About stability of the equilibrium state for a hydrodynamical model of charge transport in semiconductors Текст научной статьи по специальности «Математика»

Вычислительные технологии

Том 4, № 3, 1999

ABOUT STABILITY OF THE EQUILIBRIUM STATE FOR A HYDRODYNAMICAL MODEL OF CHARGE TRANSPORT IN SEMICONDUCTORS *

A. M. Blokhin Institute of Mathematics SB RAS, Novosibirsk, Russia e-mail: blokhin@math.nsc.ru

A. S. BüSHMANOVA Novosibirsk State University, Novosibirsk, Russia

V. Romano

Politecnico di Bari, sede di Taranto, Taranto, Italia

Доказана устойчивость и, при некоторых дополнительных предположениях на плотность легирования, асимптотическая устойчивость (по Ляпунову) состояния равновесия гидродинамической модели переноса заряда в полупроводниках в линейном приближении.

1. Preliminaries

Hydrodynamical models are widely used in mathematical modelling of physical phenomena in modern semiconductor devices. Derivation of such models is based on the study of the transport equation for the charge carriers density in an electric field. A conservation system with infinite number of equations (i. e., a system of conservation laws) is obtained from the transport equation with the help of a special technique of moments. Then, in a view of some physically justified assumptions, the system is exposed to simplification. As the result, the system of infinite number of moments equations reduces to one or another system of the hydrodynamical type.

In this paper we take, as a basis, a hydrodynamical model of charge transport in semiconductors suggested in the recently published paper [1]. In [2], this model is written as a quasilinear system of dimensionless conservation laws. Using notation from [2], below we give a nondivergent variant of this model

RT + uRs + Rus = 0,

•& 1 U

Ut + UUs + — Rs + tfs + T^Ss = Q--,

R R Tp

, + +2 (, + S^ u. + 4e. = 2 iw (1 ) +3(1 -

3 1 R) s 3R s 3 \rn 22t

*This work was supported by Russian Foundation for Basic Research, Grant No. 96-01-01560. © A.M. Blokhin, A. S. Bushmanova, V. Romano, 1999.

^ 4R— + 7S 8 4Ru2 S ST + uSs +----us + — 0s =

3 s 15 s 3tp ra 5 S— / S \ 16 0

0r + u0s + "(R— + S)—s - S—Rs + — - - Ss + 160us =--+

2 R \ R / 5 Tq

+Su I — +1 -1 ^ + " Ru— ( 1 -1 ^ + Ru! ( A -1 - A ^ - 5 Ru i - — (11)

2 V TP Tq ) 2 V 3tw Tq Tp J 2 Tw

e2^ss = R - p.

(1.2)

Here R, u, E, 0, ^ are the density, velocity, temperature, stress, heat flux and electric potential correspondingly; e2 = 1/0, 0 is a positive constant (see [2]); Q = tp = Tp(E), tw =

u2 3

t,

,(E), Ta = tct (E), Tq = Tq(E) are the relaxation times; E = — + ^the doping density

p = p(s) is a function given on [0, 1]. In the sequel we will assume that the function (p(s) — 1) is sufficiently smooth and finite, and

1 > p(s) > ¿> 0, s G [0,1].

A typical profile of the function taken in [2] is of the form:

t

—J

-^

1/6

5/6

Following [2], we will consider the well-known in physics of semiconductors problem on ballistic diode and formulate boundary conditions at s = 0,1 (see. [3, 4]):

r(t, 0) = r(t, 1) = —(t, 0) = —(t, 1) = 1, S(t, 1) = 0, <p(Tj 0) = A, <p(t, 1) = A + B,

(1.3)

(1.4)

where A, B are constants and the bias across the diode B > 0. Without the loss of generality, we assume that A = 0. Finally, as usual, at t = 0 we formulate initial data.

Following [2], we give an equivalent formulation of mixed problem (1.1)-(1.4). We will consider system (1.1) coupled with the relation

1

e2QT = y r(t,£)u(t,£) d^ — Ru

0

instead of the Poisson equation (1.2). Equation (1.2) rewritten in the form

e2Qs = R — p

(1.2')

(1.2")

will be treated as an additional stationary law which the initial data, in particular, must meet. From boundary conditions (1.4) it follows that the relation

1

/q(T,£) = B (1.5)

0

is fulfilled and the initial data also satisfy this relation. The electric potential < = <(t, s) is found from the evident equality

s

<(T,S) = / Q(T,£) d£. (1.6)

0

Thus, instead of mixed problem (1.1)-(1.4) one can consider problem (1.1), (1.2'), (1.3) with additional requirements (1.2''), (1.5), which actually are requirements on the initial data. It is easily shown that these two formulations are equivalent, at least on smooth solutions. Problem (1.1)-(1.4) has a stationary solution (the equilibrium state) at B = 0:

u(t, s) = u = 0, 0(t,s) = tf=1,

E(T,S) = E = 0, ((T,S) = (9 = 0,

R(t, s) = R(s) = e^(s), <(t, s) = 9(s), (1.7)

where <9(s) satisfies the Poisson equation

e2<p" = R — p (1.8)

with the boundary conditions

<9(0) = <9(1) = 0. (1.9)

It is obvious that, at small e, the solutions to boundary value problem (1.8), (1.9) can be presented as

<9(s) = ln p(s) + O(e). (1.10)

By this remark, we will assume in the sequel that the function <9(s) is sufficiently smooth and finite.

Remark 1.1. Let B = 0. From certain physical considerations (see [1]), the solution of (1.1)-(1.4) tends to the equilibrium state at t ^ to for any initial data, i.e.,

u(t, s) ^ 0, tf(T,s) ^ 1, E(t, s) ^ 0,

((t, s) ^ 0, R(t, s) ^ R(s), <(t,s) ^ <9(s).

Below we will prove this fact for a linearized mixed problem (though at a certain, very essential, restriction on the doping density p(s)).

Remark 1.2. Note that we can consider as the equilibrium state the following functions:

- u= 0, ^ = 1, z9 = 0, ,

( = 0, R(s)'= p(s), ' <(T,s) = lnp(s), that are obviously solution of the system (1.1) and the Poisson equation

e2<9'' = R — p,

where „ 2

p = p — e (In p) .

It is evident that functions p and p have the same profile with only the small difference on

some intervals--A <s< —+ A,--A <s< —+ A.

6 6 6 6

2. Linearization of mixed problem (1.1) —(1.4)

Let linearize original quasilinear problem (1.1)-(1.4) with respect to the equilibrium state (1.7). As the result, omitting intermediate, quite cumbersome calculations, we come to a linear system (small perturbations of the sought values are denoted with the same symbols):

rT + us + (' u = 0,

uT + rs + $s + gs + + ('g + pu = Q,

3 3

2 + us + qs + ('q + 2 = 0,

3 2 2 3

-Gt + us + - qs + - / q + 7 XG = °

4 5 5 4

2 2 2

-qT + ^s + -Gs + -Yq = pu, (2.1)

5 5 5

Here r

£2Qt = -Ru + J R(£)u(t, £) d£. 0

R S 0

R, " = R = —, ß

R

3\ . = 1 \ *

2 , ■ Tw — Tw 1 2 ,Ta

(2.2)

1 1 1 1

— > 0, v = — > 0, x = — > 0, Y = — > 0, ß = ß - y,

, Tq = Tq ^^ • It follows from [2] that the constants v, x, Y, 0 are large parameters. Boundary conditions to system (2.1) become of the form

r(t, 0) = r(t, 1) = —(t, 0) = —(t, 1) = g(t, 1) = 0. Linearization of relations (1.2''), (1.5), (1.6) results in the following conditions

£2Qs = R r,

1

[ q(t,o de = 0,

(2.3)

^(t,s)

Q(T,e) i.e.) ^s = Q.

(2.4)

The aim of this paper is the study of stability (by Lyapunov) of the trivial solution to linear problem (2.1)-(2.3).

In the next section we will derive an a priori estimate which implies stability of the trivial solution. Besides, in the last section we will prove asymptotic stability of this solution under an additional condition on the function p(s).

We rewrite system (2.1) in the vector form

AUt + BUs + DU = F,

(2.1')

1

p

w

a

q

0

s

0

where

U

D

( r \

u tf a

V q )

/00 0 ^

00 00 00

( 1 0 0 0 0 \ 0 10 0

3

A

\

0 0

3v Y

0 0

0 0

0

3x

4 0

0 0 2

0 0 0 0 0 0

0 0

0 0

2y

5 /

0

00

3

4 0

B

+ ^

0 2

5 /

01000 0 0 110 0 0 0 0 1

0 0 0 0 2

5

v 0 0 0 0 0 )

0100 10 11 0100

0100

0 0 1-

0 0 1 2

5 0

0

Q 0 0

\ /2u y

System (2.1') is symmetric t-hyperbolic (by Friedrichs) (see [5]). The matrix B has two positive, two negative and one zero eigenvalues. This means (see [5]) that one boundary condition in (2.3) is redundant. In fact, condition a(r, 1) = 0 is automatically fulfilled if the initial function a0(s) = a(0, s) possesses the property

ao(1) = 0.

Indeed, from boundary conditions

r(t, 0) = r (t, 1) = tf (t, 0) = tf (t, 1) = 0

and from the first and the third equations of system (2.1) it follows that

us(t, 0) = us(t, 1) = qs(T, 0) = qs(T, 1) = 0. (2.5)

While deriving (2.5), we took into account that <£(s) is a sufficiently smooth and finite function. By (2.5), from the fourth equation of system (2.1) we obtain

ot(t, 0) + xa(T, 0) = 0, °r (t, 1) + xa(T, 1) = 0

i. e.,

ct(t, 0) = e-XTao(0), a(T, 1) = e-XT ao(1).

Thus, boundary condition a(T, 1) = 0 is fulfilled if a0(1) = 0. Later we will suppose also that a0(0) = 0, i.e.,

ct(t, 0) = 0.

The energy integral identity for system (2.1') in the differential form is (see [5]):

(2.6)

43

{R(AU, U)}t + [R(BU, U)}s + i?(2^u2 - 2/xuq + -7q2 + 3vtf2 ^xa2) = 2i?uQ, (2.7)

52

2

where (BU, U) = 2(wr + wd + ua + +— aq), and, in a view of boundary conditions (2.3),

5

(BU, U)|s=o,i = 0. (2.8)

Integrating (2.7) by s from 0 to 1 (with account of (2.8)), we come to the energy integral identity in the integral form for system (2.1'):

1 1 dT||U(t)||A + 2 J R(pw2 - pwq + 2Yq2 + + 3Xa2) ^ = 2 J RwQd£. (2.9)

oo

1

Here ||U(t)||A = / R(AU, U) .

o

Multiplying (2.2) by 2Q, integrating the obtained expression by s from 0 to 1 (accounting 1

the relation / Q(t, £) d£ = 0, see (2.4)) and summing it up with (2.9), we finally have: o

1

(0)(t) + 2 [ R(pw2 - pwq + 2?q2 + + 3T*2) = 0, (2.10)

dT / 5 2 4

where I(0) (t) = || U(t) || A + e2 / Q2 (t, £) d£.

IA

0

Fulfillment of (2.10) means well-posedness of linear mixed problem (2.1)-(2.3) since (2.10) implies the a priori estimate:

d

-d/(0)(T) < 0, i.e., I(0)(t) < /(0)(0), t > 0 (2.11)

dT

provided that

mi < ^(2.12)

In following we will assume that inequality (2.12) is fulfilled. A priori estimate (2.11) means that

U(t, s) G L2(0,1), q(t, s) G L2(0,1)

at every t > 0. From (2.4) it follows that

IIp(t)||l2(0,I) < IIQ(T)||L2(0>1),

o

i.e., ^(t, s) gw^ (0,1) at every t > 0, and

||¥>(t)|| o 1(01) <||Q(T)||l2(0,I).

A priori estimate (2.11) also implies stability of the trivial solution to linear mixed problem (2.1)-(2.3). In the subsequent sections we will obtain more delicate a priori estimates which

0

allow to prove existence of a smooth solution to linear mixed problem (2.1) - (2.3) for all t > 0 and stability and asymptotic stability (by Lyapunov) of its trivial solution. Remark 2.1. Indeed, from (2.4), it follows that

Q(t, s) e W1(0,1),

o

<(T,s) eW22 (0,1)

for all t > 0.

Remark 2.2. Treating the Poisson equation (see (2.4))

^ss = 3Rr,

as an ordinary differential equation for the unknown function ^ with the boundary conditions

^(t, 0) = ^(t, 1) = 0,

we arrive at

1

v(t,s)= p/ g(s,£)R(e)r(T,e) de (2.13)

o

and

s 1

q(t, s) = ^s(T,s) = 3/ RR(e)r(T,e) de - 3/(1 - e)R(e)r(T,e) de. (2.14)

oo

Here g(s,e) is the Green function

e(s -1) for 0 < e < s,

g(s,e) ^ s(e -1) for s < e < 1.

It is seen that the functions Q, given by (2.13), (2.14), satisfy conditions (2.4). Remark 2.3. Differentiate (2.1') by t. Then, for UT, we have the system

A(Ut )t + B(Ut )s + DUt = Ft , (2.15)

where

FT = (0, QT, 0, 0,/ZuT)*

* stands for transposition.

For system (2.15) it is easy to derive the energy integral identity in the integral form:

1

d

dTl(1)(t) + 2 J R(uu2 - ¿UtqT + 2Yq2 + f tf? + 34Xa2) de = 0, (2.16)

o

where I(1) (t) = || Ut (t) || A + e2 / Q2 (t, e) de.

o

The a priori estimate follows from (2.16)

I (1)( t ) < i(1)(0), t > 0. (2.17)

Differentiating system (2.15) once again by t, we finally arrive at the estimate of the form:

I (2)( t ) < I(2)(0), t > 0 etc. (2.18)

Here I(2) (t) = || Utt (t) || A + e2 / Q2t (t, e)

o

Remark 2.4. Let introduce a value

1

2

h

t2 = l|Uo||W|(o,i) = R[(Uo,Uo) + (U0,U0) + (U0',U0')] de,

i. e., the squared norm of the vector of initial data U0(s) = U(0, s) = (r0(s), u0(s), $0(s), a0(s), q0(s))* in the space W22(0,1).

We can prove the following estimates

1 1 1

J Q2(0,£) d£ < K1t2, J Q (0,0 d£ < Kt2, J (0,£) d£ < Kat2, (2.19)

000

where K1, K2, K3 are the positive constants. Indeed, by the formula (2.14) we have

s 1

|Q(0, s)|2 = R(£)r(0,£) de - 0/(1 - £)R(£)r(0,£) de|2 < 00 1 1

< [20 / R(e)|r(0, e)I de]2 = 402(/ R1/2|r0|R1/2 de)2 <

00 1 1 1

< 402 j Rdej Rr2 de = 402K0 / Rr2 de < k^2, 0 0 0

1

where K0 = J Rde > 0, K1 > 0. Thus, the first of estimates (2.19) is fulfilled. Later, since 0

s 1

IQ (0, s)i < 0(i/ R(e)rT (0, e) de i + i / (1 - e)R(eK (0, e) dei < 00 1 1 1

< 20 / R(e)|rT(0, e)i de < 20(/ Rde)1/2a Rr2 de)1/2,

0 00

than, if the inequality

1

JR(e)r2(0,e) de < K4t2, K > 0 (2.20)

0

is held, than the estimate

1

[ Q (0,e) de < K2t2

0

took place.

By the analogy, under condition

R(Or2TT(0,i) di < K5t2, K5 > 0 (2.21)

we can show correctness of the third of estimates (2.19). The proof of inequalities (2.20), (2.21) is given in section 3. Finally note that

0 RQi m < 0 (0,i) < P wj Rr0 * < (222)

0 0 0

Remark 2.5. Unfortunately, we did not succeed in obtaining the additional entropy conservation law for system (1.1). Such a law is easily derived for linearized system (2.1) by exclusion of the derivatives of us, qs from the first, third, fourth equations of system (2.1):

5 5

(r + •& - 4a)T + tp'u + - 4= 0. (2.23)

Remark 2.6. We will say that the trivial solution of linear problem (2.1)-(2.3) is stable, if for any e > 0 there exists > 0 such that from the inequality

l|Uo||< S

will follow the inequality

|| U (t )||< e

for all t > 0.

Remark 2.7. We will say that the trivial solution to linear problem (2.1) - (2.3) is asymptotically stable, if, at an arbitrary initial data U0(s) from a Sobolev space, the solution U(t, s) tends to zero at t ^ to, also in the Sobolev space.

i

0

i

i

i

3. Stability of trivial solution to problem (2.1)-(2.3)

Since we assume that the function <£(s) is sufficiently smooth (see section 1), henceforth we can suggest that |<£>'(s)|, |<9"(s)| < C0, where C0 is the positive constant. From the first equation of system (2.1) it follows that

r2 = u2 + (^)2u2 + 2us(p'u < 2(u2 + C02u2),

so

R(£)rT(0,0 di < KAtT

Remark 3.1. The constants K1, K2, K3, K4 (as well as other positive constants Kj, i = 5,... , 7, Nj, j = 1,... , 3, Mk, k = 1,... , 18, which appear in the sequel in this section) are determined finally via the constants v, x, Y, 0 and the function p(s). Let us differentiate the second equation of (2.1) by s:

UTS = Qs — (rss + 0ss + ass + (¿'(0s + ^s) + + <¿"(0 + a)).

Then, with account to relation (2.22) it yields the inequality

1

J R(e)u2s(0,e) de < K7t2.

0

Now, differentiating the first equation of (2.1) by t:

rTT + Uts + (¿'«T = 0,

we arrive at the estimate

1

J R(e)r^T (0,e) de < ^. 0

Thus the inequalities (2.20), (2.21) from Remark 2.4 are valid, so the estimates (2.19) are proved.

Note that

(AUt, Ut) < N1{(U, U) + (Us, Us) + Q2} (3.1)

since at the desired functions in (2.1) we have either the constants or the known bounded functions.

From the differentiated by s system (2.1) we obtain

(AUts, Uts ) < N2{(U, U) + (Us, Us) + (Uss, Uss)}. And differentiating (2.1) by t, we finally have

(AUtt, Utt) < N2{(U, U) + (Us, Us) + (Uss, Uss) + Q2 + Q2}. (3.2)

Gathering estimates (2.11), (2.17), (2.18), (2.19), (3.1) and (3.2), we can write that 1

J R[r2 (T,e)+u2(T,e)+02(T,e) + a2 (T,e) + q2(T,e)+ 0

+r?(T, e)+«2 (t, e)+(t, e) + a^, e) + ¿(t, e)+

+r^T(T,e)+«2t(T,e)+02t(T,e) + a2T(T,e) + &(T,e)] de < M^2, T > 0. (3.3)

Completing the derivation of an a priory estimate, we deduce some auxiliary inequalities. First, we will derive estimates for the first derivatives by s of the desired functions. From the first equation of system (2.1) it follows that

2 ^ 2 , ri2 2

« < rT + C0u ,

so

1

J R(e)«2(T,e) de < M2t2, t> 0. (3.4)

0

Evaluate the derivative qs with the help of the third equation of system (2.1):

3 ' 3v

qs = - ( ^ 0t + «s + q + y 0

i. e.,

1

J Rq*2 de < M3t2, t> 0. (3.5)

0

While deriving similar results for 0s, as, rs it is necessary to have estimates for derivatives uTs, rTs, 0Ts, aTs. Differentiating the first equation of (2.1) by t, we will have

rTT + uTs + (p'uT = 0.

Whence we easily obtain

1

22

R<s de < M4t2, T > 0. (3.6)

0

Now let us use the second and fifth equation of system (2.1) differentiated by t:

rTs + 0ts + aTs = F = Qt - «TT - (¿'0T - (¿'aT - , (3.7)

0ts + 2aTs = F2 = - 2qTT - T^qT, (3.8)

5 5 5

with the help of (3.7) and (3.8) we transform the additional relation (2.23) into:

^rs + 0ts - 5ffT^ + i ^rTs + 0ts - 4= F3 = //F1 - vF2 - (¿'«ts - (¿''«T, (3.9)

5 8 5

where i = -x +--v. Multiplying (3.9) by 2R I rTs + 0Ts--aTs I and integrating by s from 0

9 45 \ 4 J

to 1, we get

dT IJ R (Vrs + 0rs - 5ar^ de j + (2i - 1) J R ^>s + 0rs - 4ar^ de < M5t2,

then for i > - we have 2

1 2

I R f rrs + 0rs - 5 arJ de < Met2.

Since from (3.7) it follows that

i

J R (rts + $TS + ^Ts)2 < M7t2, 0

than we receive

1

I R^2s dC < M12,

and after that from (3.8) and (3.7) we sequentially deduce analogous estimates for and r^. Finally

i

222

R[r2s + + <s] dC < Mgt2, t > 0. (3.10)

Later, from the first equation of system (2.1), we obtain

uss = -rTs - (^'u)s,

i. e.

Ru2s dC < M10t2, t> 0. (3.11)

In order to evaluate the derivative qss, we differentiate by s the third and the fourth equations of system (3.1). From these equations and from the last equation of system (3.1), we find

(2 16 \ 1 2 (2 16 \ (2 16 \ . 2 2 (,3V + 75Xj qss = - V^ - 5X"ss ^3V + 75Xj uss -{ 3V + u)s +5qT + + 5 Yq- ^

So, if we assume that

0 < V0 < v, 0 < X0 < X,

than we have

1

/ Rq2s dC < M11t2, t> 0. (3.12)

To evaluate the derivatives $s, as, we differentiate the third and forth equations of system (2.1) by s and multiply them by 2R$s and 2Ras, correspondingly. Finally, we have the inequalities:

3dT {/ R^2 dc| + (3V - 1)j R^2 dC < M12t2, 00

0

0

1

0

0

3[ Ra2 dd + f3? - 1 ) I iia22 de < M13t2

4 5T ./ s M V 2

i. e.,

1

J R[02 + a2] de < M14t2, T > 0. (3.13)

0

12

While deriving (3.13) we assumed that v > - and x > _ that is true (see [2]). From the second equation of system (2.1) it follows that

rs = F4 = Q - («r + 0s + as + <¿'0 + <'a + ^w),

so

1

J Rr2 de < M15t2, T> 0. (3.14)

0

From the third equation of (2.1) differentiated by t it follows that

9 9v2

2 9 2 2 2 2 9v 2

qrs < 30rr + «rs + C0 qr + — 0r,

i. e.,

1

j Rq^ de < M1612, t> 0. (3.15)

0

Finally to estimate the derivatives 0ss, rss, ass we will use again the method we have used while deriving the derivatives 0Ts, rTs,aTs. To this end we differentiate the second and fifth equations of system (2.1) by s:

rss + 0ss + ass = G1 = 0Rr - «rs - (<'0)s - (<<'a)s - P«s, (3.16)

2 2 2Y

0ss + 5ass = G2 = - 5qrs - — qs, (3.17)

and (2.23) by s twice:

rss + 0ss - 5as^ + 15v ^rss + 0ss - 5as^ = G3 = -vG2 + ^vG1 - (<<'«)ss. (3.18)

Then it follows from (3.16), (3.17), (3.18) that

1

i R[r2s + 02s + a2s] de < Mnt2, T > 0. (3.19)

1

0

0

Gathering estimates (3.3)-(3.6), (3.10) - (3.15), (3.19), we can write down the desired a priori estimate:

1

J R[(U,U) + (Ur,Ur) + (Us,Us) + (Urr,Urr) + (Urs,UTs) + (Uss,Uss)] d£< M^t2, t>0. (3.20) 0

From (3.20) it follows that

o

u(t, s) G W22(0,1), Q(t,s) G W23(0,1), ^(r,s) GW24 (0,1) for all t> 0.

It also implies stability (by Lyapunov) of the equilibrium state in the linear approximation. Remark 3.2. More exactly,

o

r(t, s), tf(T,s), a(T,s) GW2 (0,1).

Besides,

l|U(t)||W2(0,1) < M1212, T > 0,

which means (see Remark 2.5) stability (by Lyapunov) of the trivial solution to mixed problem (2.1)-(2.3). Here

1

12

h

0

i|u (t )||w22(0,i> = ) + (us,us) + (uss,uss)] de.

4. Asymptotic stability of trivial solution

Evidently, the first equation of system (2.1) can be rewritten as

(Rr)T + = 0.

It is expedient to introduce into consideration a potential ^ = ^(r, s) such that

RRr = (4.1)

First two boundary conditions in (2.3) imply

0) = ^s(r, 1) = 0. (4.2)

The remained conditions in (2.3) and assumption (2.6) give

(Êtf)(r, 0) = (Êtf)(r, 1) = 0, (4.3)

(Êa)(r, 0) = (Êa)(r, 1) = 0. (4.4)

Note that equation (2.2) with regard to (4.1) transforms into

e2Q + ^ -J ^ d£ | =0,

i. e.,

1

e2Q + ^ -J * dC = A0(s),

0

where A0(s) is an arbitrary function. On the other hand, by (2.4), we have

e2Qs = Rr, i. e., A0(s) = 0 and A0 = const; 1

since / QdC = 0, A0 = 0. 0

So,

Q = 3 ^ J tf dC -^\=ph(r,s). (4.5)

Rewrite the second condition of system (2.1) as

(Ru)t + (R r)s - -¿'Rr + (Rtf)s + (RRa)s + pRu = Rq,

and then, with the use of relations (4.1), in the form

$TT - ^ss + (Rtf)s + (RR^)s + P^T + <^s = R^h- (4.6)

By the analogy, the third, fourth, fifth equations from (2.1) can be rewritten as follows:

Q Q

2(R^)t + ^ts - <^t + (Rq)s + y(RR^) = 0, (4.7)

3(iRa)T + ^ts - <^t + q)s + 3X(R^) = 0, (4.8)

4 5 4

9 9 9 9o/

5 (RR q)T + (Rtf)s - £'(Rtf) + ^(R^)s - 5 + y (Rq) - ^ = 0. (4.9)

Later we will use equations obtained from (4.6) - (4.9) by differentiating by s:

Htt - Hss + (Rtf)ss + (RR^)ss + pH + <S'Hs + 3(2RR - p)H = (4.10)

Q Q

^(R^)ts + Hts - (^T)s + (RRq)ss + y(Rtf)s = 0, (4.11)

3(Rff)rs + Hts - (^t)s + 2(Rq)ss + 3X (R^)s = 0, (4.12)

4 5 4

9 9 9 9o/

^(Rq^s + (RR^)ss - (£'Rtf)s + ^(R^)ss - ^'R^s + y(Rq)s - № = 0, (4.13)

where H =

Remark 4.1. The aggregate h (see (4.5)) can be rewritten

1 1

h(T,s) = y *(t,£) d£ - *(T,s) = y [*(t,£) - *(t,s)] d£

00

(t, z) dz

i 5

de = / h(t,z) dz de.

(4.14)

0 s

Now we proceed to derivation of the desired estimate. With this purpose, we multiply equation (4.6) by 2^T, equation (4.7) by 2(i?$), equation (4.8) by 2(_Ra), equation (4.9) by 2(i?q), sum them up and integrate it by s from 0 to 1 (accounting boundary conditions (4.2)-

(4.4)):

d_

dT

+*2+3(R0)2+3(R ^)2q)2

dO +

+2

ß^T + y (RR0)2 + "X (RR^)2 +1? (Rq)2 + ^ - (R0) - (R^)-

4

5

2

-ß*r(iRq) - ^(iR0)(iRq) - 2£W(Rq)

5

de = 2ßj iR^r h de.

0

(4.15)

By analogy, multiplying (4.10) by 2HT, (4.11) by 2(Rtf)s, (4.12) by 2(Ra)s, (4.13) by 2(i?q)s summing up, integrating by s (accounting (4.2)-(4.4) and also (2.5)), we have:

d_

dT

H + Hs2 + 2(R0)2 + 4(iR^)2 + 2(iRq)2 + ß (2R - p)h2

de +

0

+2

ßH + (ii0)2 + "X(ii^)2 + 2Y(Rq)2 + ^'HrHs - (^)s(R0)s - (^r)s(Rr)s-

2

-ßHr(iRq)s - (^'iR0)s(iRq)s - -(^'iRa)s(iRq)

5

de = 2ßj rR'Hth de. 0

(4.16)

We will use the result of integration by s from 0 to 1 (with account to boundary conditions) of equation (4.10) multiplied by 2H:

dT ^ I [2HHr + ßH2]de }> +2

-Hr2 + Hs2 - Hs(Ê0)s-

5

1

0

s

1

0

1

0

1

1

1

0

1

1

0

0

-Hs(Ra)s + 3(3R - p)H

dC = 2^ R'HhdC.

0

(4.17)

Summing up (4.15)-(4.17), we obtain the expression:

dT

J(0) + J(1) = 23 / iR^ThdC + 23 / rR'HthdC + 23 / R'HhdC <

1

1

1 \ 2 / 1 \ 2 / 1

?2 JA 1 I ^ JA 1 I I f td„\2

< 23 \ I R dCl I I (Ru)2 dC j(Rr)2 d£| +

0 0 0

1

+23 /(R')2 dC

1

1 \ 2 / 1 \ 2 / 1

' 2 S 2 S 2

(r r)2 dC

1

2 1

(Rr)2 dC + 23 / (R')2 dC / (Rr)2 dC. (4.18)

Note that the estimate of the right hand part in (4.18) is derived with the help of the inequalities of Holder and Cauchy, and formula (4.14). Besides, in (4.18)

J(0)

3 ^ 3 ^ 2-

(R u)2 + 2(Rtf)2 + 4(R<)2 + 5(Rq)2 + (i?rT )2 +

332

+ (i?r)2 + 2(rR^)2 + 4(i? <7)2 + 5(R q)2 + + (3(2ii - p) + p + 1)(Rr)2 + 2(iir)(RrT)! dC

J(1) = 2

P(iiu)2 + (R^)2 + f (rR <)2 + f (ii q)2 +

+ (P - 1)(R r)2 + (iir)2 + ^ (Rtf)2 + ^ (R<)2 +2Y (Rq)2 +

2

4

5

3

+ 2(3/? - p)(Rr)2 - <p'(Ru)(Rr) - <p'(Ru)(R$) - <p'(Ru)(R<) -

- p(Ru)(Rq) - <p'(R$)(Rq) - 2<p'(Ro")(Rq) + -¿'(R^(Rr)s -

5

- (^'Ru)s(Rtf)s - ($Ru)8(R(j)8 + p(iir)T(iiq)s - (<p'R$)s(Rq)s -

2

— (<p'R a)s(i?q)s + (i?r)s(i?^)s + (i?r)s(i?<)i

5

dC.

Rewrite (4.18) with account to the estimate of the right hand part as the following inequality

A

dT

J(0) + J(2) < 0,

(4.19)

1

2

1

1

1

d

0

0

0

0

0

0

1

0

1

0

where

1

J (2) =J (1) - P (/ R f81/(R u)2 -

1 i 1 1 i 1

-A J R2 dM U (Rr)2 de - A J (R' )2 d^ 82 J (R r)2 de-\0 / 0 \0 /0 1 1 1 1 1 1

-A|(R')2 d£j 82 I(iir)2 de - 20 i J(R')2 d£j J(Rr)2 de.

\0 /0 \0 /0

While deriving J(2) we used the Cauchy inequality with some positive constants 81; 82.

In J(0) and J(2) the expressions under the integral sign are positive definite squared forms of the variables Rr, Ru, R$, Ra, Rq, (Rr)s, (R$)s, (Ra)s, (Rq)s, RrT, since the parameters

v, x, 7, P are sufficiently large, with / > -7 and if we choose constants 81; 82 such that

5

8, < ^, i = 1, 2,

i + i < 2/.

81 82

Here positive constant / is founded from the inequality

1

s - (J(p'(e))2 dej >/, (4.20)

which exactly is the essential restriction on the function p(s) first mentioned in Remark 1.1. Note, that /2 < 0 (see [1]) and with high accuracy, by (1.10),

3R - p = 2p.

Under fulfillment of (4.20) there exists a constant M0 > 0 such that

J(2) > M0J(0). (4.21)

By (4.21) inequality (4.19) transforms into

-^J(0) + M0J(0) < 0,

dT

i. e.,

J(0)(t) < e-MoTJ(0)(0). (4.22)

Remind relations (4.1) and rewrite (4.22) as

[r2(r, s) + u2(t, s) + $2(t, s) + a2(r, s) + q2(r, s) + r2(t, s) +

+ r2(T, s) + u2(T, s) + $2(t, s) + a2(r, s) + qs2(r, s)] d£ < M1e-M°rt2. (4.23)

i

Here M1 > 0 is a constant, t2 = ||Uo||Wi(o 1) = / i(Uo, Uo) + (U, U)] d£ is the squared norm of

2 ( ' 1 o

the vector of initial data Uo(s) = U(0, s) = (ro(s), uo(s), $o(s), ao(s), qo(s)) * in the Sobolev's space W.1(0,1).

Remark 4.2. Constants Mo, M1, as well as positive constants M2, M3, are finally determined via the constants p, v, x, Y, 3 and the function p(s).

Derivatives ur, $r, ar and qr are estimated with the help of system (2.1):

ur = Q — (rs + + + <p'$ + <pV + pu), 2 ' 3

$r = — 3(us + qs + <p q + 2v$),

2 2 3 0. ^r = — 3(us + 5qs + 5q + 4

^2 2 qr = — + 5 ^ + 5 Yq — pu)-

Consequently,

i

j [u2(t, s) + (t, s) + (t, s) + q2(T, s)] d£ < M2e-M°rt2. (4.24)

o

Combining estimates (4.23) and (4.24), we come to the desired a priori estimate: 1

/ [(U, U) + (Ur, Ur) + (Us, Us)] d£ < Mae-M°rt2, t > 0. (4.25)

o

From (4.25) it follows that

U(t, s) G W;1(0,1), Q(t, s) G Wf(0,1),

O

^(t,s) GW| (0,1), for all T > 0,

and the equilibrium state in the linear approximation is asymptotically stable (by Lyapunov). Remark 4.3. Precisely,

O

r(t, s), $(t, s), ct(t, s) GW1 (0,1).

Besides,

IIU(t)||W2i(o,ii < Mse-M°rt2, t> 0,

just this means (see Remark 2.6) asymptotic stability (by Lyapunov) of the trivial solution to mixed problem (2.1)—(2.3). Remind that

1

IIU(t)||W2i(o,1i = / [(U, U) + (Us,Us)] d£.

o

5. Conclusions

The analysis, carried out in the paper, states a very important (from the applications point of view) fact on asymptotic stability of the equilibrium state for the antidemocratic hydrodynamical model (see [1]) of charge transport in semiconductors. Indeed, in absence of the bias across the real semiconductor devices, transport of charge carriers (i.e., electric flow) must be absent. Consequently, applying hydrodynamical models in description of physical phenomena of charge transport in semiconductors, we must require of them the adequate description of these phenomena (including correct description of the transition process in semiconductor devices in absence of the bias across the diode).

Unfortunately, the fact of asymptotical stability of the equilibrium state is proved under essential restriction (4.20) on the doping density p(s) and in the linear approximation as yet. It should be noted at the same time that proof of stability of the equilibrium state does not contain any restrictions on the doping density.

We gratefully thank Prof. A. M. Anile for many helpful discussions. We also appreciate A. A. Iohrdanidy for efficient cooperation.

References

[1] Anile A.M., MuSCATO O. Improved hydrodynamical model for carrier transport in semiconductors. Physical Rev. B., 51, No. 23, 1995, 16728-16740.

[2] Blokhin A.M., Iohrdanidy A. A., Merazhov I. Z. Numerical investigation of a hydrodynamical model of charge transport in semiconductors. Preprint No. 33, Institute of Mathematics SB RAS, Novosibirsk, 1996.

[3] Gardner C. L., Jerome J. W., Rose D. J. Numerical methods for hydrodynamic device model: subsonic flow. IEEE Transactions on Computer-aided Design, 1989, 8, No. 5, 501507.

[4] Gardner C. L. Numerical simulation of a steady-state electron shock wave in a submicrometer semiconductor device. IEEE Transactions on Electron Devices, 1991, 38, No. 2, 392-398.

[5] Blokhin A. M. Energy Integrals and Their Applications to Gas Dynamic Equations. Nauka, Novosibirsk, 1986.

[6] Blokhin A. M. Elements of Theory of Hyperbolic Systems and Equations. Izd-vo NGU, Novosibirsk, 1995.

Received for publication November 24, 1998, in revised form January 6, 1999