Nonineor Anoi.vsrs. Theory, Methods & A.~/~licofions, Vol. 26, No. 4, PP. 845-856, 1996 Copyright 0 1995 Elsevier Science Ltd

Printed m Great Britaln. All rights reserved 0362-546X/% $15.00+ .OO

0362-546X(94)00326-2

ON A STEADY-STATE QUANTUM HYDRODYNAMIC MODEL FOR SEMICONDUCTORS

BO ZHANGt and JOSEPH W. JEROME$§ t Department of Mathematics, Northwestern University, Evanston, IL 60208, U.S.A. and Department of Mathematics,

Stanford University, Stanford, CA 94305, U.S.A.; and $ Department of Mathematics, Northwestern University, Evanston, IL 60208, U.S.A.

(Received 10 June 1994; received in revised form 10 ilugust 1994; received for publication 21 October 1994)

Key words and phrases: Quantum hydrodynamic model, integro-differential equation, quantum perturbation.

1. INTRODUCTION

The quantum hydrodynamic (QHD) model is a moment model, derived from the Wigner equation. It may be viewed as a quantum corrected version of the classical hydrodynamic equations, with the stress tensor and the energy density corrected by O(ti*) perturbations. Some applications, such as the resonant tunnel diode, also involve quantum well potentials. Ancona, Iafrate and Tiersten [l, 21 derived the expression for the stress tensor, and Grubin and Kreskovsky formulated a one-dimensional version of the model [3]. A derivation of the general model is carried out by Gardner in [4]. It is of considerable physical and practical importance. An example of this is furnished by the regions of negative differential resistance detected in the voltage current curve associated with the model; closely tied to this is an hysteresis effect, which has been confirmed by Chen ef al. in [5]. Earlier, Kluksdahl et al. [6] had discovered this effect in the Wigner equation. The QHD model is computationally more tractable than the Wigner formulation, however.

No complete existence theory is available for the classical or quantum hydrodynamic model for semiconductors, either in steady-state or in long time evolution. The model must be self- consistent in the sense that the electric field, which forms a forcing term in the momentum equation, is determined by the coupled Poisson equation. The classical isentropic model (or, better, the relaxed perturbation of this model) has been studied self-consistently by Gamba [7] in steady-state and by Zhang [S] in evolution. Their methods are completely different. Gamba employs artificial viscosity, while Zhang proves the convergence of Godunov’s method. Although these reduced classical models are somewhat idealized in that heat conduction is not taken into consideration (see [9, lo] for the simulation of the full classical model), they are still capable of predicting velocity overshoot. In this paper, we consider the quantum equivalent of such a reduced model, in that we analyse self-consistently the perturbation of the classical isentropic model by the relaxation and quantum stress tensor perturbations in the steady-state case. We do not consider quantum well applications discussed in [5]. The model which we study is of physical importance; in fact, a simplification of our model been characterized as a pure state, single carrier transport model in [3].

$ Supported by the National Science Foundation under grant DMS-9123208.

845

846 B. ZHANG and J. W. JEROME

Our approach is completely novel to this application area, in that we reduce the system to an integro-differential equation, with a set of boundary conditions, including a nonstandard second order boundary condition, which is equivalent to specifying the quantum potential at the (current) inflow boundary. We are able to obtain a priori estimates and an existence result via the Leray-Schauder fixed point theorem. As noted by Gamba [ll], the techniques we use to obtain our pointwise a priori estimates is analogous to division by the dependent variable, and integration along the streamlines of the Bernoulli function. There is some hope, therefore, of eventually extending these results to two spatial dimensions.

The quantum perturbation introduces a third order density perturbation in the momentum equation. Our estimates clearly show that an h + 0 strong limit is not possible. This is remin- iscent, for systems, of the oscillatory dispersion phenomenon noticed by Lax and Levermore in [12]. Their fundamental model was the evolution equation, for which a weak limit was demonstrated and studied. It is not clear whether a weak limit can be demonstrated for this steady-state model; as noted earlier, the limiting equations do have a weak solution [7].

2. EORMULATION AND SUMMARY OF RESULT

We shall present the equations for the simplified QHD model as developed in [3,4]

(wu), + &vu2 + P(P) + Q(P)), = -P& - y, (2)

4, = e(N, - NA - 18, (3)

where p is the electron density, u the velocity, and C#J the electrostatic potential. The pressure function, p = p(p), has the property that p2p’(p) is strictly monotonically increasing from [0, co) onto [0, a). A commonly-used hypothesis is [13]

P(P) = kP?> y> l,k>O.

Quantum mechanics is represented by the quantum potential [l]

Q(P) = - & Alog(/N, .

where h is the normalized (by 2n) Planck’s constant and m the effective electron mass. The classical collision term is modeled by the momentum relaxation time approximation, r = r(p, pv) in (2). Assume that r satisfies

r(p, pu) L 5” > 0. (6)

The device domain is the x-interval, I = (0, 1). The given functions No and NA are called the density of donors and the density of acceptors, respectively. They satisfy

ND, Nt, 6 L”UA N,, ~ NA f 0.

The constant e > 0 in (3) is the electronic charge unit.

(7)

Quantum hydrodynamic model 847

In this paper we investigate the steady-state case pt = (pv), = 0. Then, after the introduction of the current density j = pv, the system (l)-(3) reduces to

j(x) = const, (8)

F + P(P) + Q(P) = -p& - y,

4Xx = eW, - NA - PI. (10)

Assume that j is a specified positive constant. Since (9) is a third-order ordinary differential equation, and (10) is Poisson’s equation, three boundary conditions for (9) and two boundary conditions for (10) are prescribed as follows

P(O) = PO 7 P(l) = PI T PO P,,(O) - t Pm = P2 3 (11)

4(O) = 40, 4(l) = 4,9 (12)

where po, p1 and p2 are positive constants; 4. and $1 are the applied bias potentials. The main result of this paper is the following theorem.

THEOREM 1. Assume that

Then, for each j > 0, there exists a classical solution (p, 4) of (9)-(12), satisfying the properties that p E C;(I) and l/N 5 \/p I N for a positive constant N.

In order to prove the above theorem, we first use a Green’s function to solve Poisson’s equation (lo), and then we reduce the system (9), (10) to an integro-differential equation. We shall see that the existence of a smooth solution of the original system is equivalent to that of a smooth solution of the integro-differential equation.

The solutions of (10) with boundary data (12) is given uniquely by

I

’ I qb=e (3, OO’o - N,, - PI d< + XC+, - (bo) + 90, (13)

, 0

where G(x, 0 is the Green’s function for this problem, and is defined by

W, 0 = xtr - I), x < r, ax - I), x > <. (14)

We now transform the equation (9) to a second-order ordinary differential equation by integration. Dividing (9) by p, we have

(15)

848 B. ZHANC and J. W. JEROME

Since

then, (15) becomes

Integrating (16) from 0 to x and using the boundary data (1 l), we have

+h2p2 W2 ky y-1 12m pi 2~: y - 1 po - 40.

(16)

(17)

Let w = vz. By substituting (13) for C#J in (17), the system of equations @-(lo) reduces to an integro-differential equation with Dirichlet conditions

h2 mj2 - w,, = --y + b ” dy 6m 2w

---wzy-’ + mjw 2 Y-1 I .ow 5

where

G(x, OWn -- N.4 - w2) d< + ~(4, - 40) + b >

3

w(O) = W() 3 w(l) = WI,

(18)

(1%

If w is a smooth solution of (18), (19), then

hbMa = P2 *

That is, the third boundary condition of (11) holds for w. Hence, the existence of a smooth solution of (8)-(12) is equivalent to that of a smooth solution of (18), (19), provided p does not vanish on I.

Quantum hydrodynamic model 849

3. A PRIORI ESTIMATES AND PROOF OF THEOREM

In this section we study the problem (18), (19), since it is equivalent to (8)-(12), for the existence of smooth solutions. First, we show, in the following lemma, that any solution w of (18), (19) is nonnegative.

LEMMA 1. Assume that the boundary data and given function satisfy the following

(20)

Then, any solution w of (18), (19) is nonnegative.

Proof. Let

q(x) = max{ - w(x), 01.

Then, q E Hi(l). Also, qX = -w, on the set A(0) = (x E I: -w(x) 2 0), whereas qX = 0, elsewhere.

Multiplying (18) by v(x) and integrating over (0, l), we have

- ej: ,,(~;Gw2d5)dx.

From the first condition in (20), we have

(21)

LX 4, - 4% - ; IIND - ~AIIL”o >

2 0. Note that

-G(x, 0 > 0, (x, () E I x I.

Thus, ‘1

I -G(x, ()w’dc L 0.

.O

Moreover, by the second condition in (20), b 2 0. Therefore, (21) implies that ?j = 0; i.e. wzo. n

850 B.ZHANG and J. W.JEROME

LEMMA 2. Let f(z), f, I z < 03, be nonnegative and nonincreasing such that

where C, a, and B are positive constants with p > I. Then

where

fU0 + a = 0,

For the proof, see [14, lemma B.l, p. 631. An a priori upper bound is obtained in the following lemma.

LEMMA 3. Assume that (6), (7) and (20) hold. Then, there exists a positive constant D, depending only on the boundary data, such that

w(x) 5 D a.e. in [0, 11. (22)

Proof. For I 2 1, = max[w,, w,], define

q(x) = max]w(x) - I, 0).

Then, q(x) 1 0 and FZ E H;(Z). Note that qX = w, on the set A(Z) = (x E I; w(x) 2 Z), whereas qX = 0, elsewhere.

Multiplying both sides of (18) by q(x) and integrating over (0, l), we have

ky ”

<2(y .o I

sI - w2’, ‘qdx+ c, qdx,

/ < 0

where

c, = f IINn - NAIlI” + 4, - $0 + b,

Quantum hydrodynamic model 851

Then, by the Hiilder and Poincark inequalities, ‘1

I 6mC, ”

. 0 dd-=F rlk

\ <o

6mC, ”

= -r ! rib

~ A(I) ‘1 l/2

5 6mC, 71 meas A [/)I 1’2

(.\ > q2 ti

0

1 ‘I 5-

\

1 8m2C,2 2 db+, [meas A(I)].

.O Thus, for C, = 36m2Ci,

From this inequality and Sobolev’s inequality, we have, for a fixed constant c and c, = cc,,

for every positive integer n 2 1. If f0 I I, < f2, then A(f,) c A(f,) so that

> 2/n

(f2 - f,)2[measA(f2)]2’” I (w - f,)“dx A(b)

(S >

2/n

5 (w - I,)“& . AU,)

Consequently,

(I, - fJ2[meas A(f2)]2’” 5 2 [meas A(f,)].

meas A(f,) 5 ctf2 :)fi2y[meas A(f,)]“‘2.

That is,

We take n > 2 and apply lemma 2 to derive that

measA(D) = 0, where

D= yc y/e-2)

h2 + I,.

This implies that w I D a.e. in [0, 11. n

852 B. ZHANG and J. W. JEROME

We have already demonstrated that solutions w are necessarily nonnegative. A positive a priori lower bound is proven next.

LEMMA 4. Assume that the conditions of lemma 3 hold. Then, there exists a constant 6 > 0 such that

w(x) I 6 > 0, x E P, 11, (23)

where 6 is independent of w.

Proof. Assume that there were not a positive constant 6 such that w(x) 2 6 > 0 for x E [0, 11. Then, there exist a sequence of solutions f w,) of (18), (19) and a sequence (x,) c (0, 1) such that w,(x,,) + 0 as n -+ 00.

We claim that the sequence (w,(x,)j goes to zero exponentially. By the condition (20) in lemma 1, solutions w, of (18) satisfy the following inequality

where

J,(w,)w, 5 (w,),,, (24)

l,(w,) = 3m2j2

m)4

Consider the following linear equation

W xx = A,(W)w,

w(O) = wg 7 w(l) = WI,

where ii, is a constant satisfying 0 < W 5 D. The solution of the above equation is

(1 - em 2(1 -x)v’X,(W) )wl

(1 _ e-26m)W, w(x) =

e xJhy(3 ~ e-(2-x)d’x,(w) + ,(I -x)JX,(W) _ ,-(1+x)-’

which satisfies the inequality

(25)

WO WI w(x) 5

e 1 h,(w) _ e -(2-X)JX,(W) + e’l -x)4(W) _ e-(l+x)m’

Let W = w,(x,) in (25). Then the solutions represented by (25) decay to zero exponentially as n 4 m. By (24) and the maximum principle, (w,(x,)) decays also to zero exponentially.

Using the Green’s function defined by (14), we have an implicit expression for the solution w, of (18), (19)

Quantum hydrodynamic model 853

By lemma 3 and the fact that G(x, 0 < 0 for (x, r) E I x I, we have

3m2j2 w,(%J) 5 2 h

(26)

Since G(x,, <) is a piecewise linear function defined by (14), and (l/w,(x,)l goes to 00 exponentially, the improper integral of (26) diverges to --oo. Thus, by (26), there exists a positive integer Nsuch that w,(x,) < 0 when n 2 N. This is impossible because of lemma 1. W

The final estimate is an L, bound for the fourth derivative of w.

LEMMA 5. Assume that the conditions of lemma 3 hold and the relaxation time r satisfies

aT II-II aw L=

5 M.

Then, there exists a constant C = C(6, D) such that

where C is independent of w

Proof. Taking the square of both sides of (18), and integrating over (0, l), we have

fi4 ‘I

36m2 ,o I

‘I

w’,dx =

I

g2(w) h,

.O

where g(w) is the right-hand side of (18); i.e.

-2 ky

g(w) = yJ + ~ w 21-I + w

I’ mj

Y-l I i-dy

,OWT

(I

‘I +w e G(x, <)(ND - N4 - w2) d[ + ~(4, - cpO) + b .

. 0 >

It follows from lemmas 3 and 4 that

where C, = C,(6, D) is independent of w. Taking the second derivative of both sides of (18) and multiplying by w,,,, , we have

(27)

dx=I,+12+I,+14,

854 B. ZHANG and J. W. JEROME

where

W2Y - 1)

Y-l w2(-‘) + x(4, - $0) + b w,*w,,,,dx

‘I + “’ mj

I (! ‘I

,dy+e I G(N, - NA - w2) dt wxx wmx d-~ . 0 <ow T ,O >

I, = ” 6mj2

I( 7 + 2ky(2y - l)wZYm7 w;w,,dX,

.O > ‘1

I, = 2 I

GO’, - NA - w*) dt- wxwxxx, k . 0 >

Z4 = e I ” (N, - N, - w2)ww,,,, dx.

.O

It follows from (6), (22), (23) and (28) that

Recall the interpolation inequality

‘I “I w,’ 5 v

! w:;dx + c,

! w2 dx.

.O .O

As a consequence of this inequality and (28), we have

Thus,

I ’ I

:m dx 5 C(6, D) W

..O 73

where C is independent of w. n

Some comments are now in order. From lemma 5, we conclude w E H4(I) lies in a set with a priori bound. Thus, by the Sobolev embedding theorem, w E C;(I) also lies in a set with a priori bound. Moreover, by lemmas 3 and 4, there exists a constant N, depending only on the boundary data, such that

; 5 w(x) I N, x E [O, 11. (29)

The following lemma is a special case of the Leray-Schauder fixed point theorem. For the proof, see [15, theorem 10.3, p. 2221 (mapping continuity is assumed a priori).

Quantum hydrodynamic model 855

LEMMA 6. Let T be a compact mapping of a Banach space B into itself. Suppose there exists a constant M such that

I/~B 5 M

for all (u, a) E B x [0, l] satisfying u = T(u, 0). Then T(u, 1) has a fixed point.

We are now prepared to give the final arguments.

LEMMA 7. Assume that the conditions of theorem 1 hold. Then, there exists a solution of (181, (19).

Proof. To apply lemma 6, we construct an operator T,: C’,‘[O, 11 -+ C’*‘[O, 11 for a fixed constant cy > 0 by solving the linear equation

6m2 W xx = ~R,W,

(30)

w(O) = wo > w(l) = WI,

where

s,,(u) = f

g(u)7 u > cl!,

g(ff), u 5 CY,

for u E Co’ ‘[O, l] and g(u) is defined by (27). The linear equation (30) is uniformly elliptic. For every function u E CoP’[O, 11, there exists

a unique solution w = T,(u) satisfying

That is, T, maps bounded sets of Co, ‘[O, I] into bounded sets of C”‘[O, 11. By the Ascoli- Arzela theorem, T, is compact in C’*‘[O, 11; continuity is routine.

If w satisfies

for 0 5 o I 1, i.e. w is a solution of w = 07;,(w), then it follows from the previous lemmas that05 wlNandI(wll co.~ 5 N, . By lemma 6, we conclude the existence of a fixed point of T,; i.e. there exists a w such that

6m2 W xx = I&’

w(O) = wg 1 w(1) = w,.

856

By (29), w satisfies

B. ZHANG and J. W. JEROME

1 2 5 w(x) 5 N, x E [O, 11.

In particular, we take (Y < l/N in (30) such that g,(w) = g(w) for l/N I w I N. Therefore, such a w solves (18), (19). W

As a consequence of lemma 7, theorem 1 has been proved.

I.

2.

3.

4.

5.

6.

7.

8.

9.

10.

11. 12.

13. 14.

15.

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ANCONA M. G. & IAFRATE G. J., Quantum correction to the equation of state of an electron gas in a semiconductor, P&s. Rev. B39, 9536-9540 (1989). ANCONA M. G. & TIERSTEN H. F., Macroscopic physics of the silicon inversion layer, Phys. Rev. B35, 7959-7965 (1987). GRUBIN H. L. & KRESKOVSKY J. P., Quantum moment balance equations and resonant tunnelling structures, Solid-Stale Elec. 32, 1071-1075 (1989). GARDNER C. L., The quantum hydrodynamic model for semiconductor devices, SIAM J. appl. Math. 54, 409-427 (1994). CHEN Z., COCKBURN B., GARDNER C. L. & JEROME J. W., Quantum hydrodynamic simulation of hysteresis in the resonant tunneling diode, J. Comp. fhys. 117, 274-280 (1995). KLUKSDAHL N. C., KRIMAN A. M., FERRY D.K. & RINGHOFER C., Self-consistent study of the resonant tunneling diode, Whys. Rev. B39, 7720-7735 (1989). GAMBA I. M., Stationary transonic solutions of a one-dimensional hydrodynamic model for semiconductors, Commune partial dsff. Eqns 17, 553-577 (1992). ZHANG B., Convergence of the Godunov scheme for a simplified one-dimensional hydrodynamic model for semiconductor devices, Communs math. fhys. 157, l-22 (1993). GARDNER C. L., JEROME, J. W. & ROSE D. J., Numerical methods for the hydrodynamic device model: subsonic flow, IEEE Tram Comp. Des. Int. Circ. Sys. CAD-8, 501-507 (1989). FATEMI E., JEROME, J. & OSHER S., Solution of the hydrodynamic device model using high-order nonoscillatory shock capturing algorithms, IEEE Tram Comp. Des. Int. Circ. Sys. CAD-lo, 232-244 (1991). GAMBA 1. M., Personal communication. LAX, P. D. & LEVERMORE C. D., The small dispersion limit for the Korteweg-de Vries equation I, Communs pure appl. Math. 36, 253-290. (1983). COURANT R. & FRIEDRICHS K. O., Supersonic Flow and Shock-Waves. J. Wiley and Sons, New York (1967). KINDERLEHRER D. & STAMPACCHIA G., in Introduction to Variational Inequalities and Their Applications. Academic Press, New York (1980). GILBARG D. & TRUDINGER N. S., Ellipfic Par/is/ D(fJerential Equations of Second Order. Springer, Berlin (1977).