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The quantum Child-Langmuir problem
Pergamon
Nonlinear Analysts, Theory, Methods&Applications, Vol. 31, No. 5/6. pp. 629-648, 1998 1998 Elsevier ScienceLtd Printed in Great Britain.All rightsreserved 0362-546X/98 $19.00+0.00
PII: S0362-546X(97)OO429-X
THE QUANTUM CHILD-LANGMUIR PROBLEM1" NAOUFEL BEN ABDALLAH,:[: PIERRE DEGOND$ and PETER A. MARKOWICH§ SMIP, Laboratoire CNRS (UMR 5640), Universit~ Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France; and §Fachbereich Mathematik, Technische Universit~t Berlin, StraBe des 17 Juni 136, 10623 Berlin, Germany (Received 9 January 1996; received for publication 18 February 1997) Key words and phrases: Quantum phenomena, semiclassical limit, Wigner transform, current carrying state, monokinetic flow, Child-Langmuir limit.
1. I N T R O D U C T I O N
Vacuum electron devices like vacuum diodes, amplifier tubes, etc., are usually described by means of kinetic models of charged-particle transport like the Vlasov-Poisson, Vlasov-Maxwell or Boltzmann-Maxwell systems. The accurate modelling of a real vacuum device requires the design of appropriate boundary conditions in particular to describe the injection of particles from the cathode. The mathematical analysis of the boundary value problem for the Vlasov-Poisson and Vlasov-Maxwell systems has recently been investigated in [1,2] for the stationary case and in [3, 4] for the time dependent case. The role of the space-charge in the limitation of the maximal current that can be extracted from the cathode of an electron tube is well-known since the work of Child, Langmuir and Compton [5]. A mathematical framework to the space-charge limited current emission problem in vacuum diodes has recently been provided in [6-8]. In these references, it is shown that the space-charge limitation regime (or Child-Langmuir regime) can be described by the Vlasov-Poisson system with a singularly perturbed boundary condition at the injection boundary. This condition merely expresses that a very large number of electrons start from the cathode with a very small initial velocity (compared with the one they are expected to reach in the applied electric field), thus creating a space-charge layer in the vicinity of the cathode. Then, it has been shown in a series of work [3, 9-11] that the same framework is able to describe semi-classical electron transport phenomena in semiconductor devices such as the injection of electrons over potential barriers at junctions. In these works, it is observed that the highly doped N ÷ region behaves like the metallic cathode of a vacuum device and that the electrons crossing the junction are facing similar conditions (mutatis mutandis) as those extracted from the cathode of a vacuum device. Numerical computations [12] strongly support these observations. It is tempting to investigate if a similar theory can be developed in the framework of quantum transport. The first step is the design of boundary conditions modelling the
t This work has been supported by the "Human Capital and Mobility" project #ERBCHRXCT 930413 entitled "Nonlinear spatio-temporal structures in semiconductors, fluids and oscillator ensembles" funded by the EC, by the Groupement de Recherches " S P A R C H " of the CNRS, France, by the bilateral cooperation project Procope #96123 funded by the DAAD-APAPE and by the project "Analysis und Numerik yon Kinetischen Quantentransportmodellen" funded by the DFG under the contract number MA 1662/I-I. 629
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N. BEN ABDALLAH et al.
injection of particles at the boundary of a quantum device. This step has been realized in [13], with the following assumptions: the metallic contacts are supposed to be connected to metallic leads; electrons entering the device from the contact are identified with incoming scattering states of the Schr6dinger equation in a domain consisting of the union of the device itself and the surrounding leads. Then, the problem is restricted to the device itself by the prescription of boundary conditions which are absorbing for the considered scattering states. Assuming that electrons entering the device are distributed according to given statistics (for instance, that of a reservoir of particles which sends the electrons along the lead from infinity), a density matrix is constructed which allows to define the electric potential self-consistently taking into account the Coulomb interaction. The resulting probelm is a coupled Schr6dinger-Poisson system. In [13], the existence of solutions and the semi-classical approximation of this system are investigated. Once appropriate injection boundary conditions are derived, the second step towards a "quantum Child-Langmuir" asymptotic is to tailor these conditions to model the injection of a "large" number of electrons with a " s m a l l " incoming velocity. To this aim, we express the statistics of scattering states coming into the device with wave-vector k according to (l/e~')eP(k/e), where e is a small parameter related to the velocity scale of the incoming particles. Not all powers a give rise to a nontrivial asymptotic limit and it is proven in [13] that the correct one is ct = 3. The coupled Schr6dinger-Poisson system with a e dependent injection statistic is system (P~,h) below: ~2 ,, 2 --tl ~gk.h, t + V h , t ~ k . h , ~ = k ¢k.h,~
h¢~,h,e(O) (P~.h)
+ ik~k,h,e(O ) =
2ik
hiot~.h,c(l) = i x / - ~ - VltOk,h.~(l) - vh"~ = -~ 3 o v~.~(o) = o,
\e/ Vh.~(l) = V,
where the constant h > 0 represents a scaled Planck constant. The boundary potential V~ is assumed to be negative. In [13], the limit as e --. 0 of (Pc,h) has been proven to lead to the following problem for a monokinetic (single state) Schr6dinger-Poisson model refered to as (P0,h): --haCl'h + V h ~ h
= 0
h~(O) = 2i
(~o,h)
h~v~(1) = i~-SVl I//h(l ) -Vn" = Azl(uh(X)l2,
k2~(k)dk ,0
vh(o) = o,
Vh(1) = v~.
On the contrary, when we let e be fixed and h ~ 0, we are naturally led to the following Child-Langmuir type problem for the classical Vlasov-Poisson problem denoted by (P~,o):
The quantum Child-Langmuir problem
P~
@
631
Po.h
0
~ 0
+i 0
0
®
o,o
Fig. I. Convergence diagram.
vaf, l dV~ 0f~ = Ox 2 dx Ov l(v)
A ( o , v) = ~ *
(P~,o)
f , ( l , v) = O, -U'
= n+ =
0
,
v > o
v< 0
I ++f ~ d v
~(0) = O,
~(1) -- Vl.
Note that the usual scaling factor of the incoming distribution function is 1/• 2 instead of l / t 3. Thus, compared to the standard theory [6-8] an accumulation of particles at x = 0 is expected to occur, leading to the appearance of a concentration in the density. The limit o f (Pe,0) as e --* 0 leads to the following semi-classical monokinetic problem denoted by (Po,0) which is also the limit of (Po,n) when h --+ 0:
_
V
tt
__
J ~-v
V ( O ) = V'(O) = O,
(Po,o)
4
V(I) = V~ 3/2
J = Jct(VI) = ~(-- 1) n = ~
+ ~f8A~i(x),
=
k 2 0 ( k ) dk.
In this paper we shall deal with the convergence diagram shown in Fig. 1, linking the above detailed four models. In particular we shall prove the convergences II and III. We recall that convergences I and IV have been analysed in [13] (however, convergence IV has been proved in [l 3] for e large enough only. The proof for arbitrary e is more involved and will be presented in a future paper.)
N. BEN A B D A L L A H et al.
632 2. S E M I - C L A S S I C A L
LIMIT OF THE REDUCED
QUANTUM
PROBLEM
Let ¢'h and Vh be a solution of (PO.h) for h > 0, V~ < 0. We recall the equations satisfied by ~'h and Vh: -h2c'g + Vh~'h = 0
(2.1)
h~,~ (0) = 2i
(2.2)
h~(l)
= ivr-SV~ ~'h(1)
(2.3) (2.4)
--Vh" = ,~211],/h]2
Vh(0) = 0,
V~(1) = V~.
(2.5)
Let us define the particle current density Jh by
jh(X) = ).2h Im(~,~ (x)ff/h(x))
(2.6)
This quantity is independent of x. Let us also define the particle density (2.7)
nh(X ) = A2[~Vh(X)I 2
and the Wigner function [14] associated with ~'h by ei~(O~'h) x - ~ y
Wh(X, V) = X 2
(Off~h) X + ~ y
dy
(2.8)
,-oo
where 0 is a C OOcompactly supported function identically equal to 1 in a neighbourhood of [0, 1]. It is well known that nh and Jh are the zeroth and first order moments of Wh (with respect to v) in [0, 1]. For the analysis of the limit behaviour of Wh we introduce the test space (i = lip = q~(x,v)l(~YvqO(x, rl)~ L~(tR,;Co(~x))}, (see [141), where 5:~ denotes the Fourier transform with respect to v. The existence of solutions of (P0,h) is proven in [13]. Here we are interested in convergence of such solutions when h tends to zero (corresponding to the semi-classical limit). We now state the main theorem of this section. THEOREM 2.1. Let j, n, V, be defined by the following formulae. (i) j = ~(-Vl) 3/2 =: jcL(V,),
J
(ii) n(x) = xfE-#(x) (iii)
-V"
-
J C:V'
+ ~a6(x), V(0) = V'(0) = 0,
V ( 1 ) = V~,
(iv) W(x, v) = n(x) ® J(v - ~/-V(x)) in [0, 1] x R v. Then, as h tends to zero, Vh converges to V in C°([0, 1]) fq C~o¢((0, 1]), Wh converges in (~'w. to W, n, converges to n in the weak star topology of bounded measures on [0, 1] and Jh converges to j. Note that JcL(V~) is the Child-Langmuir limit current [8].
The q u a n t u m C h i l d - L a n g m u i r p r o b l e m
633
2.1. P r e l i m i n a r y e s t i m a t e s
In this paper we do not mention explicitly the extraction of subsequences in the limit arguments based on compactness [since the limit problem (Po,o) is uniquely solvable]. Multiplying (2.1) by ~h and integrating from 0 to x gives the following identities
v,(u)l~,n(u)l z du = h 2 Re[~(u)0h(u)l~
h2[~/,(u)l z du + ,0
(2.9)
0
2 Re(~,,(0))
= ~
I~Uh(l)l2.
(2.10)
In view of (2.3) and (2.6), equation (2.10) can be written as 2 Re(~'h(0)) = ~ .
(2.1 l)
Using the identity d ( h 2 l ~'~ (x) l2 - Vn(x)[ ~'h(x) l2) = - Vn'(x)I~h(X)[ 2 which follows immediately from the Schr6dinger equation (2.1), and the Poisson equation satisfied by the potential ~ we deduce that the function Hh(X)
:= h2llp'~(x)[ 2 -
Vh(X)Iq/h(X)[ 2 -- ~
(2.12)
is constant, i.e. Hh(O) = 4 -- ~
Hh(x )
- -
(2.13)
=
Hh(l ) = A. j h _ _
-
In order to obtain the necessary estimates to pass to the limit h --, 0, we start with the following lemma.
LEMMA 2.2. The following assertions are equivalent: (i) Jh is bounded, (ii) IIh~'~ IIL~ + IIh~'hllL~ is bounded, (iii) IIv ll.,, is bounded. Proof. •
(i) ~ (ii).
Assume that Jh is bounded. Then in view of (2.10), (2.11), ~'h(l) is bounded. It is thus sufficient to prove the boundedness of [[h~,~ [[L®. Let us set gh(X) = h2llm ~ ( x ) l 2 -
Vhllm~'h(X)l 2.
Taking into account the relation g~
=
-Vh'IIm(~,Dl 2
634
N. BEN ABDALLAH
et al.
and the concavity of Vh (in view of the Poisson equation), we conclude that gh assumes its maximum at x = 0 or x = 1 and moreover, we have
gh(O)
=
4,
gh(l)
2
<
-~jh~--VI,
Let x0 be a point at which ]Im(~g~)l achieves its maximum. If Xo is equal to 1 or to O, then it is easy to show using (2.2) and (2.3) that hllm(~'~(Xo))] -< C. If Xo is in the interior of the interval, then Im ~,~' vanishes there. Using the imaginary part of (2.1), we deduce that
gh(Xo) = h21Im(~'~(Xo))l z <- C. Therefore h Im(q/~) is bounded in L~. For the real part, we proceed analogously and obtain therefore the boundedness in L°~ of h~,~. • (ii) ~ (iii). Since jh -ff-L~ = -A2V~I~Uh(I)I 2 = h2l~u~;(l)l 2 the boundedness of [Ihw~ IlL®+ Ilh~'hllL®implies [in view of (2.13)1 that (Vh'(1))2 - (Vh'(O))z is bounded. Hence, keeping in mind that Vh is concave, it is sufficient for the proof of (iii) to prove that V~'(I) is bounded. For this aim, we first deduce from (2.9) that jl Vhl~'h]2 is bounded, which can be written after an integration by parts ,b I
t (Vh'(u))2du -
Vl Vh'(1)
is bounded.
(2.14)
,0
Also, using (2.12) and (2.13), we have Vhtx)l~,,(x)l 2 + ~
- ~
- -
= h2l~,A(x)l z
-
~jh~--Vi.
The right-hand side of this inequality is bounded in L® by hypothesis. Therefore, integrating the left-hand side between 0 and 1 and using the Poisson equation (2.4) gives 3! l - -
( V ~ ( u ) ) 2 d u + Vt
2 o
1 Vh'(1) + ~ (Vh'(l))2 is bounded.
Combining (2.14) and (2.15) implies that V~Vh'(l) - (Vh'(l))2 is bounded and yields the boundedness of Vh'(l). • (iii) ~ (i). Obvious in view of (2.13). •
(2.15)
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635
2.2. Boundedness o f the current LF.MMA 2.3. There exists C > 0 such that IJh2~llL- + IIh2~,hllL- ___ C.
Proof. At first we shall prove the estimate h2[~,~l 2 <_ C(l + Iwh(l)l%.
(2.16)
This can be done by considering again the function
gh(X)
=
hZllm ~'~(x)l 2 - Vnllm ~h(x)l 2
and its analogue with the real parts replacing the imaginary parts. These two functions take their maximum at 0 or at 1. Since this maximum is bounded by the right-hand side of (2.16), we can proceed as in the proof of Lemma 2.2 [second part of (i) = (ii)]. Now by taking the square root of this inequality and integrating between 0 and 1, we obtain C I~'h(0)- ~'h(1)l ~ ~ ( 1 4-
I h(l)l).
We deduce from this inequality and from (2.10) that I~uh(1)l4 _< Cl~,'h(O)l z < ~ ( 1 + I~Uh(I)I2) which leads to hl~,h(1)l _< C and ends the proof in view of (2.16). LEMMA 2.4. The L® norm o f Vh is bounded.
Proof. Since Vh is concave, the assertion IlvhllL- is bounded is equivalent to max Vh is bounded. We prove the lemma by contradiction. Assume that max Vh tends to +oo. Then using the concavity of Vh and its boundedness from below, we conclude that there exists et > 0 such that for h small enough Vh_> 1
onD:=
[ct, l - a].
Using the maximum principle for the elliptic equation (2.1), one can prove that there exists C > 0 such that
vx ~ t~,
I~,h(x)l < suplv/hl exp(-Cd(x, O~)/h). am
(2.17a)
Now we estimate the current density Jh at x = 1/2, taking into account the above inequality and the estimates of Lemma 2.3. We obtain C
Jh < ~ exp(-C/h). The current is bounded and Lemma 2.2 gives a contradiction.
•
636
N. BEN ABDALLAH et al.
COROtLARV 2.5. Let et > 0 be fixed. Then there exists C,~ > 0 such that
IlVh'llL'(~,l-o> --< ca.
(2.17)
For all sequence an e (0, 1 - c0 such that Vd(a h) <_ 0 the following estimate holds
IIv;IIL~<~,,_o) -< c,,.
(2.18)
P r o o f . The corollary is an obvious consequence o f the above lemma and the concavity o f Vh. •
N o w we are able to prove the main result o f this subsection. P R o a o s m o N 2.6. The current Jh is b o u n d e d . P r o o f . Let us denote by Xh the largest point where Vh vanishes. T w o cases are possible. Either xh tends to one or to a value less then one.
• First case: xh --' 1 - 2c~, c~ e (0, 2Xl. We first notice that in view o f the Poisson equation we have I-¢r I~Uh(U) Iz d u = A2(Vn'(1 _ 2o¢) - ~(1 - or)). ,, 1 - 2 c t
This term is b o u n d e d in view o f Corollary 2.5. N o w using the mean value theorem for the above integral, we conclude that 3 C > 0 V h > 0 3Zh ~ [1 - 2~, 1 - c~l : I~'h(Zn)l <- C.
T o show that the current is b o u n d e d , it is sufficient to prove that Since Vh(Xn) = 0, we have Hh(Xh) = h2lql~(Xh)[ 2 -- ~
- -
hql~(Zh) is
bounded.
= 4 --
The boundedness o f Vd(xn) (see Corollary 2.5) implies that both Vd(0) and h~'~(xh) are b o u n d e d which in turn yields the boundedness o f Hh. N o w the evaluation o f Hn at x = zh yields the boundedness o f h~u~(zh). • Second case: xh ---' 1. Since Vh is nonnegative on [0, Xh] and -h
2
Re qt~ + Vh Re qth = 0 ,
Re q/~0) = 0 , Jh Re ~ h ( 0 ) = ~-~ --> 0,
it is readily seen that Re q/h is increasing on [0, xh] ( L e m m a A. 1). As a consequence o f this, we deduce --Vh" --> ).2lRe q/,12 _> C j ~
on [0, Xh].
It is now obvious that the boundedness o f Vh in L = and the fact that xn does not tend to zero imply the boundedness o f j h . •
The quantum Child-Langmuir problem
637
2.3. The limit potential is nonpositive In this section we show that Vn has a potential barrier at x = 0 for h > 0 and that this potential barrier vanishes in the limit h --* 0+. We recall that Vn is bounded in W L*° and hence converges uniformly on [0, I]. LEMMA 2.7. There exists ho > 0 such that Vn'(0) > 0 for all h < ho. Proof. Assume that the assertion is not true. Then, we can find a subsequence such that Vn _< 0 on [0, 1]. Let V b e the C°-limit of this subsequence. Since Vn -< 0 the solution of the Schr6dinger equation (2.1) is purely oscillatory with the (local) frequency x / - V h / h . This motivates the introduction of the following function Fn(x) = 221h~,k(x) - i ~ / - Vn(x) ~,h(x)l 2.
(2.19)
We can rewrite Fn as follows Fh(x) = ½(~'(x)) 2 - 2jhx/--Vh(x) + 22Hh.
(2.20)
Let us denote by x* > 0 the greatest point where V vanishes, We have V m 0 on [0, x*] and V < 0 on (x*, 1]. The boundedness of ~ ' and of Hh in L*~ imply the uniform boundedness of Vh Vh" [cf. (2.12)]. This implies the boundedness o f Vh" in L®(x * + c~, 1) for every fixed c~. Then the convergence of Vh to V holds in Cl([x * + o~, 1]). • Reaching a contradiction. It is proved in the next paragraph that Fh converges uniformly to Fo - 0 on [x* + c~, 1]. Using this convergence and the C°([0, 1]) convergence of V~, we can find z > x* and h small enough such that 2jn - ~ n ( Z )
-< 22,
Fh(Z) <-- 22.
Using (2.20) where H h is evaluated at x = 0 [see (2.13)] we obtain the following inequality ½(Vh'(Z))2 -- ,(Vh'(O)) 2 < --2~. 2 which in view of Vh' _< 0 implies
V~(z) > v~(o) leading to an obvious contradiction with the concavity of Vh. • Showing that F n --, 0. We first notice that Fn(I) = 0. Then, in order to prove that Fn converges uniformly to zero on Ix* + (x, 1], it is sufficient to prove that Fn' is bounded in L*O(x* + (x, 1) and that its weak limit is equal to zero. Since F n ' = Vn'( Vn'+ J-----~V~Vh)
(2.21)
the boundedness of Fn' in L®(x * + ~, 1) is an obvious consequence of the boundedness of Vh V~'. To show that the limit is zero, one has to show that V " = _j/,fL---~. For this, we
N. B E N A B D A L L A H et al.
638
first deduce from (2.9) and from the boundedness of Vh" in L°°(x* + a , 1) (or equivalently that of q/h)
~t
h21~/'(u)lZdu +
X*+~
Vh(u)l~U~,(u)12 du
in L*~(x* + a, 1)
= O(h)
X*+ot
which gives after differentiation lim
+ v,l
in ff)'(x*, 1).
,l:l = o
h+O
Hence lira h2lqj~l 2 -
VV"
h~O
-2
in ~ ' ( x * + t~, 1).
A
We now pass to the limit in (2.12) and find in view of the above property and (2.13): 2VV"
- ½ V '2 = 2jx/~-V~ - ½(V'(I)) 2
on [x* + a, l].
(2.22)
Differentiation of this identity gives 2 VV"
-
V ' V" = 0
and we conclude: V" ~
V = constant.
From (2.22) we conclude that this constant is equal to - j .
•
Now having proved the existence o f a potential barrier for h > 0, we shall prove in the following lemma that this potential barrier vanishes in the limit h --* 0+. LEMMA 2.8. There exists a positive constant Ch tending to zero as h tends to zero such that V~ <_ Vntx) <- C , ,
x ~ [0, 11.
P r o o f . The proof is very similar to that of Lemma 2.4. Assume that max Vh _> 2Co with Co > 0 and let ah < Pn be the points where Vn takes the value Co (these points exist and are unique because Vh is strictly concave and the maximum of Vn is assumed in (ah,/~h)). The boundedness of Vh' in L= implies that the length of the interval [ah,/~h] does not tend to zero. Moreover Vh > Co on [O~h,Phi which yields in view of (2.17a)
lim
I~'h(u)12du = 0
h ~ O ~C~h
and implies that Vh'(C~h)-- Vh'(Bh) tends to zero. Since Vh'(c~D > O, Vh'(flh) < 0 and Vh is concave, we deduce from the above result that tends to zero which implies that iim(
sup
IVh(x)-Co[)=O
h 4 0 k,x ~ [~h,t3hl
which contradicts the hypothesis max Vh > 2C o.
•
The quantum Child-Langmuir problem
639
We end this section with the following iemma. LEMMA 2.9. If V vanishes on an interval, then the (limit) current j is equal to zero. Proof. Assume that V - 0 on [Xo, xx]. Since the maximum of Vh tends to zero, the concavity of Vh implies that Vh' converges uniformly to zero on [Xo + a, x~ - cd for every positive a. The mean value theorem yields the existence of zh ~ [Xo + or, x~ - ~t] such that "fx'-~ I~UhtU)12du = (X, - X o - 2C~)I~Uh(Zh)I2. ~, X o +
Since the above integral is nothing but 1
).7 (V,'(x0 + ~) - V,'(Xl - cOL we conclude that limh~o ~Uh(Z,~) = 0. This and Lemma 2.2(ii) imply j = lim Jh = lim jn(Zh) = O. h~O h~O
•
2.4. Characterization o f the limit problem We go now further in the analysis of the potential barrier and introduce the points xh > 0
is the solution of Vh(xh) = 0
(2.23)
Yh > 0
is the solution of Vh'(Yh) = 0
(2.24)
Vhmax = Vh(Yh)
(2.25)
Xo = lim Xh. h-0
(2.26)
In Fig. 2 we plot the points Xh, Yh and Vhmax. Vh
Vh
Vh iyh
xh
I
vI Fig. 2. The potential.
x
N. BEN ABDALLAH et al.
640
Let us begin with the following proposition. PRoPosrrioN 2.10. (i) The limit potential V < 0 on (Xo, 1] where x0 is defined by (2.26). (ii) For every fixed a > 0, the potential Vh converges to Vin C~([xo + o~, 1]. Moreover V tt ~
J
- - - -
x/.Z~
on (Xo, 11.
(iii) limh~o h21q/~(xh)l 2 = 0. (iv) Let V'(x~)) := limx.x~ V'(x). Then V'(x~) = l i m h . o Vh'(Xh).
Proof. •
(i).
Assume that there exists x~ > x o such that V(x~) = 0. Since V is concave, V - 0 on [xo, x d follows. This leads to limb ~o Vn'(xn) = 0 (because Vn is concave and Vh'(Xh) < 0). N o w evaluating Hh at x = xh and passing to the limit in (2.12) we conclude that h=
limHh>--0. h~0
On the other hand, since V = 0 on Ixo, x~], L e m m a 2.9 yields j = lira Jh = lim jh(Zh) = O. h~O
h~0
As a consequence o f this, the evaluation o f H h at x = 1 gives in the limit h = -½(V'(1)) 2 < 0 and results in a contradiction. •
(ii).
The p r o o f is exactly the same as the last part o f the p r o o f o f L e m m a 2.6 (showing that Fn ~ 0). We only have to replace x* by x o. • (iii). Let us consider the function Fn defined in (2.19). This function tends uniformly to zero on Ix0 + ~, 1]. Let us consider (2.20) where Hh is evaluated at xn. We obtain ½(Vh'(x)) 2 - ½(V~(Xh)) 2 = Fn(x) - 22hZlqt/,(Xh)12 + 2Yhx/--Vh(X),
X ~ [X0 + a, 11. (2.27)
Let us assume that lira ).2h2lv//,(xh)l 2 = 413 > O. h~0
F r o m the u n i f o r m convergence o f Fh and o f Vh we deduce that we can find x~ > x o close e n o u g h to Xo and h small enough such that
Fh(x,) <- 13,
2Yh
--X/--L-~h(X,)<-- 13,
22h21gt],(Xh)[ 2 > 313.
We deduce f r o m this and f r o m (2.27) that ½(Vh'(Xl)) 2 -- ½(Vh'(Xh)) 2 _< --13
which is in contradiction with the concavity o f Vn (note that Vn' < 0 on (xh, l]).
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641
• (iv).
Since V,' _< 0 on (Xh, 1], it is sufficient to prove that (V'(xg)) 2 = limn.o(Vh'(Xh)) 2. But this is a straightforward consequence o f (iii). It is sufficient to let successively tend h to zero and x to Xo+ in (2.27). • LEMMA 2.11. A s s u m e that limh_.o(Vh'(Xh))2= Co > 0. Then there exist four positive constants C~, C2, C3 and C 4 such that the following estimates hold (2.28)
C i x h <~ Vhmax --< C 2 Y h
C3Xh <--
(Vh'(U))2 du _< C4Yh.
(2.29)
.0
Proof. The evaluation o f Hh at x = xh yields in view o f Proposition 2.10(iii) that h = limHh<0. h--0
But since (~,)2 _ 2Vh ~,, = 222h2i~12 _ 222Hh, we deduce the existence o f B 2 > B I > 0 such that for h small enough Bi --< (V,') 2 - 2Vh Vh" < B 2 .
(2.30)
Let now x be in [0, Yh]" Since Vh' is nonnegative, multiplying (2.30) by V,'/Vh2 gives
-BI
()
1 '
-((Vh')2"Y < - B 2
<_ \ - i f / /
_
on (0, Yh).
'
This gives after integration between x and Yh
(
BI 1
(
vhma-----;]--< (Vh'(X)) 2 -----B 2 1 - -~ - ) ,
x e [O, yh].
(2.31)
W e divide these inequalities by 1 - Vh(x)/Vh rex, take the square root and integrate between x and Yh. We thus obtain x / - ~ l ( y h -- X) <-- 2Vh max ~ / 1
Vh(x)
vhmax -- ~ 2 ( Y h
- X),
X • [0, yh].
(2.32)
T a k i n g this inequality at x = 0 leads to x / ~ l y h <_%2Vhmax <_ x ~ 2 y , .
(2.33)
A n a l o g o u s calculations on the interval [ y , , Xh] give
X/~l(Xh -- Yh) <-- 2V*m~ --< X/~2(Xh -- Yh).
(2.34)
N. BEN ABDALLAH et al.
642
These two estimates give (2.28). T o prove (2.29), we deduce f r o m (2.32) and (2.28) that
,/
1
vhmu > B3 1 -
,
x E [0, Yh]"
As a consequence of this inequality and (2.31), we have
IIv~'ll~- -> (v~(x)) 2 and (2.29) is straightforward.
>-
B4(1 - x~ 2, \ Yn/
X e [0, Yhl.
(2.35)
•
COROLLARY 2.12. Let x o the point defined in f o r m u l a (2.26). Then x 0 = 0.
Proof. Assume x0 > 0. Then l/vanishes on [0, Xo] which implies in view o f L e m m a 2.9 that the current j is equal to zero. Hence h := l i m h , o H n = limh~oHh(1) < 0. But since Proposition 2.10(iii) implies lim (Vh'(Xh))z = - 2 2 2 h > 0, h~0
L e m m a 2.11 applies and yields xh -< C ~ max. Hence xh tends to zero (Vhmax tends to zero in view of L e m m a 2.8) and we have a contradiction. • To prove T h e o r e m 2.1, we need the following technical l e m m a whose p r o o f is deferred to the Appendix. LEMMA 2.13. l i m h . o Vg(xh) = O.
Proof o f Theorem 2.1. The topologies for carrying out the limit h --, 0 were already shown to be the ones mentioned in the t h e o r e m (except for Wh, which can be treated by the methods of [14]). (iii) is a direct consequence o f L e m m a 2.13, Corollary 2.12 and Proposition 2.10 (ii). (i) is a straightforward consequence o f (iii). T o prove (ii), we note that since n is a b o u n d e d positive measure on [0, 11 and n = j/x/-S-V on (0, 1], we have n = ~
J
+ no6fX),
where no is a positive constant. One can easily that n o equals limh~o(Vh'(0) -- V'(0)) = limh-.O and Proposition 2.10 (iii) is that h = limh~oHh limh~o Vh'(0) = x/8A. (iv) is shown by applying standard methods [141. •
show using the Poisson equation (2.4) Vh'(0). A consequence of L e m m a 2.13 = 0. This implies in view of (2.13) that f r o m the theory of Wigner functions
The quantum Child-Langmuir problem
643
3. e 3 - - C H I L D - L A N G M U I R A S Y M P T O T I C S
In this section we show that P r o b l e m (P~,0) " c o n v e r g e s " to (P0,0) as e tends to zero. We recall (P~,o) v Of~ 3x
1 d Ve Of~ _ 0 2 d x 3v l
f , ( 0 , v) = ~
\~j,
f , ( l , v) = 0,
v > 0
(3.36)
(3.37)
v < 0
(3.38)
- V/' = n c = 1 -*~f~ dv
(3.39)
'~ + o 0
V~(O) = O,
V~(I) = V~.
(3.40)
We set +co
,~2 =
U2~)(U) dv
(3.41)
v f e ( x , v) dr.
(3.42)
o
and j~ =
l
+o0
and we assume that * is decreasing and decays rapidly at ~ . The existence (and uniqueness of solutions is shown in [1]). We are interested in the limit e -~ 0. The main result o f this section is in the following theorem.
THEOREM 3.1. T h e functions j , , n,, V~, f , converge to j, n, V, f , resp. in ~, the weak star t o p o l o g y o f b o u n d e d measures on [0, 1] , C°([0, 1])tq Clot((0, 1 l]) and the weak star t o p o l o g y o f b o u n d e d measures on [0, 1] × [~. The limits are defined by (i) j = ~ ( - V t ) 3/2 = JcL(V1),
J + v~,~(x), (ii) n(x) - .,/-L--#(x) (iii) - V "
J
- ~=-~, V(0) = V'(0) -- 0, V(I) = V~,
(iv) f ( x , v) = n(x) ® rS(v - 4 - : - # ( x ) ) . In [8], Degond and Raviart proved that the asymptotics corresponding to the factor t - 2 in (3.37) instead to e -3 lead to the limits stated in the above t h e o r e m except that there is no Dirac m e a s u r e at x = 0 for the density. Here, the e-3 term leads to the concentration without altering the limit o f the potential and the current. Let us notice that analogous p h e n o m e n a occur for other scalings o f the V l a s o v - P o i s s o n system [10]. Hence we will not give the details o f the p r o o f o f T h e o r e m 3.1, but only give the reason o f the concentration at x = 0. The fact that n = j / x f Z - # in (0, 1) comes only f r o m u p p e r solutions estimates
64,4
N. BEN ABDALLAH et al.
[2, 7, I l] which allow also to easily prove the following assertions • f e converges weakly in b o u n d e d measures, • V e converges in C°[0, l] and Clot(0, t 1] to the potential V mentioned in the theorem. As a consequence, the density converges in the weak t o p o l o g y o f b o u n d e d measures to n -
~
J
+ no~(X),
where n o = limc,o(V~'(O) - V'((O+)) = d i m ~ , o V~'(O). It remains to prove that n o = ~ 2 . We now give a sketch o f this proof. We recall that .+oo
t
A2He :=
1
v 2 f e dv - ~ (V*'(x)) 2 -co
is an invariant. Let h be the limit o f h e. Then, since the support o f f is Iv = ~ / - V ] we have A2h(l) =
'~=
1
v 2 f d v - ~ (V'(I))2 l
= -,~Z-~l j - ~ (V'(I)) 2 = O. Hence h = 0. Let us now pass to the limit in Hr(0). We notice that the injected current at x = 0 is o f order l / e whereas the " t r a n s m i t t e d " current is the C h i l d - L a n g m u i r current (which is finite). That means that " m o s t o f the particles" are reflected. Then the kinetic energy at x = 0 is the double o f the injected kinetic energy. lim e~O
v2f~(O, v) dv = 2 lim -¢o
e-~O
v2f~(O, v) dv = 2A2 , 0
hence 0 = h = 222 - ~l clim(~'(O))2 ~o which gives n o = lim V~'(O) = v ~ . .
•
e~O
REFERENCES I. Greengard, C. and Raviart, P. A., A boundary value problem for the stationary Vlasov-Poisson equations: the plane diode. Comm. Pure Appl. Math., 1990, 43, 473-507. 2. Poupaud, F., Boundary value problems for the stationary Vlasov-Maxwell system. Forum Mathematicum, 1992, 4, 499-527. 3. Ben Abdallah, N., Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system. Math. Meth. in the Appl. Sci., 1994, 17, 451-476. 4. Guo, Y., Global weak solutions of the Vlasov-Maxwell system with boundary conditions. Com. Math. Phys., 1993, 159(2), 245-263. 6. Degond, P., The Child-Langmuir law in the kinetic theory of charged particles. Part l, Electron flows in vacuum. In Advances in Kinetic Theory, ed. B. Perthame. World Scientific, Singapore, 1994. 7. Degond, P., Jaffard, S., Poupaud, F. and Raviart, P. A., The Child-Langmuir asymptotics of the Vlasov-Poisson equation for cylindrically or spherically symmetric diodes, Parts 1 and 2. Math. Meth. in the Appl. Sci., 1996, 19, 287-312 (a), 313-340 (b).
The quantum Child-Langmuir problem
6,15
8. Degond, P. and Raviart, P. A., An asymptotic analysis of the one-dimensional Vlasoc-Poisson system: the Child-Langmuir law. A s y m p t o t i c Analysis, 1991, 4, 187-214. 9. Ben Abdallah, N. and Degond, P., The Child-Langmuir law for the Boltzmann equation of semiconductors. S I A M J. on Math. Anal., 1995, 26(2), 364-398. 10. Ben-Abdallah, N., Degond, P. and Schmeiser, C., On a mathematical model for hot carrier injection in semiconductors. Math. Meth. in the Appli. Sci., 1994, 17, 1193-1212. I 1. Ben Abdallah, N., The Child-Langmuir regime for electron transport in a plasma including a background o f positive ions. Math. Mod. Meth. Appl. Sci., 1994, 4(3), 409-428. 12. Ben Abdallah, N., Degond, P. and Yamnahakki, A., The Child-Langmuir law as a model for chargedparticle transport in semiconductors. Solid State Electron., 1996, 39(5), 737-744. 13. Ben Abdallah, N., Degond, P. and Markowich, P. A., On a one-dimensional Schr6dinger-Poisson scattering model. Z A M P , 1997, 48, 135-155. 14. Lions, P. L. and Paul, T., Sur les mesures de Wigner. Revista Mathematica lberoamericana, 1993, 9, 553-618. APPENDIX P R O O F O F L E M M A 2.13 The lemma will be proven by contradiction and the proof will be divided in several steps. The idea consists in analysing the boundary layer near x = 0. We will describe very precisely the shape and the order of magnitude o f the potential barrier. From this analysis, we will deduce the existence of an interval of the form [xh , x h + Ch2/31 on which ~1, is of order h -'/~. We deduce from this and from the Poisson equation (2.4) that Vh'(X, + Ch 2/3) - V~(xh) <_ - C I < O.
This inequality is in contradiction with Proposition 2.10. Let us begin with the following lemma which will be used widely in the proof. LEMMA A.I. Let a < c < b be three real numbers. Let U be a nonnegative locally bounded function on [a, b] such that U > 0 on (a, c), and let f be a solution of -f"
+ Uf=O.
Assume that f ( a ) >- 0
and
f ' ( a ) >_ 0
and that one of these numbers is not vanishing. Then f and f ' are increasing and f ' ( b ) >_f ' ( c ) > O.
The proof o f this lemma is straightforward. Let us now begin the analysis of the boundary layer at x = O. LEMMA A.2. Assume that lim h_0 V~(Xh) = - C o < 0. Then there exists C > 0 such that (i) !h~,,(0)l -< Cxh, (ii) x h >_ Ch 2/3. Proof. • (i). We prove this assertion by contradiction. Assume that
limsup hw,(0) = +ao. h-o I xh I Since
IIh~,,; IlL- is bounded,
(AI)
we have h[~,(x) - u/h(0)l ~ Cx which implies in view of (AI): lim sup sup ~'l,(x) h~o xe[O,xd ~ -
I I = 0.
I
(A2)
646
N. BEN A B D A L L A H et al.
Integration of the Poisson equation (2.4) between 0 and x h then gives Vg(xD - V~(O) - -,12xh[~,(0)12 which leads to lim supxh[¢,(0)! 2 = C > 0. (A3) h--O
Integrating the Schr6dinger equation (2.1) gives
1 ,t x" h(~[,(x,) - Wj,(O)) = ~
o Vh(u)~h(u) du.
When h tends to zero the left-hand side of this equation tends to - 2 i ((2.2) and Proposition 2.10(iii)) whereas in the right-hand side we can replace ~h(u) by ~,(0). Hence 2 - I~,(0)1 h
ti'
Vh(u) du _ ~C I~,,(0) lxL
(A4)
Combining (A3) and (A4). we get x , - c , h 2",
I~h(0)l
C2
h,/3
which leads to a contradiction with (AI).
• (ii). We deduce from (i) and the boundedness of
IIhw,~LIL-that
]hw,(x)l < C x , for x e [0. xh]. We conclude
I~,(u)l 2 du < lim inf C h .
0 < C~ = lira IV~(xD - Vh'(0)[ = ,t 2 h ~0
and (ii) follows.
0
h 40
•
LEMM^ A.3. Assume that l i m , _ 0 Vh'(x,) = - C < 0, then there exists C O > 0 and C~ > 0 such that (i) lira h40 h l / 3 ~ , ( 0 ) = - i C o . (ii) hl/31~h(x,)[ < C I. P r o o f . First of all we introduce a new scaling. We set
(AS)
V,(x) : * " v
X
(
(A6)
and finally xh
(A7)
~h -- h 2 / 3 .
(i) and (ii) can be rewritten in terms of Oh as follows: (i) lim h_o OhiO) = - i C o . (ii) ¢,(~,) is bounded. We can also rewrite (iii) o f Proposition 2.10: lim $~(~h) = 0.
(A8)
h~O
The advantage of the new variables is that the equations are simpler. They read
-~
+ U h ¢ , = O,
on [0, h -2/s]
(A9)
,,;(o) = 2/
(Al0)
Re ¢,(0) = h 1/3 J* 2`12
(Al I)
-vg u~(o) = o,
= ~%,12
v~(~,) = o
(AI2) (A13)
The q u a n t u m C h i l d - L a n g m u i r problem
647
Moreover
Ilug Itt-,o,h-~,) -< C,
(AI4)
and in the case limh_o Vh'(Xh) = - C < 0, there exists 0 < '~1 < ~h and C1 > 0 such that Uh(O -> C,~, Uh(¢') --> Cl(~h - 0 ,
~ ~ [0, {,l,
(Al5)
~ e [~h - ~l, ~h].
(AI6)
Let us now prove (i) by contradiction. One of the following posibilities occurs: (I) Oh(O) ~ 0. (11) Ioh(o)l -~ o0, (Ill) I~Ph(0). ---' C o > O. We will first prove that (I) and (I1) cannot happen. • (!) cannot happen. Assume that (I) holds. We can pass to the limit in the imaginary part of (A9) and obtain lm Oh - ' ¢o in C , ~ ( ~ +) where - O ~ + UOo = O
I
O{,(O) = 2
00(0) = O,
U(O >-- C~,
Hence applying L e m m a A.I yields
on [0, ~d.
O6 ~ 2 .
By the h - ' 0 limit, we can find Go > 0 such that lm Oh(Go) -> Go and l m O~(~o) >- l for every h small enough. This implies, again by applying L e m m a A. 1, that Im 0~(,~h) -> 1 and contradicts (AS). • (II) cannot hold. Since Re Oh(0) is bounded lira 0h(0)] --" 00 has to hold. We introduce 0h(g')
=
Im Oh(G) Im Oh(0)"
As above 0h converges to 0 o in Ct~(IR ÷) where -O~ + UOo = 0
I
0o(0 ) = 1,
06(0) = 0
U(O > c ~ .
Applying L e m m a A. ! as in the previous case yields the existence o f Go such that for h small enough, we have 0h(Go) ~- C o > 0 and 0g(~o) > Ct > 0. Therefore 0,~(~h) -> C~ which yields limh,o[Ok(~h)] = +o0 and contradicts (AS). Now let 0II) hold. Since the real part of (ah(0) tends to zero, the limit of Oh(0) is imaginary and the sign o f its imaginary part has to be negative otherwise we apply L e m m a A. 1 and get a contradiction with (A.8). This ends the proof of point (i). To prove point OiL we just have to prove that (II) does not hold when, we replace 0 by ~ = ~h. This can be done by studying the boundary layer in the vicinity o f ~h [using (AI6)]. The arguments are the same and the details are left to the reader. •
LEMMA A.4. A s s u m e that lim, -o Vh'(xD = - C with C > 0. Then, there exist Go, P, C~, C2 > 0 such that
(i) limh_o~, = GO (ii) limh-oiOh(~h)l = Ct (iii) V~'(Xh(1 + p)) - V~(xh) < - C 2 .
648
N. BEN A B D A L L A H et al.
Proof. •
(i).
In view of L e m m a A.2(ii), it is sufficient to prove that x h < Ch 2/~. For this, we integrate (2.9) between 0 and x h and obtain in view of L e m m a A.3: hz
Iq/~(u)[2du + ,0
V,(u)l~uh(u)12du <_ Ch zj3. 0
Since by L e m m a 2.2 .o V'(u)l~u'(u)12du = ~ .,o (V;(u)) 2 du • Cx, we conclude x~ <_ Ch 2/~. •
(ii).
Let limb., o Ch = Co > 0. Let u be positive and arbitrary for the moment and let ~o be the CLlimit of Im Sh on [0, (1 + P)~o]. Passing to the limit in (A9) and (A8) gives -O~ + U ~ o = 0
l
%(0) = lim ~h(0),
q~(0) = 2
h~0
0~(~o) = 0,
O0(C0)= lim CJ,(Ci,) h *0
We deduce from the above equations that ~o(Co) ;~ 0. Otherwise, the Cauchy-Lipschitz theorem applies to ~o and gives Oo m 0 which is not possible in view of the boundary conditions at ~ = 0. This proves (ii). •
(iii).
Since % is continuous and ~0(~0) is different from zero, there exists an interval [C0, (1 +/~)~ol on which ~o does not vanish. Hence for h small enough I¢h is bounded (from below) away from zero: 3 C > 0,
such that I ~ 2 _> C o n l~h,(l + /OChl.
Now we can integrate the Poisson equation (AI2) and get U,;(l + p)~,) - U,;(~,) ~ - C 2 < 0 which is the scaled version of (iii).
•
E n d o f the p r o o f o f L e m m a 2.13 Assume that L e m m a 2.13 does not hold, then L e m m a A.4 applies and yields Vh'(Xh(1 + bt)) - V;(xh) <_ - C 2 < O. Using the concavity of Vh and the fact that x~,(l + ,u) tends to zero as h tends to zero, we deduce that for any positive x, we have for h small enough Vh'(X) -- Vh'(Xh) <-- - c 2 . Letting successively h and x tend to zero, we obtain V'(0)
-
lim V~(xn) < - C 2 < O.
This is obviously in contradiction with Proposition 2.10(iv). Hence L e m m a 2.13 holds. •
Nonlinear Analysts, Theory, Methods&Applications, Vol. 31, No. 5/6. pp. 629-648, 1998 1998 Elsevier ScienceLtd Printed in Great Britain.All rightsreserved 0362-546X/98 $19.00+0.00
PII: S0362-546X(97)OO429-X
THE QUANTUM CHILD-LANGMUIR PROBLEM1" NAOUFEL BEN ABDALLAH,:[: PIERRE DEGOND$ and PETER A. MARKOWICH§ SMIP, Laboratoire CNRS (UMR 5640), Universit~ Paul Sabatier, 118, route de Narbonne, 31062 Toulouse Cedex, France; and §Fachbereich Mathematik, Technische Universit~t Berlin, StraBe des 17 Juni 136, 10623 Berlin, Germany (Received 9 January 1996; received for publication 18 February 1997) Key words and phrases: Quantum phenomena, semiclassical limit, Wigner transform, current carrying state, monokinetic flow, Child-Langmuir limit.
1. I N T R O D U C T I O N
Vacuum electron devices like vacuum diodes, amplifier tubes, etc., are usually described by means of kinetic models of charged-particle transport like the Vlasov-Poisson, Vlasov-Maxwell or Boltzmann-Maxwell systems. The accurate modelling of a real vacuum device requires the design of appropriate boundary conditions in particular to describe the injection of particles from the cathode. The mathematical analysis of the boundary value problem for the Vlasov-Poisson and Vlasov-Maxwell systems has recently been investigated in [1,2] for the stationary case and in [3, 4] for the time dependent case. The role of the space-charge in the limitation of the maximal current that can be extracted from the cathode of an electron tube is well-known since the work of Child, Langmuir and Compton [5]. A mathematical framework to the space-charge limited current emission problem in vacuum diodes has recently been provided in [6-8]. In these references, it is shown that the space-charge limitation regime (or Child-Langmuir regime) can be described by the Vlasov-Poisson system with a singularly perturbed boundary condition at the injection boundary. This condition merely expresses that a very large number of electrons start from the cathode with a very small initial velocity (compared with the one they are expected to reach in the applied electric field), thus creating a space-charge layer in the vicinity of the cathode. Then, it has been shown in a series of work [3, 9-11] that the same framework is able to describe semi-classical electron transport phenomena in semiconductor devices such as the injection of electrons over potential barriers at junctions. In these works, it is observed that the highly doped N ÷ region behaves like the metallic cathode of a vacuum device and that the electrons crossing the junction are facing similar conditions (mutatis mutandis) as those extracted from the cathode of a vacuum device. Numerical computations [12] strongly support these observations. It is tempting to investigate if a similar theory can be developed in the framework of quantum transport. The first step is the design of boundary conditions modelling the
t This work has been supported by the "Human Capital and Mobility" project #ERBCHRXCT 930413 entitled "Nonlinear spatio-temporal structures in semiconductors, fluids and oscillator ensembles" funded by the EC, by the Groupement de Recherches " S P A R C H " of the CNRS, France, by the bilateral cooperation project Procope #96123 funded by the DAAD-APAPE and by the project "Analysis und Numerik yon Kinetischen Quantentransportmodellen" funded by the DFG under the contract number MA 1662/I-I. 629
630
N. BEN ABDALLAH et al.
injection of particles at the boundary of a quantum device. This step has been realized in [13], with the following assumptions: the metallic contacts are supposed to be connected to metallic leads; electrons entering the device from the contact are identified with incoming scattering states of the Schr6dinger equation in a domain consisting of the union of the device itself and the surrounding leads. Then, the problem is restricted to the device itself by the prescription of boundary conditions which are absorbing for the considered scattering states. Assuming that electrons entering the device are distributed according to given statistics (for instance, that of a reservoir of particles which sends the electrons along the lead from infinity), a density matrix is constructed which allows to define the electric potential self-consistently taking into account the Coulomb interaction. The resulting probelm is a coupled Schr6dinger-Poisson system. In [13], the existence of solutions and the semi-classical approximation of this system are investigated. Once appropriate injection boundary conditions are derived, the second step towards a "quantum Child-Langmuir" asymptotic is to tailor these conditions to model the injection of a "large" number of electrons with a " s m a l l " incoming velocity. To this aim, we express the statistics of scattering states coming into the device with wave-vector k according to (l/e~')eP(k/e), where e is a small parameter related to the velocity scale of the incoming particles. Not all powers a give rise to a nontrivial asymptotic limit and it is proven in [13] that the correct one is ct = 3. The coupled Schr6dinger-Poisson system with a e dependent injection statistic is system (P~,h) below: ~2 ,, 2 --tl ~gk.h, t + V h , t ~ k . h , ~ = k ¢k.h,~
h¢~,h,e(O) (P~.h)
+ ik~k,h,e(O ) =
2ik
hiot~.h,c(l) = i x / - ~ - VltOk,h.~(l) - vh"~ = -~ 3 o v~.~(o) = o,
\e/ Vh.~(l) = V,
where the constant h > 0 represents a scaled Planck constant. The boundary potential V~ is assumed to be negative. In [13], the limit as e --. 0 of (Pc,h) has been proven to lead to the following problem for a monokinetic (single state) Schr6dinger-Poisson model refered to as (P0,h): --haCl'h + V h ~ h
= 0
h~(O) = 2i
(~o,h)
h~v~(1) = i~-SVl I//h(l ) -Vn" = Azl(uh(X)l2,
k2~(k)dk ,0
vh(o) = o,
Vh(1) = v~.
On the contrary, when we let e be fixed and h ~ 0, we are naturally led to the following Child-Langmuir type problem for the classical Vlasov-Poisson problem denoted by (P~,o):
The quantum Child-Langmuir problem
P~
@
631
Po.h
0
~ 0
+i 0
0
®
o,o
Fig. I. Convergence diagram.
vaf, l dV~ 0f~ = Ox 2 dx Ov l(v)
A ( o , v) = ~ *
(P~,o)
f , ( l , v) = O, -U'
= n+ =
0
,
v > o
v< 0
I ++f ~ d v
~(0) = O,
~(1) -- Vl.
Note that the usual scaling factor of the incoming distribution function is 1/• 2 instead of l / t 3. Thus, compared to the standard theory [6-8] an accumulation of particles at x = 0 is expected to occur, leading to the appearance of a concentration in the density. The limit o f (Pe,0) as e --* 0 leads to the following semi-classical monokinetic problem denoted by (Po,0) which is also the limit of (Po,n) when h --+ 0:
_
V
tt
__
J ~-v
V ( O ) = V'(O) = O,
(Po,o)
4
V(I) = V~ 3/2
J = Jct(VI) = ~(-- 1) n = ~
+ ~f8A~i(x),
=
k 2 0 ( k ) dk.
In this paper we shall deal with the convergence diagram shown in Fig. 1, linking the above detailed four models. In particular we shall prove the convergences II and III. We recall that convergences I and IV have been analysed in [13] (however, convergence IV has been proved in [l 3] for e large enough only. The proof for arbitrary e is more involved and will be presented in a future paper.)
N. BEN A B D A L L A H et al.
632 2. S E M I - C L A S S I C A L
LIMIT OF THE REDUCED
QUANTUM
PROBLEM
Let ¢'h and Vh be a solution of (PO.h) for h > 0, V~ < 0. We recall the equations satisfied by ~'h and Vh: -h2c'g + Vh~'h = 0
(2.1)
h~,~ (0) = 2i
(2.2)
h~(l)
= ivr-SV~ ~'h(1)
(2.3) (2.4)
--Vh" = ,~211],/h]2
Vh(0) = 0,
V~(1) = V~.
(2.5)
Let us define the particle current density Jh by
jh(X) = ).2h Im(~,~ (x)ff/h(x))
(2.6)
This quantity is independent of x. Let us also define the particle density (2.7)
nh(X ) = A2[~Vh(X)I 2
and the Wigner function [14] associated with ~'h by ei~(O~'h) x - ~ y
Wh(X, V) = X 2
(Off~h) X + ~ y
dy
(2.8)
,-oo
where 0 is a C OOcompactly supported function identically equal to 1 in a neighbourhood of [0, 1]. It is well known that nh and Jh are the zeroth and first order moments of Wh (with respect to v) in [0, 1]. For the analysis of the limit behaviour of Wh we introduce the test space (i = lip = q~(x,v)l(~YvqO(x, rl)~ L~(tR,;Co(~x))}, (see [141), where 5:~ denotes the Fourier transform with respect to v. The existence of solutions of (P0,h) is proven in [13]. Here we are interested in convergence of such solutions when h tends to zero (corresponding to the semi-classical limit). We now state the main theorem of this section. THEOREM 2.1. Let j, n, V, be defined by the following formulae. (i) j = ~(-Vl) 3/2 =: jcL(V,),
J
(ii) n(x) = xfE-#(x) (iii)
-V"
-
J C:V'
+ ~a6(x), V(0) = V'(0) = 0,
V ( 1 ) = V~,
(iv) W(x, v) = n(x) ® J(v - ~/-V(x)) in [0, 1] x R v. Then, as h tends to zero, Vh converges to V in C°([0, 1]) fq C~o¢((0, 1]), Wh converges in (~'w. to W, n, converges to n in the weak star topology of bounded measures on [0, 1] and Jh converges to j. Note that JcL(V~) is the Child-Langmuir limit current [8].
The q u a n t u m C h i l d - L a n g m u i r p r o b l e m
633
2.1. P r e l i m i n a r y e s t i m a t e s
In this paper we do not mention explicitly the extraction of subsequences in the limit arguments based on compactness [since the limit problem (Po,o) is uniquely solvable]. Multiplying (2.1) by ~h and integrating from 0 to x gives the following identities
v,(u)l~,n(u)l z du = h 2 Re[~(u)0h(u)l~
h2[~/,(u)l z du + ,0
(2.9)
0
2 Re(~,,(0))
= ~
I~Uh(l)l2.
(2.10)
In view of (2.3) and (2.6), equation (2.10) can be written as 2 Re(~'h(0)) = ~ .
(2.1 l)
Using the identity d ( h 2 l ~'~ (x) l2 - Vn(x)[ ~'h(x) l2) = - Vn'(x)I~h(X)[ 2 which follows immediately from the Schr6dinger equation (2.1), and the Poisson equation satisfied by the potential ~ we deduce that the function Hh(X)
:= h2llp'~(x)[ 2 -
Vh(X)Iq/h(X)[ 2 -- ~
(2.12)
is constant, i.e. Hh(O) = 4 -- ~
Hh(x )
- -
(2.13)
=
Hh(l ) = A. j h _ _
-
In order to obtain the necessary estimates to pass to the limit h --, 0, we start with the following lemma.
LEMMA 2.2. The following assertions are equivalent: (i) Jh is bounded, (ii) IIh~'~ IIL~ + IIh~'hllL~ is bounded, (iii) IIv ll.,, is bounded. Proof. •
(i) ~ (ii).
Assume that Jh is bounded. Then in view of (2.10), (2.11), ~'h(l) is bounded. It is thus sufficient to prove the boundedness of [[h~,~ [[L®. Let us set gh(X) = h2llm ~ ( x ) l 2 -
Vhllm~'h(X)l 2.
Taking into account the relation g~
=
-Vh'IIm(~,Dl 2
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N. BEN ABDALLAH
et al.
and the concavity of Vh (in view of the Poisson equation), we conclude that gh assumes its maximum at x = 0 or x = 1 and moreover, we have
gh(O)
=
4,
gh(l)
2
<
-~jh~--VI,
Let x0 be a point at which ]Im(~g~)l achieves its maximum. If Xo is equal to 1 or to O, then it is easy to show using (2.2) and (2.3) that hllm(~'~(Xo))] -< C. If Xo is in the interior of the interval, then Im ~,~' vanishes there. Using the imaginary part of (2.1), we deduce that
gh(Xo) = h21Im(~'~(Xo))l z <- C. Therefore h Im(q/~) is bounded in L~. For the real part, we proceed analogously and obtain therefore the boundedness in L°~ of h~,~. • (ii) ~ (iii). Since jh -ff-L~ = -A2V~I~Uh(I)I 2 = h2l~u~;(l)l 2 the boundedness of [Ihw~ IlL®+ Ilh~'hllL®implies [in view of (2.13)1 that (Vh'(1))2 - (Vh'(O))z is bounded. Hence, keeping in mind that Vh is concave, it is sufficient for the proof of (iii) to prove that V~'(I) is bounded. For this aim, we first deduce from (2.9) that jl Vhl~'h]2 is bounded, which can be written after an integration by parts ,b I
t (Vh'(u))2du -
Vl Vh'(1)
is bounded.
(2.14)
,0
Also, using (2.12) and (2.13), we have Vhtx)l~,,(x)l 2 + ~
- ~
- -
= h2l~,A(x)l z
-
~jh~--Vi.
The right-hand side of this inequality is bounded in L® by hypothesis. Therefore, integrating the left-hand side between 0 and 1 and using the Poisson equation (2.4) gives 3! l - -
( V ~ ( u ) ) 2 d u + Vt
2 o
1 Vh'(1) + ~ (Vh'(l))2 is bounded.
Combining (2.14) and (2.15) implies that V~Vh'(l) - (Vh'(l))2 is bounded and yields the boundedness of Vh'(l). • (iii) ~ (i). Obvious in view of (2.13). •
(2.15)
The quantum Child-Langmuir problem
635
2.2. Boundedness o f the current LF.MMA 2.3. There exists C > 0 such that IJh2~llL- + IIh2~,hllL- ___ C.
Proof. At first we shall prove the estimate h2[~,~l 2 <_ C(l + Iwh(l)l%.
(2.16)
This can be done by considering again the function
gh(X)
=
hZllm ~'~(x)l 2 - Vnllm ~h(x)l 2
and its analogue with the real parts replacing the imaginary parts. These two functions take their maximum at 0 or at 1. Since this maximum is bounded by the right-hand side of (2.16), we can proceed as in the proof of Lemma 2.2 [second part of (i) = (ii)]. Now by taking the square root of this inequality and integrating between 0 and 1, we obtain C I~'h(0)- ~'h(1)l ~ ~ ( 1 4-
I h(l)l).
We deduce from this inequality and from (2.10) that I~uh(1)l4 _< Cl~,'h(O)l z < ~ ( 1 + I~Uh(I)I2) which leads to hl~,h(1)l _< C and ends the proof in view of (2.16). LEMMA 2.4. The L® norm o f Vh is bounded.
Proof. Since Vh is concave, the assertion IlvhllL- is bounded is equivalent to max Vh is bounded. We prove the lemma by contradiction. Assume that max Vh tends to +oo. Then using the concavity of Vh and its boundedness from below, we conclude that there exists et > 0 such that for h small enough Vh_> 1
onD:=
[ct, l - a].
Using the maximum principle for the elliptic equation (2.1), one can prove that there exists C > 0 such that
vx ~ t~,
I~,h(x)l < suplv/hl exp(-Cd(x, O~)/h). am
(2.17a)
Now we estimate the current density Jh at x = 1/2, taking into account the above inequality and the estimates of Lemma 2.3. We obtain C
Jh < ~ exp(-C/h). The current is bounded and Lemma 2.2 gives a contradiction.
•
636
N. BEN ABDALLAH et al.
COROtLARV 2.5. Let et > 0 be fixed. Then there exists C,~ > 0 such that
IlVh'llL'(~,l-o> --< ca.
(2.17)
For all sequence an e (0, 1 - c0 such that Vd(a h) <_ 0 the following estimate holds
IIv;IIL~<~,,_o) -< c,,.
(2.18)
P r o o f . The corollary is an obvious consequence o f the above lemma and the concavity o f Vh. •
N o w we are able to prove the main result o f this subsection. P R o a o s m o N 2.6. The current Jh is b o u n d e d . P r o o f . Let us denote by Xh the largest point where Vh vanishes. T w o cases are possible. Either xh tends to one or to a value less then one.
• First case: xh --' 1 - 2c~, c~ e (0, 2Xl. We first notice that in view o f the Poisson equation we have I-¢r I~Uh(U) Iz d u = A2(Vn'(1 _ 2o¢) - ~(1 - or)). ,, 1 - 2 c t
This term is b o u n d e d in view o f Corollary 2.5. N o w using the mean value theorem for the above integral, we conclude that 3 C > 0 V h > 0 3Zh ~ [1 - 2~, 1 - c~l : I~'h(Zn)l <- C.
T o show that the current is b o u n d e d , it is sufficient to prove that Since Vh(Xn) = 0, we have Hh(Xh) = h2lql~(Xh)[ 2 -- ~
- -
hql~(Zh) is
bounded.
= 4 --
The boundedness o f Vd(xn) (see Corollary 2.5) implies that both Vd(0) and h~'~(xh) are b o u n d e d which in turn yields the boundedness o f Hh. N o w the evaluation o f Hn at x = zh yields the boundedness o f h~u~(zh). • Second case: xh ---' 1. Since Vh is nonnegative on [0, Xh] and -h
2
Re qt~ + Vh Re qth = 0 ,
Re q/~0) = 0 , Jh Re ~ h ( 0 ) = ~-~ --> 0,
it is readily seen that Re q/h is increasing on [0, xh] ( L e m m a A. 1). As a consequence o f this, we deduce --Vh" --> ).2lRe q/,12 _> C j ~
on [0, Xh].
It is now obvious that the boundedness o f Vh in L = and the fact that xn does not tend to zero imply the boundedness o f j h . •
The quantum Child-Langmuir problem
637
2.3. The limit potential is nonpositive In this section we show that Vn has a potential barrier at x = 0 for h > 0 and that this potential barrier vanishes in the limit h --* 0+. We recall that Vn is bounded in W L*° and hence converges uniformly on [0, I]. LEMMA 2.7. There exists ho > 0 such that Vn'(0) > 0 for all h < ho. Proof. Assume that the assertion is not true. Then, we can find a subsequence such that Vn _< 0 on [0, 1]. Let V b e the C°-limit of this subsequence. Since Vn -< 0 the solution of the Schr6dinger equation (2.1) is purely oscillatory with the (local) frequency x / - V h / h . This motivates the introduction of the following function Fn(x) = 221h~,k(x) - i ~ / - Vn(x) ~,h(x)l 2.
(2.19)
We can rewrite Fn as follows Fh(x) = ½(~'(x)) 2 - 2jhx/--Vh(x) + 22Hh.
(2.20)
Let us denote by x* > 0 the greatest point where V vanishes, We have V m 0 on [0, x*] and V < 0 on (x*, 1]. The boundedness of ~ ' and of Hh in L*~ imply the uniform boundedness of Vh Vh" [cf. (2.12)]. This implies the boundedness o f Vh" in L®(x * + c~, 1) for every fixed c~. Then the convergence of Vh to V holds in Cl([x * + o~, 1]). • Reaching a contradiction. It is proved in the next paragraph that Fh converges uniformly to Fo - 0 on [x* + c~, 1]. Using this convergence and the C°([0, 1]) convergence of V~, we can find z > x* and h small enough such that 2jn - ~ n ( Z )
-< 22,
Fh(Z) <-- 22.
Using (2.20) where H h is evaluated at x = 0 [see (2.13)] we obtain the following inequality ½(Vh'(Z))2 -- ,(Vh'(O)) 2 < --2~. 2 which in view of Vh' _< 0 implies
V~(z) > v~(o) leading to an obvious contradiction with the concavity of Vh. • Showing that F n --, 0. We first notice that Fn(I) = 0. Then, in order to prove that Fn converges uniformly to zero on Ix* + (x, 1], it is sufficient to prove that Fn' is bounded in L*O(x* + (x, 1) and that its weak limit is equal to zero. Since F n ' = Vn'( Vn'+ J-----~V~Vh)
(2.21)
the boundedness of Fn' in L®(x * + ~, 1) is an obvious consequence of the boundedness of Vh V~'. To show that the limit is zero, one has to show that V " = _j/,fL---~. For this, we
N. B E N A B D A L L A H et al.
638
first deduce from (2.9) and from the boundedness of Vh" in L°°(x* + a , 1) (or equivalently that of q/h)
~t
h21~/'(u)lZdu +
X*+~
Vh(u)l~U~,(u)12 du
in L*~(x* + a, 1)
= O(h)
X*+ot
which gives after differentiation lim
+ v,l
in ff)'(x*, 1).
,l:l = o
h+O
Hence lira h2lqj~l 2 -
VV"
h~O
-2
in ~ ' ( x * + t~, 1).
A
We now pass to the limit in (2.12) and find in view of the above property and (2.13): 2VV"
- ½ V '2 = 2jx/~-V~ - ½(V'(I)) 2
on [x* + a, l].
(2.22)
Differentiation of this identity gives 2 VV"
-
V ' V" = 0
and we conclude: V" ~
V = constant.
From (2.22) we conclude that this constant is equal to - j .
•
Now having proved the existence o f a potential barrier for h > 0, we shall prove in the following lemma that this potential barrier vanishes in the limit h --* 0+. LEMMA 2.8. There exists a positive constant Ch tending to zero as h tends to zero such that V~ <_ Vntx) <- C , ,
x ~ [0, 11.
P r o o f . The proof is very similar to that of Lemma 2.4. Assume that max Vh _> 2Co with Co > 0 and let ah < Pn be the points where Vn takes the value Co (these points exist and are unique because Vh is strictly concave and the maximum of Vn is assumed in (ah,/~h)). The boundedness of Vh' in L= implies that the length of the interval [ah,/~h] does not tend to zero. Moreover Vh > Co on [O~h,Phi which yields in view of (2.17a)
lim
I~'h(u)12du = 0
h ~ O ~C~h
and implies that Vh'(C~h)-- Vh'(Bh) tends to zero. Since Vh'(c~D > O, Vh'(flh) < 0 and Vh is concave, we deduce from the above result that tends to zero which implies that iim(
sup
IVh(x)-Co[)=O
h 4 0 k,x ~ [~h,t3hl
which contradicts the hypothesis max Vh > 2C o.
•
The quantum Child-Langmuir problem
639
We end this section with the following iemma. LEMMA 2.9. If V vanishes on an interval, then the (limit) current j is equal to zero. Proof. Assume that V - 0 on [Xo, xx]. Since the maximum of Vh tends to zero, the concavity of Vh implies that Vh' converges uniformly to zero on [Xo + a, x~ - cd for every positive a. The mean value theorem yields the existence of zh ~ [Xo + or, x~ - ~t] such that "fx'-~ I~UhtU)12du = (X, - X o - 2C~)I~Uh(Zh)I2. ~, X o +
Since the above integral is nothing but 1
).7 (V,'(x0 + ~) - V,'(Xl - cOL we conclude that limh~o ~Uh(Z,~) = 0. This and Lemma 2.2(ii) imply j = lim Jh = lim jn(Zh) = O. h~O h~O
•
2.4. Characterization o f the limit problem We go now further in the analysis of the potential barrier and introduce the points xh > 0
is the solution of Vh(xh) = 0
(2.23)
Yh > 0
is the solution of Vh'(Yh) = 0
(2.24)
Vhmax = Vh(Yh)
(2.25)
Xo = lim Xh. h-0
(2.26)
In Fig. 2 we plot the points Xh, Yh and Vhmax. Vh
Vh
Vh iyh
xh
I
vI Fig. 2. The potential.
x
N. BEN ABDALLAH et al.
640
Let us begin with the following proposition. PRoPosrrioN 2.10. (i) The limit potential V < 0 on (Xo, 1] where x0 is defined by (2.26). (ii) For every fixed a > 0, the potential Vh converges to Vin C~([xo + o~, 1]. Moreover V tt ~
J
- - - -
x/.Z~
on (Xo, 11.
(iii) limh~o h21q/~(xh)l 2 = 0. (iv) Let V'(x~)) := limx.x~ V'(x). Then V'(x~) = l i m h . o Vh'(Xh).
Proof. •
(i).
Assume that there exists x~ > x o such that V(x~) = 0. Since V is concave, V - 0 on [xo, x d follows. This leads to limb ~o Vn'(xn) = 0 (because Vn is concave and Vh'(Xh) < 0). N o w evaluating Hh at x = xh and passing to the limit in (2.12) we conclude that h=
limHh>--0. h~0
On the other hand, since V = 0 on Ixo, x~], L e m m a 2.9 yields j = lira Jh = lim jh(Zh) = O. h~O
h~0
As a consequence o f this, the evaluation o f H h at x = 1 gives in the limit h = -½(V'(1)) 2 < 0 and results in a contradiction. •
(ii).
The p r o o f is exactly the same as the last part o f the p r o o f o f L e m m a 2.6 (showing that Fn ~ 0). We only have to replace x* by x o. • (iii). Let us consider the function Fn defined in (2.19). This function tends uniformly to zero on Ix0 + ~, 1]. Let us consider (2.20) where Hh is evaluated at xn. We obtain ½(Vh'(x)) 2 - ½(V~(Xh)) 2 = Fn(x) - 22hZlqt/,(Xh)12 + 2Yhx/--Vh(X),
X ~ [X0 + a, 11. (2.27)
Let us assume that lira ).2h2lv//,(xh)l 2 = 413 > O. h~0
F r o m the u n i f o r m convergence o f Fh and o f Vh we deduce that we can find x~ > x o close e n o u g h to Xo and h small enough such that
Fh(x,) <- 13,
2Yh
--X/--L-~h(X,)<-- 13,
22h21gt],(Xh)[ 2 > 313.
We deduce f r o m this and f r o m (2.27) that ½(Vh'(Xl)) 2 -- ½(Vh'(Xh)) 2 _< --13
which is in contradiction with the concavity o f Vn (note that Vn' < 0 on (xh, l]).
The quantum Child-Langmuir problem
641
• (iv).
Since V,' _< 0 on (Xh, 1], it is sufficient to prove that (V'(xg)) 2 = limn.o(Vh'(Xh)) 2. But this is a straightforward consequence o f (iii). It is sufficient to let successively tend h to zero and x to Xo+ in (2.27). • LEMMA 2.11. A s s u m e that limh_.o(Vh'(Xh))2= Co > 0. Then there exist four positive constants C~, C2, C3 and C 4 such that the following estimates hold (2.28)
C i x h <~ Vhmax --< C 2 Y h
C3Xh <--
(Vh'(U))2 du _< C4Yh.
(2.29)
.0
Proof. The evaluation o f Hh at x = xh yields in view o f Proposition 2.10(iii) that h = limHh<0. h--0
But since (~,)2 _ 2Vh ~,, = 222h2i~12 _ 222Hh, we deduce the existence o f B 2 > B I > 0 such that for h small enough Bi --< (V,') 2 - 2Vh Vh" < B 2 .
(2.30)
Let now x be in [0, Yh]" Since Vh' is nonnegative, multiplying (2.30) by V,'/Vh2 gives
-BI
()
1 '
-((Vh')2"Y < - B 2
<_ \ - i f / /
_
on (0, Yh).
'
This gives after integration between x and Yh
(
BI 1
(
vhma-----;]--< (Vh'(X)) 2 -----B 2 1 - -~ - ) ,
x e [O, yh].
(2.31)
W e divide these inequalities by 1 - Vh(x)/Vh rex, take the square root and integrate between x and Yh. We thus obtain x / - ~ l ( y h -- X) <-- 2Vh max ~ / 1
Vh(x)
vhmax -- ~ 2 ( Y h
- X),
X • [0, yh].
(2.32)
T a k i n g this inequality at x = 0 leads to x / ~ l y h <_%2Vhmax <_ x ~ 2 y , .
(2.33)
A n a l o g o u s calculations on the interval [ y , , Xh] give
X/~l(Xh -- Yh) <-- 2V*m~ --< X/~2(Xh -- Yh).
(2.34)
N. BEN ABDALLAH et al.
642
These two estimates give (2.28). T o prove (2.29), we deduce f r o m (2.32) and (2.28) that
,/
1
vhmu > B3 1 -
,
x E [0, Yh]"
As a consequence of this inequality and (2.31), we have
IIv~'ll~- -> (v~(x)) 2 and (2.29) is straightforward.
>-
B4(1 - x~ 2, \ Yn/
X e [0, Yhl.
(2.35)
•
COROLLARY 2.12. Let x o the point defined in f o r m u l a (2.26). Then x 0 = 0.
Proof. Assume x0 > 0. Then l/vanishes on [0, Xo] which implies in view o f L e m m a 2.9 that the current j is equal to zero. Hence h := l i m h , o H n = limh~oHh(1) < 0. But since Proposition 2.10(iii) implies lim (Vh'(Xh))z = - 2 2 2 h > 0, h~0
L e m m a 2.11 applies and yields xh -< C ~ max. Hence xh tends to zero (Vhmax tends to zero in view of L e m m a 2.8) and we have a contradiction. • To prove T h e o r e m 2.1, we need the following technical l e m m a whose p r o o f is deferred to the Appendix. LEMMA 2.13. l i m h . o Vg(xh) = O.
Proof o f Theorem 2.1. The topologies for carrying out the limit h --, 0 were already shown to be the ones mentioned in the t h e o r e m (except for Wh, which can be treated by the methods of [14]). (iii) is a direct consequence o f L e m m a 2.13, Corollary 2.12 and Proposition 2.10 (ii). (i) is a straightforward consequence o f (iii). T o prove (ii), we note that since n is a b o u n d e d positive measure on [0, 11 and n = j/x/-S-V on (0, 1], we have n = ~
J
+ no6fX),
where no is a positive constant. One can easily that n o equals limh~o(Vh'(0) -- V'(0)) = limh-.O and Proposition 2.10 (iii) is that h = limh~oHh limh~o Vh'(0) = x/8A. (iv) is shown by applying standard methods [141. •
show using the Poisson equation (2.4) Vh'(0). A consequence of L e m m a 2.13 = 0. This implies in view of (2.13) that f r o m the theory of Wigner functions
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3. e 3 - - C H I L D - L A N G M U I R A S Y M P T O T I C S
In this section we show that P r o b l e m (P~,0) " c o n v e r g e s " to (P0,0) as e tends to zero. We recall (P~,o) v Of~ 3x
1 d Ve Of~ _ 0 2 d x 3v l
f , ( 0 , v) = ~
\~j,
f , ( l , v) = 0,
v > 0
(3.36)
(3.37)
v < 0
(3.38)
- V/' = n c = 1 -*~f~ dv
(3.39)
'~ + o 0
V~(O) = O,
V~(I) = V~.
(3.40)
We set +co
,~2 =
U2~)(U) dv
(3.41)
v f e ( x , v) dr.
(3.42)
o
and j~ =
l
+o0
and we assume that * is decreasing and decays rapidly at ~ . The existence (and uniqueness of solutions is shown in [1]). We are interested in the limit e -~ 0. The main result o f this section is in the following theorem.
THEOREM 3.1. T h e functions j , , n,, V~, f , converge to j, n, V, f , resp. in ~, the weak star t o p o l o g y o f b o u n d e d measures on [0, 1] , C°([0, 1])tq Clot((0, 1 l]) and the weak star t o p o l o g y o f b o u n d e d measures on [0, 1] × [~. The limits are defined by (i) j = ~ ( - V t ) 3/2 = JcL(V1),
J + v~,~(x), (ii) n(x) - .,/-L--#(x) (iii) - V "
J
- ~=-~, V(0) = V'(0) -- 0, V(I) = V~,
(iv) f ( x , v) = n(x) ® rS(v - 4 - : - # ( x ) ) . In [8], Degond and Raviart proved that the asymptotics corresponding to the factor t - 2 in (3.37) instead to e -3 lead to the limits stated in the above t h e o r e m except that there is no Dirac m e a s u r e at x = 0 for the density. Here, the e-3 term leads to the concentration without altering the limit o f the potential and the current. Let us notice that analogous p h e n o m e n a occur for other scalings o f the V l a s o v - P o i s s o n system [10]. Hence we will not give the details o f the p r o o f o f T h e o r e m 3.1, but only give the reason o f the concentration at x = 0. The fact that n = j / x f Z - # in (0, 1) comes only f r o m u p p e r solutions estimates
64,4
N. BEN ABDALLAH et al.
[2, 7, I l] which allow also to easily prove the following assertions • f e converges weakly in b o u n d e d measures, • V e converges in C°[0, l] and Clot(0, t 1] to the potential V mentioned in the theorem. As a consequence, the density converges in the weak t o p o l o g y o f b o u n d e d measures to n -
~
J
+ no~(X),
where n o = limc,o(V~'(O) - V'((O+)) = d i m ~ , o V~'(O). It remains to prove that n o = ~ 2 . We now give a sketch o f this proof. We recall that .+oo
t
A2He :=
1
v 2 f e dv - ~ (V*'(x)) 2 -co
is an invariant. Let h be the limit o f h e. Then, since the support o f f is Iv = ~ / - V ] we have A2h(l) =
'~=
1
v 2 f d v - ~ (V'(I))2 l
= -,~Z-~l j - ~ (V'(I)) 2 = O. Hence h = 0. Let us now pass to the limit in Hr(0). We notice that the injected current at x = 0 is o f order l / e whereas the " t r a n s m i t t e d " current is the C h i l d - L a n g m u i r current (which is finite). That means that " m o s t o f the particles" are reflected. Then the kinetic energy at x = 0 is the double o f the injected kinetic energy. lim e~O
v2f~(O, v) dv = 2 lim -¢o
e-~O
v2f~(O, v) dv = 2A2 , 0
hence 0 = h = 222 - ~l clim(~'(O))2 ~o which gives n o = lim V~'(O) = v ~ . .
•
e~O
REFERENCES I. Greengard, C. and Raviart, P. A., A boundary value problem for the stationary Vlasov-Poisson equations: the plane diode. Comm. Pure Appl. Math., 1990, 43, 473-507. 2. Poupaud, F., Boundary value problems for the stationary Vlasov-Maxwell system. Forum Mathematicum, 1992, 4, 499-527. 3. Ben Abdallah, N., Weak solutions of the initial-boundary value problem for the Vlasov-Poisson system. Math. Meth. in the Appl. Sci., 1994, 17, 451-476. 4. Guo, Y., Global weak solutions of the Vlasov-Maxwell system with boundary conditions. Com. Math. Phys., 1993, 159(2), 245-263. 6. Degond, P., The Child-Langmuir law in the kinetic theory of charged particles. Part l, Electron flows in vacuum. In Advances in Kinetic Theory, ed. B. Perthame. World Scientific, Singapore, 1994. 7. Degond, P., Jaffard, S., Poupaud, F. and Raviart, P. A., The Child-Langmuir asymptotics of the Vlasov-Poisson equation for cylindrically or spherically symmetric diodes, Parts 1 and 2. Math. Meth. in the Appl. Sci., 1996, 19, 287-312 (a), 313-340 (b).
The quantum Child-Langmuir problem
6,15
8. Degond, P. and Raviart, P. A., An asymptotic analysis of the one-dimensional Vlasoc-Poisson system: the Child-Langmuir law. A s y m p t o t i c Analysis, 1991, 4, 187-214. 9. Ben Abdallah, N. and Degond, P., The Child-Langmuir law for the Boltzmann equation of semiconductors. S I A M J. on Math. Anal., 1995, 26(2), 364-398. 10. Ben-Abdallah, N., Degond, P. and Schmeiser, C., On a mathematical model for hot carrier injection in semiconductors. Math. Meth. in the Appli. Sci., 1994, 17, 1193-1212. I 1. Ben Abdallah, N., The Child-Langmuir regime for electron transport in a plasma including a background o f positive ions. Math. Mod. Meth. Appl. Sci., 1994, 4(3), 409-428. 12. Ben Abdallah, N., Degond, P. and Yamnahakki, A., The Child-Langmuir law as a model for chargedparticle transport in semiconductors. Solid State Electron., 1996, 39(5), 737-744. 13. Ben Abdallah, N., Degond, P. and Markowich, P. A., On a one-dimensional Schr6dinger-Poisson scattering model. Z A M P , 1997, 48, 135-155. 14. Lions, P. L. and Paul, T., Sur les mesures de Wigner. Revista Mathematica lberoamericana, 1993, 9, 553-618. APPENDIX P R O O F O F L E M M A 2.13 The lemma will be proven by contradiction and the proof will be divided in several steps. The idea consists in analysing the boundary layer near x = 0. We will describe very precisely the shape and the order of magnitude o f the potential barrier. From this analysis, we will deduce the existence of an interval of the form [xh , x h + Ch2/31 on which ~1, is of order h -'/~. We deduce from this and from the Poisson equation (2.4) that Vh'(X, + Ch 2/3) - V~(xh) <_ - C I < O.
This inequality is in contradiction with Proposition 2.10. Let us begin with the following lemma which will be used widely in the proof. LEMMA A.I. Let a < c < b be three real numbers. Let U be a nonnegative locally bounded function on [a, b] such that U > 0 on (a, c), and let f be a solution of -f"
+ Uf=O.
Assume that f ( a ) >- 0
and
f ' ( a ) >_ 0
and that one of these numbers is not vanishing. Then f and f ' are increasing and f ' ( b ) >_f ' ( c ) > O.
The proof o f this lemma is straightforward. Let us now begin the analysis of the boundary layer at x = O. LEMMA A.2. Assume that lim h_0 V~(Xh) = - C o < 0. Then there exists C > 0 such that (i) !h~,,(0)l -< Cxh, (ii) x h >_ Ch 2/3. Proof. • (i). We prove this assertion by contradiction. Assume that
limsup hw,(0) = +ao. h-o I xh I Since
IIh~,,; IlL- is bounded,
(AI)
we have h[~,(x) - u/h(0)l ~ Cx which implies in view of (AI): lim sup sup ~'l,(x) h~o xe[O,xd ~ -
I I = 0.
I
(A2)
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N. BEN A B D A L L A H et al.
Integration of the Poisson equation (2.4) between 0 and x h then gives Vg(xD - V~(O) - -,12xh[~,(0)12 which leads to lim supxh[¢,(0)! 2 = C > 0. (A3) h--O
Integrating the Schr6dinger equation (2.1) gives
1 ,t x" h(~[,(x,) - Wj,(O)) = ~
o Vh(u)~h(u) du.
When h tends to zero the left-hand side of this equation tends to - 2 i ((2.2) and Proposition 2.10(iii)) whereas in the right-hand side we can replace ~h(u) by ~,(0). Hence 2 - I~,(0)1 h
ti'
Vh(u) du _ ~C I~,,(0) lxL
(A4)
Combining (A3) and (A4). we get x , - c , h 2",
I~h(0)l
C2
h,/3
which leads to a contradiction with (AI).
• (ii). We deduce from (i) and the boundedness of
IIhw,~LIL-that
]hw,(x)l < C x , for x e [0. xh]. We conclude
I~,(u)l 2 du < lim inf C h .
0 < C~ = lira IV~(xD - Vh'(0)[ = ,t 2 h ~0
and (ii) follows.
0
h 40
•
LEMM^ A.3. Assume that l i m , _ 0 Vh'(x,) = - C < 0, then there exists C O > 0 and C~ > 0 such that (i) lira h40 h l / 3 ~ , ( 0 ) = - i C o . (ii) hl/31~h(x,)[ < C I. P r o o f . First of all we introduce a new scaling. We set
(AS)
V,(x) : * " v
X
(
(A6)
and finally xh
(A7)
~h -- h 2 / 3 .
(i) and (ii) can be rewritten in terms of Oh as follows: (i) lim h_o OhiO) = - i C o . (ii) ¢,(~,) is bounded. We can also rewrite (iii) o f Proposition 2.10: lim $~(~h) = 0.
(A8)
h~O
The advantage of the new variables is that the equations are simpler. They read
-~
+ U h ¢ , = O,
on [0, h -2/s]
(A9)
,,;(o) = 2/
(Al0)
Re ¢,(0) = h 1/3 J* 2`12
(Al I)
-vg u~(o) = o,
= ~%,12
v~(~,) = o
(AI2) (A13)
The q u a n t u m C h i l d - L a n g m u i r problem
647
Moreover
Ilug Itt-,o,h-~,) -< C,
(AI4)
and in the case limh_o Vh'(Xh) = - C < 0, there exists 0 < '~1 < ~h and C1 > 0 such that Uh(O -> C,~, Uh(¢') --> Cl(~h - 0 ,
~ ~ [0, {,l,
(Al5)
~ e [~h - ~l, ~h].
(AI6)
Let us now prove (i) by contradiction. One of the following posibilities occurs: (I) Oh(O) ~ 0. (11) Ioh(o)l -~ o0, (Ill) I~Ph(0). ---' C o > O. We will first prove that (I) and (I1) cannot happen. • (!) cannot happen. Assume that (I) holds. We can pass to the limit in the imaginary part of (A9) and obtain lm Oh - ' ¢o in C , ~ ( ~ +) where - O ~ + UOo = O
I
O{,(O) = 2
00(0) = O,
U(O >-- C~,
Hence applying L e m m a A.I yields
on [0, ~d.
O6 ~ 2 .
By the h - ' 0 limit, we can find Go > 0 such that lm Oh(Go) -> Go and l m O~(~o) >- l for every h small enough. This implies, again by applying L e m m a A. 1, that Im 0~(,~h) -> 1 and contradicts (AS). • (II) cannot hold. Since Re Oh(0) is bounded lira 0h(0)] --" 00 has to hold. We introduce 0h(g')
=
Im Oh(G) Im Oh(0)"
As above 0h converges to 0 o in Ct~(IR ÷) where -O~ + UOo = 0
I
0o(0 ) = 1,
06(0) = 0
U(O > c ~ .
Applying L e m m a A. ! as in the previous case yields the existence o f Go such that for h small enough, we have 0h(Go) ~- C o > 0 and 0g(~o) > Ct > 0. Therefore 0,~(~h) -> C~ which yields limh,o[Ok(~h)] = +o0 and contradicts (AS). Now let 0II) hold. Since the real part of (ah(0) tends to zero, the limit of Oh(0) is imaginary and the sign o f its imaginary part has to be negative otherwise we apply L e m m a A. 1 and get a contradiction with (A.8). This ends the proof of point (i). To prove point OiL we just have to prove that (II) does not hold when, we replace 0 by ~ = ~h. This can be done by studying the boundary layer in the vicinity o f ~h [using (AI6)]. The arguments are the same and the details are left to the reader. •
LEMMA A.4. A s s u m e that lim, -o Vh'(xD = - C with C > 0. Then, there exist Go, P, C~, C2 > 0 such that
(i) limh_o~, = GO (ii) limh-oiOh(~h)l = Ct (iii) V~'(Xh(1 + p)) - V~(xh) < - C 2 .
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Proof. •
(i).
In view of L e m m a A.2(ii), it is sufficient to prove that x h < Ch 2/~. For this, we integrate (2.9) between 0 and x h and obtain in view of L e m m a A.3: hz
Iq/~(u)[2du + ,0
V,(u)l~uh(u)12du <_ Ch zj3. 0
Since by L e m m a 2.2 .o V'(u)l~u'(u)12du = ~ .,o (V;(u)) 2 du • Cx, we conclude x~ <_ Ch 2/~. •
(ii).
Let limb., o Ch = Co > 0. Let u be positive and arbitrary for the moment and let ~o be the CLlimit of Im Sh on [0, (1 + P)~o]. Passing to the limit in (A9) and (A8) gives -O~ + U ~ o = 0
l
%(0) = lim ~h(0),
q~(0) = 2
h~0
0~(~o) = 0,
O0(C0)= lim CJ,(Ci,) h *0
We deduce from the above equations that ~o(Co) ;~ 0. Otherwise, the Cauchy-Lipschitz theorem applies to ~o and gives Oo m 0 which is not possible in view of the boundary conditions at ~ = 0. This proves (ii). •
(iii).
Since % is continuous and ~0(~0) is different from zero, there exists an interval [C0, (1 +/~)~ol on which ~o does not vanish. Hence for h small enough I¢h is bounded (from below) away from zero: 3 C > 0,
such that I ~ 2 _> C o n l~h,(l + /OChl.
Now we can integrate the Poisson equation (AI2) and get U,;(l + p)~,) - U,;(~,) ~ - C 2 < 0 which is the scaled version of (iii).
•
E n d o f the p r o o f o f L e m m a 2.13 Assume that L e m m a 2.13 does not hold, then L e m m a A.4 applies and yields Vh'(Xh(1 + bt)) - V;(xh) <_ - C 2 < O. Using the concavity of Vh and the fact that x~,(l + ,u) tends to zero as h tends to zero, we deduce that for any positive x, we have for h small enough Vh'(X) -- Vh'(Xh) <-- - c 2 . Letting successively h and x tend to zero, we obtain V'(0)
-
lim V~(xn) < - C 2 < O.
This is obviously in contradiction with Proposition 2.10(iv). Hence L e m m a 2.13 holds. •