ž1.28., such that Ž r y U . g L2 ŽŽ0, T ., Y . l L`ŽŽ0, T ., L`Ž V .. l C Žw0, T x, L2 Ž V .. with t r g L2 ŽŽ0, T ., Y *. for all T g Ž0, `., and ŽE, H. g C Žw0, `., L2 Ž V .., where Ž1.18. ] Ž1.20. and the boundary condition Ž1.24. are fulfilled in the sense that `
H0 HV r
k
s
Ž t . t w Ž t . dx dt `
H0 HV
k
D k Ž x, r Ž t . . =r k Ž t . q Ž y1 . m k r k Ž t . E Ž t . =w Ž t . dx dt
`
H0 HV R Ž x, r Ž t . . w Ž t . dx dt
q
for all w g C0`ŽŽ0, `., Y .. The weak formulation of the Maxwell system Ž1.21., Ž1.22. supplemented by the boundary conditions Ž1.25., Ž1.26. is gi¨ en by d dt
HVEŽ t . ? c dx s HVh Ž t . ? curl c dx q HV Žj Ž t . y j Ž t . y j Ž t . . ? c dx 2
1
0
for all c g WE , and d dt
HVh Ž t . ? c dx s yHVEŽ t . ? curl c dx q HV Ž H t
G
Ž t . . ? v dx
for all c g WH . 3. GLOBAL EXISTENCE OF SOLUTIONS The proof of existence of a weak solution to Ž1.18. ] Ž1.28. consists of the following steps: Firstly, we introduce a sequence of regularization opera-
660
F. JOCHMANN
tors R n : L2 Ž V . ª C`Ž V ., such that R n u ª u for n ª ` in L2 Ž V . strongly for all u g L2 Ž V .. Then the regularized system
t r kŽ n , M . s = ? D k x, Ž R n r Ž n , M . .
ž
k
q
/ =r
Žn, M . k
Ž 3.41.
q Ž y1 . m k min M, Ž R n r kŽ n , M . .
½
y R x, min M, Ž R n r 1Ž n , M . .
½
ž
q
q
5E
Žn, M .
5 , min ½ M, Ž R
n
r 2Ž n , M . .
q
5/,
k g 1, 2 4
t E Ž n , M . s curl hŽ n , M . y jŽ n , M . y j 0
Ž 3.42.
t hŽ n , M . s ycurl E Ž n , M . y t H G ,
Ž 3.43.
where jŽ n , M . s yD 1 x, Ž R n r Ž n , M . .
ž
q
q D 2 x, Ž R n r Ž n , M .
ž
/ =Ž R r . . / =Ž R r . Žn, M . 1
n
q
Žn, M . 2
n
q m 1 min M, Ž R n r 1Ž n , M . .
½
q
Ž 3.44.
5 qm
2
min M, Ž R n r 2Ž n , M . .
½
q
5
=E Ž n , M .
r Žn, M . s U
on GD
ª
Ž 3.45.
n D k x, Ž R n r Ž n , M . .
ž
k
q
/ =r
Žn, M . k
q Ž y1 . m k min M, Ž R n r kŽ n , M . .
½
q
5E
Žn, M .
s0
Ž 3.46.
on GN for k g 1, 24 . ª
n n EŽ n , M . s 0
on GD
Ž 3.47.
ª
on GN
Ž 3.48.
n n hŽ n , M . s 0
r Ž n , M . Ž 0, x . s r 0 Ž x . E
Žn, M .
Ž 0, x . s E 0 Ž x . ,
Ž 3.49. h
Žn, M .
Ž 0, x . s h 0 Ž x .
Ž 3.50.
is solved by using the contraction mapping principle in a suitable subset of C Žw0, T x. L2 Ž V ... Next, we prove a priori estimates for r Ž n, M . y U in L2 ŽŽ0, T ., Y ., t r Ž n, M . in L2 ŽŽ0, T ., Y *. and for ŽE Ž n, M ., hŽ n, M . . in C Žw0, T x, L2 Ž V .., which are independent on n.
661
THE DRIFT DIFFUSION MODEL
Thus, we obtain a subsequence Ž r Ž n m , M ., E Ž n m , M ., hŽ n m , M . . m g N , which converges weakly to some Ž r Ž M ., E Ž M ., hŽ M . . with r Ž M . y U g L2 ŽŽ0, T ., Y ., t r Ž M . g L2 ŽŽ0, T ., Y *. and ŽE Ž M ., hŽ M . . g L`ŽŽ0, T ., L2 Ž V ... By a compactness argument the convergence r Ž n m , M . ª r Ž M . for m ª ` is strong in L2 ŽŽ0, T ., L2 Ž V .., so that we can conclude that Ž r Ž M ., E Ž M ., hŽ M . . solves the system
t r kŽ M . s = ? D k Ž x, Ž r Ž M . .
q
q
. =r kŽ M . q Ž y1. k mk min ½ M, Ž r kŽ M . . 5 E Ž 3.51.
y R x, min M, Ž r 1Ž M . .
½
ž
q
5 , min ½ M, Ž r
ŽM. 2
.
q
5/, k g 1, 2 4
t E
y j0
Ž 3.52.
t hŽ M . s ycurl E Ž M . y t H G ,
Ž 3.53.
ŽM.
s curl h
ŽM.
yj
ŽM.
where jŽ M . s yD 1 Ž x, Ž r Ž M . .
q
q
. =r 1Ž M . q D2 Ž x, Ž r Ž M . . . =r 2Ž M .
q m 1 min M, Ž r 1Ž M . .
½
rŽM. s U
q
5 qm
2
min M, Ž r 2Ž M . .
½
q
Ž 3.54.
5E
on GD
ª
n ? D k Ž x, Ž r Ž M . .
q
ŽM.
Ž 3.55. q
. =r kŽ M . q Ž y1. k mk min ½ M, Ž r kŽ M . . 5 EŽ M .
s0
Ž 3.56. on GN for k g 1, 24 , ª
n n EŽ M . s 0
on GD
Ž 3.57.
ª
on GN
Ž 3.58.
n n hŽ M . s 0
r Ž M . Ž 0, x . s r 0 Ž x . E Ž M . Ž 0, x . s E 0 Ž x . ,
Ž 3.59. hŽ M . Ž 0, x . s h 0 Ž x . .
Ž 3.60.
Finally, we prove the positivity of r kŽ M . and L` estimates for r Ž M ., which are independent of M by applying suitable testing functions to Ž3.51.. Then it follows immediately that Ž r Ž M ., E Ž M ., hŽ M . . is a solution of Ž1.18. ] Ž1.28. if we choose M large enough. Now, we start with some
662
F. JOCHMANN
preliminaries concerning the Maxwell system Ž1.21., Ž1.22. supplemented by the initial boundary conditions Ž1.25., Ž1.26., Ž1.28.. We define an unbounded operator B in L2 Ž V, R 6 . by Bu s Ž curl Ž u 4 , u 5 , u 6 . y curl Ž u 1 , u 2 , u 3 . . with domain DŽ B . s WE = WH , where WE , WH are the spaces defined in Section 2. Since C0`Ž V, R 3 . ; WE l WH , B is densely defined. LEMMA 1. Ži. B is skew self-adjoint in L2 Ž V, R 6 ., i.e., B* s yB. Žii. Let w g Y. Then Ž =w , 0, 0, 0. g ker B. Proof. From the definition of WE , WH it follows immediately that B is skew-symmetric, i.e., B ; yB*. Now, let u g DŽ B*.. For w g WE we have w g Žw, 0, 0, 0. g DŽ B . and hence
HV Ž u
4,
u 5 , u 6 . ? curl w dx s y² u, Bw : s y² B*u, w : sy
Hv Ž g , g 1
2,
g 3 . ? w . dx
with g s B*u. Thus Ž u 4 , u 5 , u 6 . g WH with curlŽ u 4 , u 5 , u 6 . s yŽ g 1 , g 2 , g 3 .. Now let w g C`ŽR 3 , R 3 . with GN l supp w s B. Then w g WH and hence w s Ž0, 0, 0, w. g DŽ B .. Thus
HV Ž u , u 1
2,
u 3 . ? curl w dx s ² u, Bw : s ² B*u, w : s
HV Ž g
4,
g 5 , g 6 . ? w dx.
Therefore Ž u1 , u 2 , u 3 . g WE with curlŽ u1 , u 2 , u 3 . s Ž g 4 , g 5 , g 6 .. Thus, we have shown u g DŽ B . and Bu s yg s yB*u. In order to prove Žii., let w g Y and c g C`ŽR 3, R 3 . such that the support of c does not intersect GN . We choose a sequence wn , n g N in C0`ŽR 3 _ GD . with 5 wn y w 5 H 1 Ž V . ª 0 for n ª `. Since =wn n c s 0 on a neighbourhood of V, we obtain lim H Ž =w . ? curl c dx HV Ž =w . ? curl c dx s nª` V n
s lim
nª`
HV divŽ Ž =w . n c . dx s 0.
Hence =w g WE and the assertion follows.
n
663
THE DRIFT DIFFUSION MODEL
Next, the weak formulation of the Maxwell system Ž1.21., Ž1.22., Ž1.25., Ž1.26., and Ž1.28. is given. Let j g L2 ŽŽ0, T ., L2 Ž V, R 3 ... ŽE, h. g C Žw0, T x, L2 Ž V, R 6 .. is called a weak solution of
t E s curl h y j 0 y j
Ž 3.61.
t h s ycurl E y t H G
Ž 3.62.
ª
n n E s 0 on GD , EŽ 0. s E 0 ,
ª
n n h s 0 on GN
Ž 3.63.
h Ž 0. s h 0
Ž 3.64.
if d dt
HVEŽ t . ? c dx s HVh Ž t . ? curl c dx y HV Žj Ž t . q j Ž t . . ? c dx 0
Ž 3.65.
for all c g WE , and d dt
HVh Ž t . ? c dx s yHVEŽ t . ? curl c dx y HV Ž H t
G
Ž t . . ? c dx Ž 3.66.
for all c g WH , and EŽ 0. s E 0 ,
h Ž 0. s h 0 .
Ž 3.67.
Note that Ž3.65., Ž3.66. include the boundary conditions Ž3.63. in a generalized sense. By setting uŽ t . s ŽEŽ t ., hŽ t .., f j Ž t . s yŽjŽ t . q j 0 Ž t ., t HŽ t .., Ž3.65. ] Ž3.67. can be reformulated as d dt
² u Ž t . , w: s y² u Ž t . , Bw: q² f j Ž t . , w: s² u Ž t . , B*w: q² f j Ž t . , w: u Ž 0. s Ž E 0 , h 0 .
for all w g DŽ B ., i.e., u g C Žw0, T x, L2 Ž V, R 6 .. is a mild solution of the L2 Ž V, R 6 . valued evolution equation d dt
u Ž t . s Bu Ž t . q f j Ž t . ,
u Ž 0. s Ž E 0 , h 0 . .
664
F. JOCHMANN
This Cauchy problem has a unique weak solution u g C Žw0, T x, L2 Ž V, R 6 .. given by
u Ž t . s exp Ž tB . Ž E 0 , h 0 . q
t
H0 exp Ž Ž t y s . B . f Ž s . ds, j
Ž 3.68.
where expŽ tB ., t g R is the C0 group of unitary operators in L2 Ž V . generated by B w1, 10x. Equation Ž3.68. yields the energy estimate 1 d 2 dt
uŽ t .
2 L2 Ž V .
s² f j Ž t . , u Ž t .:
Ž 3.69.
HVj Ž t . ? EŽ t . dx y HV Žj Ž t . ? EŽ t . q Ž H Ž t . . ? h Ž t . . dx.
sy
t
0
Next, it is shown that the solution ŽE, h. of the Maxwell system Ž3.21. ] Ž3.24. obeys =E s r , ª n ? E s 0 on GN weakly. LEMMA 2.
Let r g C ŽŽ0, T ., L2 Ž V .., j g L2 ŽŽ0, T ., L2 Ž V .. with y
HVE
0
? =w dx s
HVr Ž 0. w dx
and d
HVj Ž t . =w dx s dt HVr Ž t . w dx for all w g Y, i.e., = ? j s y t r and ª n ? j s 0 on GN . Then
HVEŽ t . =w dx s HVr Ž t . w dx
y for all w g Y.
665
THE DRIFT DIFFUSION MODEL
Proof. Let w g Y. By Lemma 1Žii. we have c s Ž =w , 0, 0, 0. g ker B. Thus, we obtain from Ž3.68.
HVEŽ t . =w dx s² u Ž t . , c: s² Ž E 0 , h 0 . , exp Ž ytB . c: q s² Ž E 0 , h 0 . , c: q
t
H0² f Ž s . , exp Ž ysB . c: ds j
t
H0² f Ž s . , c: ds j
t
s
HVE =w dx y H0 HV Žj Ž s . q j Ž s . . ? =w dx ds
s
HVE =w dx y H0 HVj Ž s . ? =w dx ds
s
HVE =w dx y HV
0
0
t
0
0
r Ž t . y r Ž 0 . w dx
HVr Ž t . w dx.
sy
Now, we prove existence and uniqueness for the regularized system Ž3.41. ] Ž3.50.. The regularization operars R n are defined as follows: Since V is a Lipschitz-domain, we find open sets U1 , . . . , UN ; R 3, k 1 , . . . , k N g R 3 , such that V ; D Njs1Uj and x q t k j g V for x g V l Uj , t g Ž0, 1.. Now, let G1 , . . . , GN g C`ŽR 3 . with supp Gj ; Uj and Ý Njs1Gj Ž x . s 1 for all x g V. Since V l supp Gj is a compact subset of the open set x g R 3 : x q Ž1rn.k j g V 4 , n g N, there exists some dn g Ž0, 1rN . with x y y q Ž1rn.k j g V for all x g V l supp Gj , n g N, y g R 3 with < y < F dn . Choose v g C0`ŽR 3 . with HR 3 v dx s 1, supp v ; B1Ž0.. Now, we define for u g L2 Ž V ., n g N, x g V, N
Ž R n u. Ž x . s
Ý Gj Ž x . H
u xq
Bd nŽ0 .
js1
ž
1 n
k j y y v jŽ n. Ž y . dy,
/
2Ž . Ž . where v jŽ n. Ž y . s dy3 n v yrd n . For u g L V we have
N
Rn ª
Ý xj u s u js1
for n ª `
Ž 3.70.
666
F. JOCHMANN
in L2 Ž V . strongly. From Young’s inequality we obtain constants k 0 , k n g Ž0,’`., where k n depends only on n g N, such that 5 R n u 5 C 1 Ž V . F k n 5 u 5 L2 Ž V .
Ž 3.71.
5 R n u 5 L2 Ž V . F k 0 5 u 5 L2 Ž V .
Ž 3.72.
and
for all u g L2 Ž V .. Now, let u g H 1 Ž V .. Then we have by the definition of R n N
=Ž R n u . Ž x . s
Ý Ž =Gj Ž x . . H
Bd nŽ0 .
js1
1
u xq
ž
n
k j y y v jŽ n. Ž y . dy
/
N
q
u Ž z . =v jŽ n. x q
Ý Gj Ž x . H
ž
Bd nŽ xq Ž1rn .k j .
js1
1 n
k j y z dz
/
N
s
Ý Ž =Gj Ž x . . H
u Ž x q Ž 1rn . k j y y . v jŽ n. Ž y . dy
Bd nŽ0 .
js1 N
q
Ý Gj Ž x . H
Bd nŽ xq Ž1rn .k j .
js1
1
Ž =u Ž z . . v jŽ n. x q k j y z dz
ž
n
/
since Bd nŽ x q Ž 1rn . k j . ; V for x g V l supp Gj . Therefore, Young’s inequality yields the estimate 5 R n u 5 2H 1 Ž V . F k 0 5 u 5 2H 1 Ž V .
Ž 3.73.
for all u g H 1 Ž V .. Next, problem Ž3.41. ] Ž3.50. will be solved by using a fixed point argument. Let r g C Žw0, T x, L2 Ž V, R 2 .. and ŽE, h. g C Žw0, T x, L2 Ž V, R 6 .. the unique weak solution of
t E s curl h y j 0 y j r
Ž 3.74.
t h s ycurl E y t H G
Ž 3.75.
©
n n E s 0 on GD , EŽ 0. s E 0 ,
©
n n h s 0 on GN
h Ž 0. s h 0
Ž 3.76. Ž 3.77.
where j r s yD 1 Ž x, Ž R n r . q
q
q
. =R n r 1 q D 2 Ž x, Ž R n r . . =R n r 2 q q m 1 min M, Ž R n r 1 . 4 q m 2 min M, Ž R n r 2 . 4
E. Ž 3.78.
667
THE DRIFT DIFFUSION MODEL
Now, we define AŽ n, M .r s r g C Žw0, T x, L2 Ž V, R 2 .., where r is the unique solution of the linear parabolic problem
t r k s y= ? D k Ž x, Ž R n r .
q
q . =r k q Ž y1. k mk min M, Ž R n r k . 4 E
y R Ž x, min M, Ž R n r 1 . ª
n ? D k Ž x, Ž R n r .
q
q
4 , min M, Ž R n r 2 . q 4 .
Ž 3.79.
q . =r k q Ž y1. k mk min M, Ž R n r k . 4 E
s 0 Ž 3.80.
on GN for k g 1, 24
rŽ t. y U g Y
Ž 3.81.
r Ž 0. s r 0 .
Ž 3.82.
By assumption D k Ž x, Ž R n r .q. is uniformly positive. It follows from the standard theory of monotone operators in the space L2 ŽŽ0, T ., Y ., that Ž3.79. ] Ž3.82. has a unique solution r g C Žw0, T x, L2 Ž V .. with r y U g L2 ŽŽ0, T ., Y . and t r g L2 ŽŽ0, T ., Y *.. Now, it is clear that r g C Žw0, T x, L2 Ž V .. is a solution to the regularized problem Ž3.41. ] Ž3.50. iff r s AŽ n, M .r . This fixed point problem is solved by using the contraction mapping principle. For this purpose we need the estimates 1 d 2 dt
rŽ t. y U
2 L2 Ž V .
2
sy
ÝH
ks1 V
D k Ž x, Ž R n r .
q
. =r k ? =Ž r k y U .
k
q Ž y1 . m k min M, Ž R n r k . y
HVR Ž x, min M, Ž R
n
r1 .
q
q
4 E ? =Ž r k y U .
dx
4 , min M, Ž R n r 2 . q 4 .
= Ž r 1 q r 2 y U1 y U2 . dx F yd =r Ž t . Fy
d 2
rŽ t.
2 L2 Ž V .
q k 2 , M Ž1 q EŽ t .
L2 Ž V .
2 H 1Ž V .
q k 3 , M 1 q EŽ t .
2 L2 Ž V .
ž
.
rŽ t.
H 1Ž V .
q rŽ t.
2 L2 Ž V .
/.
Ž 3.83.
668
F. JOCHMANN
The energy estimate Ž3.69. yields d dt
2 L2 Ž V .
EŽ t .
q hŽ t .
2 L2 Ž V .
F k 4, M Ž 1 q = Ž R n r Ž t . . Fy
d 4
=Ž R n r Ž t . .
2 L2 Ž V .
L2 Ž V .
.
2 L2 Ž V .
EŽ t .
q hŽ t .
q k 5, M 1 q E Ž t .
ž
2 L2 Ž V .
1r2 2 L2 Ž V .
q hŽ t .
2 L2 Ž V .
/,
Ž 3.84. where k 0 is the constant occurring in Ž3.73., and k 2, M , . . . , k 5, M are independent of r and n. By Ž3.83. and Ž3.84. there exists a constant k 1, M depending only on M, such that d dt
rŽ t. y U F
d 4 k0
2 L2 Ž V .
q EŽ t .
=Ž R n r Ž t . .
2 L2 Ž V .
q k 1, M 1 q E Ž t .
2 L2 Ž V .
y
2 L2 Ž V .
d 2
q hŽ t .
rŽ t.
q hŽ t .
2 L2 Ž V .
2 H 1Ž V .
2 L2 Ž V .
q rŽ t. y U
2 L2 Ž V .
.
Ž 3.85. According to Ž3.71., we find constants L n, M , k n, M, 3 g Ž0, `., such that AŽ n, M .r g Sn, M for r g Sn, M , where Sn, M is the set of all r g C Žw0, T x, L2 Ž V .. with
rŽ t. y U
2 L2 Ž V .
F k n , M , 3 exp L n , M t.
Ž 3.86.
By Ž3.85., Ž3.86. there exists a constant k n, M , 4 , such that for all r g S n, M
Ž AŽ n , M .r . y U
L2 ŽŽ 0, T . , Y .
q 5E 5 CŽŽ0 , T ., L2 Ž V .. q 5h 5 CŽŽ0 , T ., L2 Ž V .. F k n , M , 4 ,
Ž 3.87. where ŽE, h. g C Žw0, T x, L2 Ž V .. is the solution of the Maxwell system Ž3.74. ] Ž3.77.. Next, it is shown that AŽ n, M . is a contraction in C Žw 0, T x , L 2 Ž V .. w ith re spe ct to th e n orm A r A s sup t g w0, T x 5 r Ž t .5 L2 Ž V . exp Ž l n, M t . with some l n, M g Ž0, `. depending only on n, M. Let r , r g Sn, M , r s AŽ n, M .r , r s AŽ n, M .r . Then we obtain from
669
THE DRIFT DIFFUSION MODEL
Ž3.87. by using r Ž t . y r Ž t . g Y, 1 d
rŽ t. y rŽ t.
2 dt
2 L1 Ž V .
2
sy
ÝH
ks1 V
D k Ž x, Ž R n r .
q
. =rk y Dk Ž x, Ž R n r . q . =rk
k
q
k
q
q Ž y1 . m k min M, Ž R n r k . q Ž y1 . m k min M, Ž R n r k . y R Ž x, min M, Ž R n r 1 .
q
F yd r Ž t . y r Ž t .
4E
? = r k y r k dx
4 , min M, Ž R n r 2 . q 4 .
y R x, min M, Ž R n r 1 .
ž
4E
q
4 , min M, Ž R n r 2 . q 4 /
2 H 1Ž V .
qk n , M , 5 D k Ž x, R n Ž r Ž t . . . y D k Ž x, R n Ž r Ž t . . . rŽ t. y rŽ t. qk n , M , 5
`
H 1Ž V .
rŽ t. y rŽ t.
rŽ t. y rŽ t.
H 1Ž V .
rŽ t. y rŽ t.
L2 Ž V .
F kn, M , 6
rk y rk
L2 Ž V .
q EŽ t . y EŽ t .
q kn, M , 5 r Ž t . y r Ž t .
rŽ t. y rŽ t.
2 L2 Ž V .
L2 Ž V .
L2 Ž V .
q EŽ t . y EŽ t .
2 L2 Ž V .
.
Ž 3.88. Here ŽE, h., ŽE, h. are the solutions of Maxwell system Ž3.74. ] Ž3.77. corresponding to r , r , respectively. From Ž3.68., Ž3.87., and Ž3,78. we obtain EŽ t . y EŽ t . F
t
H0
L2 Ž V .
jr Ž s. y jr Ž s.
F kn, M , 8
t
H0
L2 Ž V .
ds
r Ž s. y r Ž s.
L2 Ž V .
q EŽ s . y EŽ s .
L2 Ž V .
ds.
Hence, there exists some k n, M , 9 g Ž0, `. depending only on n, M, such that EŽ t . y EŽ t .
2 L2 Ž V .
F kn, M , 9
t
H0
r Ž s. y r Ž s.
2 L2 Ž V .
ds.
Ž 3.89.
670
F. JOCHMANN
From Ž3.88., Ž3.89., we obtain gn, M g Ž0, 1. and l n, M g Ž0, `., which depend only on n, M, such that A AŽ n , M .r y AŽ n , M .r A F gn , M A r y r A , for all r , r g Sn, M , where the norm A ? A in C Žw0, T x, L2 Ž V .. is given by A r A s sup t g w0, T x 5 r Ž t .5 L2 Ž V . exp l n, M t. Hence, AŽ n, M . has a unique fixed point r Ž n, M . g Sn, M , i.e., the regularized problem Ž3.41. ] Ž3.50. has a unique solution r Ž n, M . g C Žw0, T x, L2 Ž V .. with r Ž n, M . y U g L2 ŽŽ0, T ., Y ., and t r Ž n, M . g L2 ŽŽ0, T ., Y *.. Next, we prove some estimates for r Ž n, M ., which are independent of n in order to pass to the limit n ª `. From Ž3.73. and Ž3.85. we obtain d
r Žn, M .Ž t . y U
dt
Fy
d 4
2 L2 Ž V .
r Žn, M .Ž t . .
2 L2 Ž V .
q EŽ n , M . Ž t .
q hŽ n , M . Ž t .
2 L2 Ž V .
2 H 1Ž V .
q k 1, M 1 q E Ž n , M . Ž t .
2 L2 Ž V .
q hŽ n , M . Ž t .
2 L2 Ž V .
q rŽn, M .Ž t . .
2 L2 Ž V .
.
Hence, there exists a constant k 6, M g Ž0, `., such that
r Žn, M . y U
L2 Ž0, T . , Y
q hŽ n , M . Ž t .
q EŽ n , M . Ž t .
L`ŽŽ 0, T . , L2 Ž V ..
Ž 3.90.
L`ŽŽ0 , T . , L2 Ž V ..
F k 6, M and by Ž3.41.
t r Ž n , M .
L2 ŽŽ0, T . , Y * .
F k 6, M .
Ž 3.91.
By the reflexivity of L2 ŽŽ0, T ., Y . there exists a subsequence Ž r Ž n m , M . . m g N and Ž r Ž M . , E Ž M . , h Ž M . . with r Ž M . y U g L2 ŽŽ0, T ., Y ., t r Ž M . g L2 ŽŽ0, T ., Y *., and ŽE Ž M ., hŽ M . . g L`ŽŽ0, T ., L2 Ž V .., such that
r Ž nm , M . y r Ž M . ª 0
for m ª `
Ž 3.92.
2 ŽŽ
in L
0, T ., Y . weakly,
r Ž n m , M . ª r Ž M .
for m ª `
Ž 3.93.
2 ŽŽ
in L
0, T ., Y *. weakly,
Ž E Ž n m , M . , hŽ n m , M . . ª Ž E Ž M . hŽ M . . `ŽŽ
in L 0, T ., L V .. weak *. 2Ž
for m ª `
Ž 3.94.
671
THE DRIFT DIFFUSION MODEL
By Ž3.90., Ž3.91., and a standard compactness result, see w4, 6x, the sequence Ž r Ž n m , M . . m g N is precompact to L2 ŽŽ0, T ., L2 Ž V .. s L2 ŽŽ0, T . = V .. Hence
r Ž nm , M . ª r Ž M .
for m ª `
Ž 3.95.
in L2 ŽŽ0, T ., L2 Ž V .. strongly. Relation Ž3.70. and the dominated convergence theorem yield R n mŽ r Ž n m , M . . y r Ž M .
L2 ŽŽ 0, T . , L2 Ž V ..
F R n mŽ r Ž n m , M . y r Ž M . . q ª0
ž
T
H0
L2 ŽŽ 0, T . , L2 Ž V .. 1r2
2
R n mŽ r Ž M . Ž t . . y r Ž M . Ž t .
L V. 2Ž
dt
/
for m ª `
Ž 3.96.
and therefore R n mŽ r Ž n m , M . Ž s . . Ž x . ª r Ž M . Ž s, t .
for m ª `
Ž 3.97.
a.e. in Ž0, T . = V after extracting a further subsequence. Moreover, from Ž3.94. and Ž3.97. we obtain min M, Ž R n m r kŽ n m , M . .
½
q
5E
Ž nm , M .
ª min M, Ž r kŽ M . .
½
q
5E
ŽM.
for m ª `
Ž 3.98.
in L2 ŽŽ0, T ., L2 Ž V .. weakly. Hence, we have jŽ n m , M . ª jŽ M .
for m ª `
Ž 3.99.
in L2 ŽŽ0, T ., L2 Ž V .. weakly, where jŽ n , M . s yD 1 x, Ž R n r Ž n , M . .
ž
q
q D 2 x, Ž R n r Ž n , M .
ž
/ =R r . / =R r
Žn, M . 1
n
q
n
q m 1 min M, Ž R n r 1Ž n , M . .
½
q
qm 2 min M, Ž R n r 2Ž n , M . .
½
Žn, M . 2
5 q
5E
Žn, M .
672
F. JOCHMANN
and jŽ M . s yD 1 Ž x, Ž r Ž M . .
q
q
. =r 1Ž M . q D2 Ž x, Ž r Ž M . . . =r 2Ž M .
q m 1 min M, Ž r 1Ž M . .
½
q
5 qm
2
min M, Ž r 2Ž M . .
½
q
5E
ŽM.
.
From Ž E Ž n m , M . Ž t . , h Ž n m , M . Ž t .. s exp Ž tB .Ž E 0 , h 0 . q H0t exp ŽŽ t y s . B .f jŽ n, M . Ž s . ds and Ž3.99. we obtain t
Ž EŽ n m , M . , hŽ n m , M . . ª exp Ž tB . Ž E 0 , h 0 . q H exp Ž Ž t y s . B . f jŽ M . Ž s . ds 0
for m ª ` in L V . weakly for all t g w0, T x. According to Ž3.94. we get 2Ž
t
Ž EŽ M . , hŽ M . . s exp Ž tB . Ž E 0 , h 0 . q H exp Ž Ž t y s . B . f jŽ M . Ž s . ds, 0
i.e., ŽE , h g C Žw0, T x, L V .. is the weak solution of the Maxwell system Ž3.52., Ž3.53., Ž3.57., Ž3.58., and Ž3.60.. From Ž3.41. ] Ž3.50., Ž3.92. ] Ž3.94., and Ž3.97. it follows easily that for all w g C0`ŽŽ0, T ., Y ., ŽM.
T
H0 HV r
Ž M ..
ŽM. k
s
2Ž
t w dx dt T
H0 HV
D k Ž x, Ž r Ž M . . k
q
. =r kŽ M .
q Ž y1 . m k min M, Ž r kŽ M . .
½
q
q
5E
ŽM.
=w dx dt q
T H0 HV R ž x, min ½ M, Ž r . 5 , min ½ M, Ž r . 5 / w dx dt
q
ŽM. 1
ŽM. 2
Moreover, we have
rŽM .Ž t. y U g Y for all t g Ž0, T ., and
r Ž M . Ž 0. s r 0 . Hence, Ž r Ž M ., E Ž M ., hŽ M . . is a solution of the system Ž3.51. ] Ž3.60.. Next, the positivity of r Ž M . is shown. For this purpose we choose some h g C`ŽR. with h9 G 0, h0 F 0, hŽ t . s 0 for t G 0 and hŽ t . - 0 for t - 0. Since U G 0 and r Ž M . Ž t . y U g Y, it follows easily by the definition of Y
673
THE DRIFT DIFFUSION MODEL
that hŽ r Ž M . Ž t .. g Y. Hence, we obtain from Ž3.51. d dt
HVh Ž r
ŽM. k
Ž t . . dx
HVh0 Ž r .
sy
ŽM. k
D k Ž x, Ž r Ž M . .
q
k
. =r kŽ M .
q Ž y1 . m k min M, Ž r kŽ M . .
½
ŽM. q 1
q
5E
ŽM.
q
=r kŽ M . Dx
HVR ž x, min ½ M, Ž r . ˙ 5 , min ½ M, Ž r . 5 / h9 Ž r . dx
y
ŽM. 2
q
q
ŽM. k
HVR ž x, min ½ M, Ž r . 5 , min ½ M, Ž r . 5 / h9 Ž r . Dx G 0
Gy
ŽM. 1
ŽM. 2
ŽM. k
since RŽ x, u. F 0 if u1 s 0 or u 2 s 0 by the assumptions on R. Hence, we have HV hŽ r kŽ M . Ž t .. dx G 0 and this implies r kŽ M . Ž t . G 0 in V a.e. Finally, we prove a L` estimate for r Ž M ., which is independent of M. Therefore, if we choose M larger than this bound, Ž r Ž M ., E Ž M ., hŽ M . . is a solution to the original problem Ž3.58. ] Ž3.68.. Assume that h g C`ŽR. such that h, h9 are bounded, h9 G 0, and hŽ t . s 0 for t F max5 U 5 ` , 5 r 0 5 `4 . With g Ž t . s H0t hŽ s . ds and HM Ž t . s
t
H0 min M, s4 h9 Ž s . ds F th Ž t .
we obtain from Ž3.51. and hŽ r kŽ M . Ž t .. g Y, d dt
HV
ŽM. ŽM. my1 Ž t . . q my1 Ž t . . dx 1 g Ž r1 2 g Ž r2
2
sy
Ý H my1 k ž
ks1 V
D k Ž x, r Ž M . . =r kŽ M . k
q Ž y1 . m i min M, r kŽ M . 4 E Ž M . =h Ž r kŽ M . . qR Ž x, min M, r 1Ž M . 4 , min M, r 2Ž M . 4 . h Ž r kŽ M . . dx
/
F
HV min M, r 4 =h Ž r . ŽM. 1
ŽM. 1
674
F. JOCHMANN
ymin M, r 2Ž M . 4 =h Ž r 2Ž M . . E Ž M . dx q K0 F
HV Ž1 q r
ŽM. 1
q r 2Ž M . . h Ž r 1Ž M . . q h Ž r 2Ž M . . dx
HM Ž r 1Ž M . . y HM Ž r Ž M . . E Ž M . dx
HV=
HV Ž1 q r . h Ž r . q Ž1 q r . h Ž r .
qk
ŽM. 1
ŽM. 1
ŽM. 2
ŽM. 2
dx, Ž 3.100.
since Ž u1 q u 2 .w hŽ u1 . q hŽ u 2 .x F 4w u1 hŽ u1 . q u 2 hŽ u 2 .x for u g w0, `. 2 , because h is nonnegative and nondecreasing. By setting r Ž t . s r 1Ž M . y r 2Ž M . q C we obtain from Ž3.51. ] Ž3.54., Ž3.60., Ž2.39., and Lemma 2,
HV=
HM Ž r 1Ž M . . y HM Ž r 2Ž M . . E Ž M . dx sy
HM Ž r 1Ž M . . y HM Ž r 2Ž M . .
Fy
HM Ž r 1Ž M . . y HM Ž r 2Ž M . . C dx
F 5 C 5`
Ý H r kŽ M . h Ž r kŽ M . . dx
HV HV
Ž r 1Ž M . y r 2Ž M . q C . dx
2 ks1 V
and hence by using Ž3.100. d
2
H gŽ r . q gŽ r . dt V ŽM. 1
ŽM. 2
dx F k 0, 1
Ý H Ž 1 q r kŽ M . . h Ž r kŽ M . . dx
ks1 V
Ž 3.101. with some k 0, 1 g Ž0, `. independent of M. In order to prove a Ln estimate for r Ž M . Ž t ., we introduce the functions hn , N Ž u. s nŽ N y A. hn , N Ž u. s nŽ u y A. hn , N Ž u. s 0 gn , N Ž u. s
ny 1
ny 1
if u - A u
H0 h
n, N
Ž s . ds,
if u ) N f u g w A, N x
675
THE DRIFT DIFFUSION MODEL
where A s max5 r 0 5 ` , 5 U 5 `4 and N ) A. By approximation with smooth functions Ž3.101. can be applied to h n, N and g n, N . A simple calculation yields uh n , N Ž u . F ng n , N Ž u . q Ah n , N Ž u .
Ž 3.102.
for u g w A, `.. Since h n, N Ž u. F n for u F A q 1 and h n, N Ž u. F ng n, N Ž u. for u ) A q 1 we have
HVh Ž r n, N
ŽM. k
Ž t . . dx F n < V < q H
r kŽ M . Ž t .GAq1 4
F n< V < q n
HVg Ž r n, N
h n , N Ž r kŽ M . Ž t . . dx
ŽM. k
Ž t . . dx.
Ž 3.103.
From Ž3.101. ] Ž3.103. we obtain the Gronwall-type estimate d dt
HV g Ž r . q g Ž r . ŽM. 1
HV g Ž r . q g Ž r .
dx F k 2 n q k 2 n
ŽM. 2
ŽM. 1
ŽM. 2
dx
Ž 3.104. and hence
HV g Ž r
ŽM. 1
Ž t . . q g Ž r 2Ž M . Ž t . . dx F 2 k 2 Tn exp 2 k 2 tn.
Ž 3.105.
Since g n, N Ž u. ª ŽŽ u y A.q. n for n ª `, we obtain from the monotone convergence theorem and Ž3.46.,
r kŽ M . Ž t . g Ln Ž V . and
Ž r kŽ M . Ž t . y A .
q Ln Ž V .
F Ž 2 k 2 Tn .
1rn
exp 2 k 2 T .
Now, letting n ª ` yields
r kŽ M . Ž t . g L` Ž V . and
Ž r kŽ M . Ž t . y A .
q L`Ž V .
F exp 2 k 2 T .
Hence, Ž r Ž M ., E Ž M ., hŽ M . . is a solution of the system Ž3.58. ] Ž3.68., if M ) A q exp 2 k 2 T.
676
F. JOCHMANN
REFERENCES 1. J. M. Ball, Strongly continuous semigroups, weak solutions and the ¨ ariation of constant formula, Proc. Amer. Math. Soc. 63 Ž1977., 370]373. 2. H. Gajewski, On existence, uniqueness and asymptotic behavior of solutions of the basic equations for carrier transport in semiconductors, Z. Angew. Math. Mech. 65 Ž1985., 101]108. 3. H. Gajewski and K. Groger, On the basic equations for carrier transport in semiconductors, J. Math. Anal. Appl. 113 Ž1986., 12]35. 4. H. Gajewski, K. Groger, and K. Zacharias, ‘‘Nichtlineare Operatorengleichungen und Operatordifferentialgleichungen,’’ Akademie Verlag, Berlin, 1974. 5. A. Juengel, On existence and uniqueness of transient solutions of a degenerate nonlinear drift diffusion model for semiconductors, submitted for publication. 6. J. L. Lions, ‘‘Quelques methods de resolution des problems aux limites non lineaires,’’ DunodrGauthier]Villars, Paris, 1969. 7. P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, ‘‘Semiconductor Equations,’’ Springer-Verlag, New YorkrBerlin, 1990. 8. M. S. Mock, An initial value problem from semiconductor device theory, SIAM J. Math. Anal. 5 Ž1974., 597]612. 9. M. S. Mock, Asymptotic behavior of solutions of transport equations for semiconductor devices, Math. Anal. Appl. 49 Ž1975., 215]225. 10. Pazy, ‘‘Semigroups of Linear Operators and Applications to Partial Differential Equations,’’ Springer-Verlag, New York, 1983. 11. T. I. Seidman and G. M. Troianiello, Time dependent solutions of a nonlinear system arising in semiconductor theory, Nonlinear Anal. 9 Ž1985., 1137]1157. 12. V. Roosbroeck, Theory of flow of electrons and holes in Germanium and other semiconductors, Bell System Tech. J. 29 Ž1950., 560]607.