Simulation of the Transient Behavior of a One ... - UT Mathematics
Simulation of the Transient Behavior of a One-Dimensional Semiconductor Device II Author(s): Irene Martinez Gamba and Maria Cristina J. Squeff Source: SIAM Journal on Numerical Analysis, Vol. 26, No. 3 (Jun., 1989), pp. 539-552 Published by: Society for Industrial and Applied Mathematics Stable URL: http://www.jstor.org/stable/2157670 . Accessed: 12/05/2011 16:42 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at . http://www.jstor.org/action/showPublisher?publisherCode=siam. . Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected].
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SIAM J. NUMER. ANAL.
1989 SocietyforIndustrialand Applied Mathematics
Vol. 26, No. 3, pp. 539-552, June 1989
003
SIMULATION OF THE TRANSIENT BEHAVIOR OF A ONE-DIMENSIONAL SEMICONDUCTOR DEVICE II* IRENE MARTINEZ GAMBAt
AND
MARIA CRISTINA J. SQUEFFt
Abstract.A numericalmethodbased on treatingthe potentialby a mixed finite-element methodand versionof a modificationof the methodof the electronand hole densityequations by a finite-element characteristics is introducedto simulatethe transientbehaviorof a semiconductordevice, forwhichthe of the conductivityequations on the electricfield is considered and the dependence of the coefficients Einsteinrelationsare not assumed. L2-errorestimatesthatare independentof Lw-errorestimatesforthe approximateelectricfieldare derivedfora singlespace variablemodel. Keywords.semiconductorsimulation,mixedfiniteelements,modifiedmethodof characteristics AMS(MOS) subjectclassification.65
0. Introduction. We assume the reader is familiarwith the work of Douglas, Martinez, and Squeff [4]. General referencesto the literaturewere given in the bibliographyof [4] and theyare givenagain herein.We statebrieflythe problemin thissection. We consider[1], [8]-[11], [13] thenonlinearparabolic/elliptic systemofequations thatdescribethe transientbehaviorof a semiconductordevice in a closed intervalof R' (ax=/alax, etc.): -p -c),
(0.la)
axq = -ax(c(x)
(0. 1b)
Qate-ax
(0.1c)
Qt3tp ax(Jp)= QR(e, p),
= -Q(e
(Je) = QR (e, p),
wherethe electricfield q and the carrierdensitiese and p are relatedthroughthe currentdensities Je=
8-t
QI-ee(q)eq + QDe( q)dxe,
Jp=
a 'QgP(q)pq
-
QDp(q)adxp,
with? and Q being positiveconstants.We assume Dirichletboundaryconditions (0.2a)
I&(0, t) = r0(t),
qf(1, t) = rl(t),
(0.2b)
e(O, t) =fo(t),
e(l, t) =f1(t),
(0.2c)
p(O, t) = go(t),
p(,
t) = gi(t),
and the initialconditions (0.2d)
e(x, 0) = eo(x),
p(x, 0) =p?(x)
If bte De, lip,Dp are assumed to be positive constants,the equations (0.1) are quasilinearand have been treatedby Douglas, Martinez,and Squeff[4]. In thispaper we generalizetheirmethodto thenonlinearsystemthatresultsfromassumingthatSte, De, lip, and Dp are functions of the electric field q. Actually, these assumptions on thecoefficients are morerealistic,sincethemobilities/leand ,upare theproportionality factorsof the driftvelocitiesto the electricfield: vd = 1xpq. Vd=_Leq, *
Received by the editorsAugust5, 1987; accepted forpublication(in revisedform)June7, 1988. of Chicago, Chicago, Illinois 60201. t Departmentof Mathematics,University t Departmentof Mathematics,New JerseyInstituteof Technology,Newark,New Jersey07102. 539
540
I. MARTINEZ GAMBA AND M. C. J. SQUEFF
Also theyallow us to improvethe estimatesof [4]. Here we derive L2-normerror estimatesforthecarrierdensitiesthatare independentof the L'-norm of theapproximation qh to the electricfieldq. Appropriatemodels forthe mobilitiesare given in Selberherr[13], and theysuggestthat a and Da for a = e or p can be assumed to satisfythe followingconditions. There existpositiveconstantsD*, M1, L1, L2, and KD such that q e R, i = e, p},
(0.3a)
D* ? inf{Di(q):
(0.3b)
IA.L(q)qL
(0.3c)
J/a(qI)qDq-/a(q2)q2-LllJql
(0.3d)
jDa(ql)-Da(q2)1
qc R,
M,,
Ipt(qj)
-q2I,
-
CL2/qI-q21, ta(q2)1 J
' KDlql-q2j,
for a = e or p. Let us also assume, for this case of single space variable problem, that (0.3)
I(aq#a(ql))q.
I at,a (q)) q 'IM2,
-
L31qI - q21,
(aqAa (q2) )q2I
fora = e orp, whereM2 and L3 are positiveconstants.If ,a is a Lipschitzfunctionas, forexample, in Markowich[10], S
S
AeP 1e
+ (
e lq j1 /ve)
1+
(s,upI qll
vP )
wherev' and vP are the saturationvelocitiesand lus and ipsstandforone of the field independent,scatteringmobilitymodels,thenassumption(0.3e) can be verifiedprovided that q and its derivativesare bounded. In the single space variable problemq and its derivativesare bounded. So, in particular,it followsfromthe firstinequality of (0.3e) that (0.3f)
Jax(Att(q)q)J M3,
a=e
orp,
whereM3 is a positiveconstantdependingon M2, the bound forthe right-hand side of the potentialequation,and thebound for,ts(q). Withoutloss of generalitywe can take L3_ M3. We willnotassumetheEinsteinrelationsforthemobilityand diffusion coefficients. In ? 1 we describethe proposed numericalprocedure.In ? 2 we deriveL2-norm errorestimatesfortheapproximateelectricfieldqh. Finally,in ? 3 we obtainL2-norm errorestimatesforthe approximationseh and Ph to theirrespectivecarrierdensitiese and p. 1. Descriptionof thenumericalprocedure.If (0.1) are scaled as in [10], [11], [15], then (1.la)
(l.lb) (1l.lc)
axq = -axxq = -z(e -p - c), Ate- UT1tte(q)qaxe -ax(De(q)axe)
- qeax(UT,te(q))
+ UTzye(q)e(e
-p - c)
= R(e, p) atp
+
UTAp(q)q&xp
-al(Dp(q)axp)
+ qpdx( UTAp(q))
-
UTzAp(q)p(e
-p
-
c)
= R (e P), where z is the inverse square of the normed characteristic Debye length of the device [10].
As in thepreviouswork[4] we willuse a mixedfinite-element methodto approximate q and C simultaneouslyand a modifiedmethodof characteristics [3], [5], [6], [12] to approximatethe densitiese and p.
SEMICONDUCTOR
SIMULATION
541
II
To introducethe modifiedmethodof characteristics for e and p let re =,Te(X, t) be theunitvectorin thedirection(- UT,Le(q)q, 1) and r, theunitvectorin thedirection fora = e or p. Then thederivativesin the (UT8Up(q)q, 1). Set fp = [1 + ( UTla(q)q)2]1/2 Ta directionsare givenby (Pea/are = at-
UT/.e(q)q
pp&/&rp= t+ UT21p (q)qax,
x,
so thatequations (1.lb) and (1.lc) can be writtenin the followingform: (1.2a)
'peae/lTe-
.x(De(q)&xe)-qeax(
+ UTte(q)ze(e-p-c)
U,UTte(q))
= R(e
'pap/lrTp-a,(Dp(q)aXp) + qpax(UT,p(q))-UTttp(q)zp(e-p-c) Now, for a = e or p,
(1.2b)
a,tt (q) = aqla
(q) *d9xq= -(
qg
=
p),
R(e, p).
(q)) z(e - p - C),
so thatthe weak formulationfor (1.2a) and (1.2b) is given as the determination of maps e and p of the timeintervalJ = [0, T] into H1(fl) such that (1a ( 1.3a)
(9e/aere,
+ (De(q)axe,
(1.3b)
(qdapl/arTp
+ (Dp(q)axp,
= (R, ;),
ax) + UTz([
ax;)
-
.Le(q)+ qq(qLe(q))]e(e-p-c),
UTz([,up(q) +
;)
qaq(ap(q))]p(e -p - c), ;)
for ; e H'(fl), and such that the boundaryand initial conditions(0.2a)-(0.2d) are satisfied. First, consider a partition of J into subintervals [tm1, tim],t"
and a partitionoffl intosubintervals [xi-1,xi], 0 =
xi1) = hd. Let
Zh =
xO < X
1,
At otherwise.
Also, note thatim and im cannotbe evaluated exactly.So, let x'm,Atm,xm and At'm relationswhenqm = q(x, tm) is replacedby qhmph be definedby thecorresponding
542
GAMBA
I. MARTINEZ
AND
M.
C.
J. SQUEFF
Then, Let eh and po lie in Zh and approximatee(x, 0) and p(x, 0), respectively. We for m'1 define e^-1= eh('em(x), tm-Ate), and Ph P=Ph(X (X) tm-Atp tm'). Now, we defineem and pm herethat,if Atm= At thene^ = eh(X(x) remark as the unique solutionof the following(algebraicallylinear) equations: ((em
(1.6a)
Uz([ge(q-1 e(q q7(e ?1-eh . ((h pm _
(1.6b)
g
eAM-1)/ltm
-
(h
~
I
h
eqp(q
~
l)]e
-/
-
)
Cm) p
)axPh, a,4)
)+qa? q
UTZ([ A(qh A
)?
a,x)
1)dxem,
(De(q
; ? (D,(qm
-l)/+t
-
?
~
m
)]p e
~
c)
P-fm1
(eh
~
;)
-
For later convenience,we define approximationsto the derivativealong the by approximatecharacteristics he,tJ h+x/= ~~ (1.7a) ~~~~ (1.7b)
~
=a1emUT,e(qh-)qIeQ ~~~Mx=z--c) ~~~ ? /Atp,h &Dtpm
(Pe,heR/areZh
(Pp,h(P
1
teMAMJ,
e
M
m,ax-)
methodto approximateq and +f Finally,we describe the mixed finite-element simultaneously,definedas follows. Firstwritethe potentialequation (l.la) in the followingform:
) andL2(h, (1.7b) againstone in Then, if (1.7a) is testedagainsta functionin H1(qm we findthe mixed weak form: (1.8a)
max (yp
(q, v) - (dv/dx,+f)= rv(1)- rv(O),
[Yai,yri], Let l be partitionedinto subintervals
vE
1()
0 Yo t
Society for Industrial and Applied Mathematics is collaborating with JSTOR to digitize, preserve and extend access to SIAM Journal on Numerical Analysis.
http://www.jstor.org
?
SIAM J. NUMER. ANAL.
1989 SocietyforIndustrialand Applied Mathematics
Vol. 26, No. 3, pp. 539-552, June 1989
003
SIMULATION OF THE TRANSIENT BEHAVIOR OF A ONE-DIMENSIONAL SEMICONDUCTOR DEVICE II* IRENE MARTINEZ GAMBAt
AND
MARIA CRISTINA J. SQUEFFt
Abstract.A numericalmethodbased on treatingthe potentialby a mixed finite-element methodand versionof a modificationof the methodof the electronand hole densityequations by a finite-element characteristics is introducedto simulatethe transientbehaviorof a semiconductordevice, forwhichthe of the conductivityequations on the electricfield is considered and the dependence of the coefficients Einsteinrelationsare not assumed. L2-errorestimatesthatare independentof Lw-errorestimatesforthe approximateelectricfieldare derivedfora singlespace variablemodel. Keywords.semiconductorsimulation,mixedfiniteelements,modifiedmethodof characteristics AMS(MOS) subjectclassification.65
0. Introduction. We assume the reader is familiarwith the work of Douglas, Martinez, and Squeff [4]. General referencesto the literaturewere given in the bibliographyof [4] and theyare givenagain herein.We statebrieflythe problemin thissection. We consider[1], [8]-[11], [13] thenonlinearparabolic/elliptic systemofequations thatdescribethe transientbehaviorof a semiconductordevice in a closed intervalof R' (ax=/alax, etc.): -p -c),
(0.la)
axq = -ax(c(x)
(0. 1b)
Qate-ax
(0.1c)
Qt3tp ax(Jp)= QR(e, p),
= -Q(e
(Je) = QR (e, p),
wherethe electricfield q and the carrierdensitiese and p are relatedthroughthe currentdensities Je=
8-t
QI-ee(q)eq + QDe( q)dxe,
Jp=
a 'QgP(q)pq
-
QDp(q)adxp,
with? and Q being positiveconstants.We assume Dirichletboundaryconditions (0.2a)
I&(0, t) = r0(t),
qf(1, t) = rl(t),
(0.2b)
e(O, t) =fo(t),
e(l, t) =f1(t),
(0.2c)
p(O, t) = go(t),
p(,
t) = gi(t),
and the initialconditions (0.2d)
e(x, 0) = eo(x),
p(x, 0) =p?(x)
If bte De, lip,Dp are assumed to be positive constants,the equations (0.1) are quasilinearand have been treatedby Douglas, Martinez,and Squeff[4]. In thispaper we generalizetheirmethodto thenonlinearsystemthatresultsfromassumingthatSte, De, lip, and Dp are functions of the electric field q. Actually, these assumptions on thecoefficients are morerealistic,sincethemobilities/leand ,upare theproportionality factorsof the driftvelocitiesto the electricfield: vd = 1xpq. Vd=_Leq, *
Received by the editorsAugust5, 1987; accepted forpublication(in revisedform)June7, 1988. of Chicago, Chicago, Illinois 60201. t Departmentof Mathematics,University t Departmentof Mathematics,New JerseyInstituteof Technology,Newark,New Jersey07102. 539
540
I. MARTINEZ GAMBA AND M. C. J. SQUEFF
Also theyallow us to improvethe estimatesof [4]. Here we derive L2-normerror estimatesforthecarrierdensitiesthatare independentof the L'-norm of theapproximation qh to the electricfieldq. Appropriatemodels forthe mobilitiesare given in Selberherr[13], and theysuggestthat a and Da for a = e or p can be assumed to satisfythe followingconditions. There existpositiveconstantsD*, M1, L1, L2, and KD such that q e R, i = e, p},
(0.3a)
D* ? inf{Di(q):
(0.3b)
IA.L(q)qL
(0.3c)
J/a(qI)qDq-/a(q2)q2-LllJql
(0.3d)
jDa(ql)-Da(q2)1
qc R,
M,,
Ipt(qj)
-q2I,
-
CL2/qI-q21, ta(q2)1 J
' KDlql-q2j,
for a = e or p. Let us also assume, for this case of single space variable problem, that (0.3)
I(aq#a(ql))q.
I at,a (q)) q 'IM2,
-
L31qI - q21,
(aqAa (q2) )q2I
fora = e orp, whereM2 and L3 are positiveconstants.If ,a is a Lipschitzfunctionas, forexample, in Markowich[10], S
S
AeP 1e
+ (
e lq j1 /ve)
1+
(s,upI qll
vP )
wherev' and vP are the saturationvelocitiesand lus and ipsstandforone of the field independent,scatteringmobilitymodels,thenassumption(0.3e) can be verifiedprovided that q and its derivativesare bounded. In the single space variable problemq and its derivativesare bounded. So, in particular,it followsfromthe firstinequality of (0.3e) that (0.3f)
Jax(Att(q)q)J M3,
a=e
orp,
whereM3 is a positiveconstantdependingon M2, the bound forthe right-hand side of the potentialequation,and thebound for,ts(q). Withoutloss of generalitywe can take L3_ M3. We willnotassumetheEinsteinrelationsforthemobilityand diffusion coefficients. In ? 1 we describethe proposed numericalprocedure.In ? 2 we deriveL2-norm errorestimatesfortheapproximateelectricfieldqh. Finally,in ? 3 we obtainL2-norm errorestimatesforthe approximationseh and Ph to theirrespectivecarrierdensitiese and p. 1. Descriptionof thenumericalprocedure.If (0.1) are scaled as in [10], [11], [15], then (1.la)
(l.lb) (1l.lc)
axq = -axxq = -z(e -p - c), Ate- UT1tte(q)qaxe -ax(De(q)axe)
- qeax(UT,te(q))
+ UTzye(q)e(e
-p - c)
= R(e, p) atp
+
UTAp(q)q&xp
-al(Dp(q)axp)
+ qpdx( UTAp(q))
-
UTzAp(q)p(e
-p
-
c)
= R (e P), where z is the inverse square of the normed characteristic Debye length of the device [10].
As in thepreviouswork[4] we willuse a mixedfinite-element methodto approximate q and C simultaneouslyand a modifiedmethodof characteristics [3], [5], [6], [12] to approximatethe densitiese and p.
SEMICONDUCTOR
SIMULATION
541
II
To introducethe modifiedmethodof characteristics for e and p let re =,Te(X, t) be theunitvectorin thedirection(- UT,Le(q)q, 1) and r, theunitvectorin thedirection fora = e or p. Then thederivativesin the (UT8Up(q)q, 1). Set fp = [1 + ( UTla(q)q)2]1/2 Ta directionsare givenby (Pea/are = at-
UT/.e(q)q
pp&/&rp= t+ UT21p (q)qax,
x,
so thatequations (1.lb) and (1.lc) can be writtenin the followingform: (1.2a)
'peae/lTe-
.x(De(q)&xe)-qeax(
+ UTte(q)ze(e-p-c)
U,UTte(q))
= R(e
'pap/lrTp-a,(Dp(q)aXp) + qpax(UT,p(q))-UTttp(q)zp(e-p-c) Now, for a = e or p,
(1.2b)
a,tt (q) = aqla
(q) *d9xq= -(
qg
=
p),
R(e, p).
(q)) z(e - p - C),
so thatthe weak formulationfor (1.2a) and (1.2b) is given as the determination of maps e and p of the timeintervalJ = [0, T] into H1(fl) such that (1a ( 1.3a)
(9e/aere,
+ (De(q)axe,
(1.3b)
(qdapl/arTp
+ (Dp(q)axp,
= (R, ;),
ax) + UTz([
ax;)
-
.Le(q)+ qq(qLe(q))]e(e-p-c),
UTz([,up(q) +
;)
qaq(ap(q))]p(e -p - c), ;)
for ; e H'(fl), and such that the boundaryand initial conditions(0.2a)-(0.2d) are satisfied. First, consider a partition of J into subintervals [tm1, tim],t"
and a partitionoffl intosubintervals [xi-1,xi], 0 =
xi1) = hd. Let
Zh =
xO < X
1,
At otherwise.
Also, note thatim and im cannotbe evaluated exactly.So, let x'm,Atm,xm and At'm relationswhenqm = q(x, tm) is replacedby qhmph be definedby thecorresponding
542
GAMBA
I. MARTINEZ
AND
M.
C.
J. SQUEFF
Then, Let eh and po lie in Zh and approximatee(x, 0) and p(x, 0), respectively. We for m'1 define e^-1= eh('em(x), tm-Ate), and Ph P=Ph(X (X) tm-Atp tm'). Now, we defineem and pm herethat,if Atm= At thene^ = eh(X(x) remark as the unique solutionof the following(algebraicallylinear) equations: ((em
(1.6a)
Uz([ge(q-1 e(q q7(e ?1-eh . ((h pm _
(1.6b)
g
eAM-1)/ltm
-
(h
~
I
h
eqp(q
~
l)]e
-/
-
)
Cm) p
)axPh, a,4)
)+qa? q
UTZ([ A(qh A
)?
a,x)
1)dxem,
(De(q
; ? (D,(qm
-l)/+t
-
?
~
m
)]p e
~
c)
P-fm1
(eh
~
;)
-
For later convenience,we define approximationsto the derivativealong the by approximatecharacteristics he,tJ h+x/= ~~ (1.7a) ~~~~ (1.7b)
~
=a1emUT,e(qh-)qIeQ ~~~Mx=z--c) ~~~ ? /Atp,h &Dtpm
(Pe,heR/areZh
(Pp,h(P
1
teMAMJ,
e
M
m,ax-)
methodto approximateq and +f Finally,we describe the mixed finite-element simultaneously,definedas follows. Firstwritethe potentialequation (l.la) in the followingform:
) andL2(h, (1.7b) againstone in Then, if (1.7a) is testedagainsta functionin H1(qm we findthe mixed weak form: (1.8a)
max (yp
(q, v) - (dv/dx,+f)= rv(1)- rv(O),
[Yai,yri], Let l be partitionedinto subintervals
vE
1()
0 Yo t
