AD-Ri54 824 A SECOND ORDER DIFFERENCE SCHEME FOR TRANSIENT i/iSEMICONDUCTOR DEVICE SIMULATION(U) WISCONSINUNIV-MADISON MATHEMATICS RESEARCH CENTER C A RINGHOFERUNCLASSIFIED MAR 85 MRC-TSR-280i DRRG29-8-C-04i F/G 9/1 NL

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in A i'C) (WRDER If)IPP;Id:NCI: SCHEME

FOR TRANSIENTr SEMICONDUC'TORDEVICE SIMULATION

Christian A. Rinqhofer

Mathematics Research CenterUniversity of Wisconsin-Madison

* . 610 Walnut Street

Madison, Wisconsin 53705

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* Approved for public releaseDistribution unlimited

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UNIVERSITY OF WISCONSIN-MADISON

MATHEMATICS RESEARCH CENTER

A SECOND ORDER DIFFERENCE SCHEME FOR

TRANSIENT SEMICONDUCTOR DEVICE SIMULATION

Christian A. Ringhofer *

Technical Summary Report #2801

March 1985

ABSTRACT

A second order scheme for the solution of the transient fundamental

semiconductor device equations is presented which does not suffer from

timestep restrictions due to the stiffness of the analytical problem. The

* second order accuracy as well as the stability properties are 31eionstrated on

the simulation of a p-n-junction diode.

Accession For

NTIS GRA&IDTIC TABUlanrounced 0Justificatio

copy_ Distribution/Availabiity Codes" Av Il and/orspecial

AMS (MOS) Subject Classifications: 35G25, 35M05, 65M1Key Words: Semiconductors, Singular Perturbations, Finite Differences

Work Unit Number 3 (Numerical Analysis and Scientific Computing)

*Department of Mathematics, Arizona State University, Tempe, Arizona 85287

Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.

0--

_ _ _" .- .

SIGNIFICANCE AND EXPLANATION

In recent years there has been an increasing amount of interest in the

calculation of A-C properties of silicon-based semiconductor devices, mainly

S." due to the rapid developments in fabrication technology and the resulting

miniaturization. This makes the numerical solution of the so-called transient

fundamental semiconductor equations an absolute necessity. Most existing

numerical methods for this problem are only first-order accurate in time

because of the quite complicated anlaytical structure of the equations

('stiffness'). In this paper we present a new method which is based on an

asymptotic analysis of this structure. In contrast to existing methods, it is

genuinely of second order and does not suffer from any timestep restrictions.

q It therefore improves the overall efficiency of A-C calculations significantly

compared to the existing methods.

The responsibility For the wording and views expressed in this descriptivesummary lies with MRC, and not with the author of this report.

-;

* A SECOND ORDER DIFFERENCE SCHEME FORIP. TRANSIENT SEMICONDUCTOR DEVICE SIMULATION

Christian A. Ringhofer

k. 0. Introduction

In this paper %ve prescnL d finite difference discretization method for the

bo~ic sec1iccndictor device equattons in the transient case. These equations-

describing the fieLd, carrier - and current concentration in a semiconductor

* material -are given by (see c.f. Von [oosbroeck [1950])

* (01) =q(n-p-C)

(.) a) V n = q[n t+r, b) V *3 = -qI[p t R]

d 03 ) 3 n q[D 1Vn - -I nV'T], b) 3p = -q[D p .+ j pv'Y]

1he iutdf- ,ndcnt variables in (0.1) -(0.3) have the following meaning:

'V electric potential

.-71Y electric field

-. n negative (electron-) carrier density

p positive (hole-) carrier density

j n electron current density

3 hole current densityp

* c anid q in (0.1) -(0.3) denote the dielectric permiLivity and the unit charge

drId .ire m~aterial conistants. D) n D0 and , denote the di ffusion coefficlents

Cand mobilities of electrons and holes which in general are Functions of the

retric ficld -VTv. It Is assunted that Einstein's relit ion

*Department of Mathematics, Arizona State University, Tempe, Arizona 85287

Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.

(04 u4p L

holds where u T - the tliermal voltage -is only depcndent on temperature and

therefor a co.-stant. C(X+) in (0.1) denotes the doping concentration (the

preconcentration of electrons and holes in the semiconductor). The regions

where C(X+) > 0 anid C(X+) < 0 holds are called the n and p regions resp~ectively

and the manifolds given by C(x) = 0 are called the p-n junctions. R in (0.2)-

denotes the generation - recombination rate of eltectrons and holes anid will in

general be a nonlinear funIction of n,p arid V'P. (See Sze [1969] or Sellcrherc

[1984] for aifferent models for R). (0.1) -(0.3) will be subject to the

initial conditions

(0.5) T~(X+,) TY(') nOx,O) n n(X+), pOX 0) p1 (c+)

* where

*(0.6) CAT' I n1 - Pi C

has to hold because of comipatibility wNith (0.1). This paper is concerned with

the twodjinensional model only. So (0.1) -(0.3) will hold for +x E C R 2. At

Cach point of the boundary 'S2 three boundary conditions are imposed which we

* *rite in the form

1 he shape of' Q arid the 1)01inda ry comidi .ons Bi will depond on the parL icila r

°.- - 7

-3-

device under consideration. A typical case is the following:

(0.8) a= aQ1 U DQ2

(0.9) a) T = V(t), b) n - p n2 c) n - p - C = 0 for x c

(0.10) a) VT • n = 0, b) 3 n n = 0, c) 3 • n 0 for x c 3Q2

where 321 represents Ohmic contacts and a92 an insulator. ,' denotes the normal

vector on the boundary and n is a given constant. V(t) In (0.10) represents

the voltage applied to the device which is varied in time. (i.e. the input is

'switched'.)

The scope of this paper is to develop a finite difference method for (0.1)

- (0.3) which is second order accurate in time. As shown by Ringhofer [19541 b)

the proule is extremely stiff in time and at the sane tinc exhibits an internal

layer behavior In the spacial directions. This causes simple one-step methods

based on central differencing (l.f. Crank-Nicholson) to suffer from prohibitive

timestep restrictions. First we scal the problem appropriately in Sec. 1.

Then - by maKing use of a singular perturbation analysis for the steady state -

and the transient problem carried out in Markorwich and Ringhofer [1984],

4 arkorwich [1984] and Rinyjhofer ['IWj4] b) we develop a dtscretization scheme

which d es not suffcr fre'n iny timIPSt.rp tcstr Lcticns .,nd Olthnujh of fir-; t order

formally is second order accurate for the range of timesteps one wishes to use

in practice. In on unscaled form It Is of the form

I I

L'

* .. ..

-4-

(0.11) Mk<1 q[nkil -k+l C]

(0.12) 1 (nki - n ) 1 k 1 [3 J k+l + k+ 3k+1 1<1 1 N + Rk+]

Atk 4q n n p P 2

(0.13) 1 (pk+l P k 1 [33k+l + + k 1 [R kRk+1Atk 4q P p n n

(J1 n = tDVn" - n ,n '] , 2.: k, k + 1

(0.15) J - q[D'V) + p 7Y] , k, k + 1p p p p

iv,here the sip"rseript 2 denotes a variable at the discrete timelevel t and

At t, t holds. R D e.t.c. denote R, Dn, n e.t.c. with their

arguments taken at t = t . The spacial differential operators in (0.11 - (0.15)

are discretized by standard exponentially fitted finite differences due to

.- ScharfeLter and Gummel [1969]. The spacial discretization is given in Sec. 3.

In Sec. 4. the time discretization is derived and an argument is given that

(0.11) -(0.15) is second order accurate as long as the timesteps are larger

than the dielectric relaxation time. The second order accuracy is verified

§,1 ~i aty in The. 6.

.- .•S=•". - o ."" - -

* 1. Scaling and Singular Perturbation Formulation

i

In this section we scale (0.1) - (0.3) appropriately and reformulate it as a

s inqular perturbation problem. We assume

(1.1) 11 = pi, const

for this and the next two sections. It should be pointed out that (1.1) is

probably a highly unrealistic assumption and therefor inadequate. Assumption

(1.1) is only made for the sake of notational simplicity and in Sec. 5. the

Iscaling and the scheme is given for the case of a general model for the

,nob)iLitics. We scale the independent variables x and t to 0(1). That means

that after scaling the size of Q and the speed with which V(t) in (0.9) is

v.;ri,d is 0MI.

+ 12) ' + z2 (~TJ(1.2) x = PX sl t = u )-ts

Here Z denotes some characteristic device length. We scale n and p such that

their maximal sLze at Ohmic contacts !s 0(1) in (0.9):

(1.3) n = Cn., p = Cp., C:= maxlC(x)j.

For " J nnd J we use the scallng

(1.4 ) IV u 3'V jn qCuT -13 n , = qCu 1j p PS S

........... .'.......... i• .,.., ,.-.

-6-

After scaling (0.1) - (0.3) assumes the form

(1.5) X2A's :ns -p = - Cs, (Cs(X) C (+)

= -P - 5 s b) mapst X)x

(1.6) a) nst =V n - R b) p Ps R s

(1.7) a) 3 = Vn - nsVT b) 3 = - - Vns s PS P s

. The subscripts s will be onmitted from hereon. The initial and boundary

• conditions have to be scaled accordingly. X in (1.5) is a small constant and

i acts as a singular perturbation parameter.

(1.8) (CuT ) 1/2 (q2 2 C)-1 1 2

For practical devices X << 1 will hold. (For instance for a 11i silicon device

with a maximal doping = 1019 cm- 3 X = 0(10- 3 ) will hold.) X, which is

proportional to the minimal Debey length, has been used as a perturbation

parameter in a singular perLurbation analysis of (1.5) - (1.7) in the steady

st.ate case by [arkorwich and Ringhofer [1984] and Markorwich [1934] and for the

Lrinsient problem by Rigjhofer [1984] b). Since we rely heavily on their

re-51jts for the development of our mthod we briefly sketch them in the

following section.

*' ~ -~- - . -. 7 -. 7 'k -. 1.~

-7-

. Analytical Results

In this section we discuss some analytical. properties of the solution of

(1.5) - (1.7). These results will be used in the following secti),:s to motivate

the choice of the discretization scheme. We are motivated by a singular

perturbation analysis carried out previously for the problem (1.5) - (1.7). We

therefor focus on the information about the structure of the solution obtained

by this asymptotic analysis. For the precise results as well as the proofs or

these results we refer the reader to the corresponding literature. It turns out

that the problem (1.5) - (1.7) is singularly perturbed in both the space and the

time dimension i.e. the solution exhibits a layer structure in tihe spacial

variables and can evolve on several different timescales simultaneously. Away

from t = 0 and the layer regions the solution of (1.5) - (1.7) will be given up

to terms of order O(X) by the outer solution w (T',npilJ) of

(2.1) 0 = n - p - C

S "-R , b) P .

(2.2) a) nt V.5 n P

(2.3) a) J Vn - nV'i' b) Jp = -Vp -PVT'.n p

So Poisson's equation degenerates for X + 0 into the condition of vanishing

space charqc. Bccause of this drop in order w can not be expected to satkfy

all boundary conditions (0.7) automatically. It turns out how;,ever that (2.1) is

consistent with (0.9) c) and one of the conditions (0.10) a), b) is satisfied

automatically as soon as the other one is because VC.n = 0 holds on the

.- - .> • -. . . . . . . . . . ..... - . . .

ap))rpriate parts of the boundary 3Q. rherefor all boundary conditions (0.9) -

MO. 1()) can he satisfied by w arid no boundary layers occur (see Markowich

DlWIo4]). We are confronted however with an internal layer structure around the

p-n junctions. The doping profile C() will vary very rapidly across the

p-n-junctLons and in the extreme case is even modelled as discontinuous there.

In either case n and p have to be discontinuous across the p-n-junctions in order

to satisfy (2.1) and this causes the 'full' solution w to develop a layer

there. Fecause w is discontinuous (2.2) - (2.3) can not hold at the

p-n-junction and the reduced problem has to be supplemented by 4 jump conditions

there. These conditions have been derived in Markowich and Ringhofer [1984] and

Markowich, k injhofer et. al. [1983] and are of the form:

-I? Y +( W . 1 ) e , c , J • , P ,

are continuous across the p-n-junction where n denotes the unit normal vector on

the p-n-junction. These conditions say that the currents and the

Pqii-fcrni-puLentLals are preserved across the junctions. Inside the

p-n-jurction Irs ', n and p vary in the 'fast' spacial variable & , d/X %here

(0 1. ,.; die [) We {rrpHdieular distance of a point x from the p-n-junction. They

Srn qk'. by i , p no 1%fying

3.' , ,) p B(s,t)e

-9-

Mihcre s is some pardITltrization of tile p-n-junction and A and Bl are determined

5v the sobit. ion sv of the reduced problem (2.1) - (2.4). J and J do not11 p

exhibit any layer behavior. The spacial. structure of the solution is depicted

graphical.ly in the figure belov.

p - l- jufct ion

(x(s),y(s)) *d

(x1n =A(s,t)e'

p =J w~sx~eyn)

Sn n WXY

J jp p

5o far %c have described the structure of the solution away from, t =0. If thle

(iescriibed asymptotic structure is to hold for all t > 0, the initial functions

,n I and p1[ apparently hdve to satisfy the relations (2.1) and (2.6). Moreover

dif 1ercntiating (2.1) wich respect to t and using (2.2) a) b) yields

.ij ~ ~ 0/* 4J i

Oi ich q(J I an additional condit ion on v, n I anid 1 .*In Harkowich ad

U r'Jr~)Y[ ~ aind rarkowich [199341 it is showi thatL 1-hese condit 1008 aore

,,. i-i if yP n ,, p1 are the solu1t ions ,)f the stt cady state jprobl) m

-23-

6. Numerical Results

In this sect ion vo verify ie stability and conwergr(:e properties of the

discussed s(heme for ti (ise of a p-1- junction diode. The lengthI and depth of

the device are taken to be 11, and 0.25 j respectively. The doping profile C is

taken to the constant 1011 Ch-3 in the n-region and -101 cm- " in the p-reglion.

(So we consider an abrupt p-n-junction). After employing the scaling of Sec. 1

we obtain the system (1.5) - (1.7) posed on a rectangular coman of tire form

Cathode Contact

/ /

/ // p-n-junct ion // / Insulator/ // i/I // // /___ __

Anode Contact

The length ard width of the doma4in are 1 and 0.25 respectively. X turns out to

oe 5.10 - 3 in this case. The contacts correspond to the part l of the

5 ouIfia 1 , . So tle bh urnd it'y coriditions (U . ) ire imposed tihere. "is insoiator

correspnrids to "2 arid (U.10) is imposed on th is part of the boundary'. The

voltirjc' ippLed to Lhe u ld is tljiven by the uif' '(r ,c(i in 1 ' ,el lwer the trio

cuiftulcts cal ibru.,d iji thfe- built, irl i' il. ii , ( .Y.f . S,e 11969J). So tIrei).

I

ui() ui( l r\ f '0 i !ijt,iori f r' i i (J. 1) a) imiluni', ilii' liiru r

.' , - (w t l oi "

-22-

t np

+; i U (vil - liv.:) b)s-)(/ t

Ti p 1 p V p l'

.nrL(5.14) -(5.6) into the discret.itation method (4.13) -(4.1 7),

- .I .-* 7) v to be rtpl aced by

7 A n hiI

1) 1) (x 11 r n

-21-

5. Variable Hobi lities

%s ,m''i.t Lo,"1 in Sec. 1 the assumipt. ion n p const and D D p const

Ps (pite ,reia[Li V. s piinted out in Hock [1981] It might sometimes be

.jP;t 'ydbhL !n steadly stdte simu]lations hilie it is probably inadequate In the

Lransient ease. It is honever rCdsonahle to assume that Einstein's relation

I) D( 5 1) - p - T

r irn p

holds Ahure u1 only depends o tempetLure (and therefore is constant for our

Iwrposrs). \arious models for p n and lp have been proposed (see c.f. Selherherr

1934] Fur an ovurviev). In the 'itost general case p and ip %II1 depend on x

dnd on ,he ..Icctric field - 2; i.e.

)r C lT r r = n,p

To include this Fona of the paramters D nD, pn and p into the model the

scaLim h.s tP he slightly nodified first: i and pp are scaled by some

cldaat''r istic vlcjs and-I t,j and 3 are then scaled bh

t X g2i6-. q 6E~-3 -D:np n p1 .7 n' p p

(5 .3) i: -. (t ([j* )1 /2

(I.',) - ( 1 .7) 1.1 fUn t, o: s

S . . . . "

-20-

(4.10) in the given form since it involves d fourth order difference operdtor

in npace and it L s diff icrilt to impl(mnt nonconstant mobilittes 1n and ip In

this form. We there for rewrite (4.7) - (4.10) using the original variable set

I(:.,n,p,J ,3) and othtain

np

(4.13) dv 'i -- k+1 - k+1 - C

1 kfIl k 1 k+1 k k+1 k k k+1(4.14) (n -n ) - [divh (33 + + + ) - 2R - 2 ]

4 n p p

1 pk+1 k 1 31Mk+1 3k 3k+1 k k 2Rk+1(415 (p -p ) =- _LdiVh( 33 + +3 - 3) +2R k +R

, 4 h p p n n

V) 2. [ ~ . - ( ) yhjI. ]

n L K 1"i )hn - (yII

.=k, k + 1

(4.17) 3 = [K Ip 9 + (lp9")vhT t]

p 11 h

Vie have alreay arAj ked heuristically that the rthod is second order accurate In

L ilor as lorng Is A 2 << At kholds. Thi s will be verifiled numerical ly in Sec. 6.

ior actual cor;putations (4.16) and (4.17) are inserted into (4.14) and (4.15)k k Pk

.11i i J arld . re Jp )or:q ted afLer tthe Vact. Given Y , n 'Ind solving (4.13) -

k+1 k+1 k+0

(4. 17) : L to I viny 3 coupled Pt.D. . 's for I , n arml p at, acth

I. jl;.rc, trp lich ic; Lhe same anmoun t of libor as is rttliren by hackward

FVi 'l' r u '; it. eich t. J ie-, . 1).

-19-

idea: ' c use backward (If'erences for (4.?) a) and centered differences (the

trapezoidal rule) For (4.,') b). .o we choose a nesh 0 = t 0 < t[ ... and

kmeshsizes Atk:= tk+1 - tk' k = 0,I, .... and approximate the gridfunctionsk kk~k-

(Y,u,JJ3) at t = tk by (.w,u,,J ) (4.2) a) - b) is now discretized by:

(4.7) X_ diVh (Vk+l -_k) diV k+1

Atk

(4.b) 1 (u k+ u k ) I div (3k + j!+) k _ Rk+lAtK 2 h u u

(4.9) jk+1 (Mukkl)vh k+1 + K k+1 [h (C+X2 div hV h')]

(4.10) 3u = KV u - M(C + X2 div hVh'') . V 2h .

. k, k + 1 where

(4.11) R2. R( h, n z, p X), KP, K( V ),. zk k + 1

holds. The local discretization error L in (4.7) and (4.8) is given by

(4.12) L = (X 2 At div V 'Ftg At2luk h ii t k ttt

Thus iIL1I const (W2 At + At 2 ) holds. Since e aim to use sVepsizes At >> X2

k k k

:Lii O(Lt) Iolds for 41 prart iral purposes. (lhis Is verified numericallyk

in 5 e.6.) 1 or ntiinr'r icopij rpr! r, it is i nconveniejnt to ootdual ly folve (4.7)-

i

• -1,8-

. i;. j ,.j call be recovred by virtue of

1 X2 1(u; divVh' + C) p - (u- 2 div Vh' - C)

!(j+ j) j 1n 2 u P

the t)brindary conditions (3.17) have to be transformed appropriately. In this

for,' it is clear Uhat (4.2) a) together with (4.3) constitutes the fast time

s,:c1e corix nent hille (4.2) b) together with (4.4) constitutes the slow Lime

.caIe component. Inserting (4.3) into (4.2) a) gives in leading order

(4.6) 12 div hV h -div h[(MU)V h Y] +

S;hirh is the discrete equivalent of (2.12). Since u n + p will be positive

i.od hounded away from zero throughout most of the device Y can evolve on the

0 WWa~b"Ie T = t/A 2 .

.i tie other land u apparently evolves on the ('slow') t-time-scale only

_Inni-i.h ao, is the scale on nhich the applied voltage V in (0.9) is varied. ,e

n refo r ,Mpi oy a sLraLegy which has been successfully enployed by lkreiss and

-hicli i l?751 and iin jiofer 11M4] a). Mat is to use loocr order, onesided,

,-.. A di ii ili: e's for tie" fiHiL i'oinor cits M or the sy;t.emn and to use symmetric

i, lie '',IA;' ilij) (-iiL,', Whi ch d( of hiiglher order ht only A-stahle.

ii ener l he e'.imecd, thaL the ntrnnq sahl itLy properties of the

•: m od (, , h-,][I 1w rt, nd "h il e Us lov.,r orduer acecuracy does not destroy

I, ,",rol ,I w.,rmriv. Ve r,;I riel ourvlves I no lA ,.i11)"t. %er.inm of h is

. ..

-17-

." 4. Dilscretization in Time

',',e noo driC i (iiscrtizatLon scheme for the system of ordinary

differential eq,ations (3.14) - (3.16). It turns out that second oruer methods

based on centered differencing (i.e. the trapezoidal rule or the box-scheme)

c\pcrLence severe stability problems. This is due to the presence of several

different time scales in the analytical problen (1.5) - (1.7) (see Sec. 2). So

the O.D.E. system (3.14) - (3.16) Is stiff. In order to circumvent this problem

(3.14) - (3.16) is transformed into an equivalent system which allows the

separation into a 'fast' and a 'slow' component. For the fast component we use

an L-stable 1st order method (i.e. backward differences). For the slow

componcnt we use an A-stable 2 nd order method (i.e. the trapezoidal rule). This

will avoid any restrictions on the timcsteps. On the other hand it Is well

k,,imon thait in such a situation the use of a lower order method for the fast

component does not destroy the over all (second order) accuracy of the scheme

precisely because of Lhc stiffness (see c.f. Hinghofer [i984J a). So we obtain

a second order method which does not suffer from prohb it ive tLiestep

restrictions. In order to separate the fast and the slow components of (3.14) -

(3.16) we differentialte (3.14) with respect to t and introduce the new variahles

(4.1) u =n + P, J + J, 3 33 3n u n p

we obtain

(4.2) ) A2 divl 7 h t div b) u divQ -N

(4.3) J -(+U)7, 1+ C+ A2 K7 div h V11

r n 1if Ui h h Md h VhAV1Y('.4) Ju ri ii1,

' - (t1")7h'P - A2 (flI mtvh Vh iO l,

- L. i -h

-16-

(3. 14 (dt) J <7n -(*NOV ~h) h

(3. 1) (b) J, P + CAP) Vhi'I

for (X ,yj) E C1

(3.17) 6 TP f pVh ,j, = 0 for x1 fyj) C G~

Thus wve have replared the mixed parabolic-elliptic system (1.5) -(1.7) by the

ODE systemn (3.15) - (3.16) coupled to the algebraic system (3.14). The scope of

0 the next sceLion %%III be to discreLize (3.14) -(3.16) in time by a L-stable

second order finite dif ference scheme.

-15-

_ The fitting factor wiatrix K is a function of Vh ' and is given by

- . .)0

i+lj ij

- (3.11) 0j° ij+l ?ij

z z (ez -1(3.12) c(z):= (e + 1)(ez_ 1)

2

The discretization (3.8) - (3.9) can be motivated in the following way:

For a certain set of boundary conditions (corresponding to the equilibrium

case) J and J will vanish identically and n and p will be of the formI n p

* (3.13) n =Ae , p =Ae - Y

for some constant A. In this case (1.7) a), b) is solved exactly by (3.8) -

(3.9). In Markowich Ringhofer et al [1983] a singular perturbation analysis of

the steady state problem has been carried out and it has bcn shown that K in

(3.11) is precisely the correct fitting factor if (1.5) - (1.7) is viewed in the

framework of exponentially fitted schemes for singularly perturbed problems (see

c.f. Doolan et.al. [1960]. (3.6) - (3.7) together "ith (3.8) - (3.9) together

with the boundary conditions (0.7) now give the discret7,ition of (1.5) - (1.7):

(3.14) X2 dlvh 7hY = n - p - C

(3.15) a) - n div 3 + I , h) L. p -iv +I

vi n 3L h ,,I

-14-

djSicret ized by ecertured di f'furcnces in a straijltforward manner

3) x2 div Y - n p Ch --

(.7) a) n div h - R , h) -t p -divhJ - R

(x.,y.) " CIy G I

' lrc R ij (ni.p..) holds. It Is the cmrent rel-.tions (1.7) h), h) wIch

h have to he discret Lzed wiLh care. As mentioned in the previous section layers

'an be 'xpected to occur in ', n adri p across tne p-n-junction (where C(x,y)

harirjcS its sign rapidly). ilarkowich and Rlingllofer et.al. [19'3] [ oirit out that

a mesh which equidistributes the local discretization error wilL inevitably

"or,itai .1 proh ihitive cinourit of points if the p-n-junction does not lie

Pardll Hi to the gridline direct ions (whic-h can not be expected in general).

* mrfetter and Cummel [1969] sggested the following discretization of (1.7)

, )

;. ) 3 [ ,hn - (!111) • 1 ''

n Ih

,.. ; , r !, r 'I i-. ,f,' il I 5v

0') l i j. i Ij "ij0 n p- tJ p r1(i[h(iJ ' ' j I 'I ij

riiei ~

0 ""* . ' " . 1

-13-

x X. j(3.) J s s ' 2 si+i/2, J

Jy( J + J+) jy (t), s = n,p..'.2 Si,j+1/2

In order to write the discrete approximations of the spacial differential

operators in (1.5) - (1.7) in a convenient form we introduce the following

dirf',rence operators: For a vectorfield V given on the intermediate points of

the grid G we define the discrete divergence operator by

* (3.4) div V' 2(6x f-Sx.i_)- 1(v + ,- v ) +

2 (Syj+-yj_ 1)1 (vy 1 i 2 - vy', ) wherej j- (vJ,11 ,j-112

,'Y ) holdsv ij (V (i+I/2,j' 'i,j+1/2 hod

for a scalar :ridfunction u we define the discrete gradiend 7h by6h

.'" (3.5) VhUi: (dhU +l2 J dyui~+

11 d i/ i I i+'I/,j' '

h Hi+ j 6x (I i+l,j- uij

Fedyu 6~y, (u Uh i,j+1/2"= 3 i,j+lu-ij

Poisson's equdt.ion (1.5) and the continuity equations (1.6) a), b) are noo

-• -- -- 12-

3. Spacidl Discretization

In this sect ion *e di',ciss the discret za Lion of (1 .5) - (1.7) in the+

spdCial variable x leavirij iic time t continuous. Thus (1.5) - (1.7) Is

replaced hy a large sparse system of ordinary differential equations. We

,utilize an expoenLially fitted finite difference scheme due to Scharfetter andCummel [1969] which has been widely used for the solution of the steady state

problem (2.8) - (2.10). First we choose a nonuniform but rectangular mesh in

space:

(3.1) C {(xi,yj), i = O)N, j = 0(1)} with

0 x 0 < x1 < .... < x -

0 YU < Yl < .... < yt : d

6x x - x , i O(1)N-1, 6y y - y , j = 0(1)M - 1i i+1 i S i+l S

For obvious reasons it is convenient to distinguish between boundary and

interior points. So we define

t (3.2 Gr= {x.,yj), 1 z 1(1)N - I, j 1(i)ri - 1, GB G I

'Y, n aind p) arc ipproximated hy their values at thte gridpoints (xi ,y .)0J

I .

L p

[--., p itd J are approxite at the midpoints etne t'o0

in leading order

(2.12) 0 - (n+p)AT + X + 21 Z.o.t.

t/X

In Ringhofer [1984] b) a complete asymptotic expansion of the solution w on the

fast timescale T = tIX 2 is given in the case of 1 space dimension. It turns out

that only f and the drift currents -nVY, -pVY can evolve on the fast tirnescale

while n and p can only evolve on the 'slow' t-tinescale.

n p-C

".-. -10-

. 2.8t) XA n -p - C

(2.9) a) V.3 R , b) 7-3 -Rn 'p

- - (2.10) a) Jn Vn - nVY , b) V-3 : -Vp - pV'

In Hinghofer [1984] b) it is shown that each time (2.7) is violated the sol.jtion

w will evolve on an additional 'fast' timescale t/X 2 Which corresix)nds to the

dielectric relaxation time in unscaled forii. This is of importance For the

. uiumerj.cal solution of (1.5) - (1.7) because it means that the problen is stiff

in tine and special stability (i.e. L - stability) properties will be required

of discretization methods. In !inghofer [19841 b) an asymptotic expansion for

the solution on all timescales and for general initial functions has been

* carried out. We only briefly sketch the results here: It is beneficial Lo

* transform the system (1.5) - (1.7): Differentiating (1.5) with respect to time

yields

(2.11) X2 AT t 7.(Jn +

(<?.11) staites the conservation of the total current consisting of the drift and

di I'FuS ion curirenL for el(ct'Uons and holes ( :J -) and of the displacement

,jrrent -A' '. it is ki knoln that the dis iac1CiIt currCnt can evolv on

the timescale of the dlelectic relaxat[on time. Whenever V.[j n +Jp O(x2 )

," t jls v,ty from the p-n-juncLion, %%here All is bounded for X40, ' will evolve on

-"lI inesci e t X [nert ifng the expressitons for a arid 3 from (1.7) givesn p

L- •

-24-

(.1) ) (x,t) v1Tb.(X) - (t) X at the cdthodeb 2

Ne siil11tec a siitch from u -- 0 (equilibrium) to u = -0.5V (forward bias) In a

t i;iciutrvat Iof 10- q sec which corresponds to 0.1 in our scaling. For the

( recombination ratu the ShockLy Head IHall - recombination turin

.p - n2.

(5.?) ix -

r kfl u .I + T I I

has heen tuscd. (n., Iud r are given constants.) The nonlinear system of

1,1:r,' ,,' ) arising in (4.13) - (4.17) at each timestep has

h eeiS sowved by I UV, ton's meUthod using a damping strategy due to Deuflhard

L )74J. lie purpose of this tesloxample is twofold: Firstly it should verify

the st,i)11ity proper:ties mntioned in Sec. 4, i.e. we should observe that no

oscillations occur if At >> X2 is chosen at any point in time. Secondly itk

shotld verify the second order accuracy in time. Therefor a fixed mesh is

chosen iq the spacial direction and it is verified numerically how well the

-, sy stm of ordiniary differential equations (3.14) - (3.16) is approximated by

-.. (4.13) - (4.17) by varying the timnesteps Atk , So we choose a rectangluar

0

(,,.;) {:(x.,y.) , i -O( )n , j = U(I)in

.," I.,( 1r, idt ie~lre ,:lustered around the p-n-junct'ion. The maximal mesh

_t",~~ ~~ ~ ~ ~ ~~~~~~ .. ........-.... ......- ::.(. - ..

ii 1111 (1I~l5(fI a

.77

-25-

6x:= rx{6×:x x[+1 -. , I = O(1)n - 1} 0.05

." (6 )

6y:-: r yj "yj Yj+ j o(1)m - 11 0.05

Then a solution is computed on a quite course mesh In time which is obtained by

hounding the growth in the drift - and diffusion current density: I.e.

1 k+1 _ IkI2 < wljJ k 12

(6.5)

j2 =j + = k, k + 11 p

is required (it each .tep Yiith some suitable parameter w. (Here l1 -12 denotes the

r,Ate L2 -nurm.) Using this strategy the coarsest mesh

(6.6) TO:-- {t? : i = O(1)K}

i is obtained. lnen the meshspacings are halved successively to obtain the meshes

".-. [[Fl T 2 . .

.7;t : i = O(1)2{K} ,

t I = 2jK -,( + ) = 2j 1

[ +"

020

then the ol'jtion on the fine st mesh is rc(lr(ed as 'exact' solu , iM and Lhe

re.-tive differences (in the discrete 1.2 no rm) to the coarser-mesh sol,,tions are

romp ut.ed. If Lhe method is of second order Lhey shoul.d decay asymptotically by

a :ctor- 4 from I to I ]able 1 gives the relative errors in the L2 norm:2.-. 2.+1 "icsL

-ABLE 1

PSI N P 3j + 3p

0 U.33E-01 0.89F +00 0. 15 1-0O O. 58L+00 0.39E+00

1 0.10L-01 0.171 .00 0.461-01 19 ".-0) 0.12?1400

S? .1 6L -02 .271 -01 O.74[-02 0.291.L-01 0.20E-01

3 0.17L-03 0.6?E-02 0.77E-03 0.31E-02 0.20E-02

4 0.39E-04 0.11611-02 0.18E-03 0.80E-03 0.45E-03

5 0.84E-05 0.38[-03 0.38E-04 0.21E-03 0.94E-04

A, d test (3.14) - (3.16) has been solved on the same meshes in the spacial and

the Lime direction by using the same spacial discretization but simple backward

diffe'r -ences for nt and Pt in (3.15) a) b). The relative differences to the

finest-nrid-solut ions for Lhis case should decay by a Factor 2 from T to T

e'aise of the first order accuracy. They are given in Table 2:

[ABLE 2

PSI P 3=3 +3 31P

flp

* .1 i -01 0.131401 0.521 -01 0.201 UO 0. I'L 4 0

U) *; ,'i -I),? U. 75! 0 0. )i -01 0. I,4I00 i,. M - I

0. 37i -W? 0. 19lH-00 0.17L-01 0.641--01 0.45L-01

? 1.'I '00 0.831 -0 0.3?1 -01 0.)3i -0 1

0" 11 -03 H .31 -01 0. 37[ -0? 0.141 -01 0"101 -01

) I A l -(3 0. 1 -()1 "1 -0? 0.4F1 -0 0. 341 -0'

"0. .- '.. . . , : % . : " . % % " . . . : " : : " . . o - - o

-27-

Comparing Tables 1 and 2 it can also be seen that to compute C.f. 3 3 n 3n p

with a relative aiccurdcy of 1% 4 times as many timesteps are required by

backward differences shown by the proposed method. In Figures 1 - 9 plots of

the solution at various time levels. The switch from u O.v to v 0.5v is

- performed in the interval from t = 0.1 to t = 0.2.

Figure 1 shows the initial solution 't'(x,O) which has been taken as the

steady state solution for u = O.v (equilibrium).

SFigures 2 - 8 show the solution at t = 0.2 (tile end of the switching interval).

In particular they show

Figure 2: '( ,0.2)

Figure 3: n( ,0.2)

SFiqure 4: p( , 0.?)

Figures 5 and 6: The x and y components of

j n(X,0.2)

Figures 7 and 8: the x and y components of

2p (,0.2)

Figure 9 shows the total current flow through the cathode contact as a

function of time:

STo t(t) f L + ) dL ca thI o de 1

ohcre n Is the norma I vector to the cathode.

[0 .

-28-

10.

4 -29-

0

//

0.7

I9

~O- D.

-30-

Il 1Ui)I I N

1.10 -

0.88

0.66~

.4 q ~<

0.221

0.00/

0

.00-31-

0.00

0.002

S 0.008

rl

-32-

I 1.60]

1.28 -

I 0.09

08b

-33-

0.40 -

0.08

-34-

TI IfVZ 7 (.T ]Dv)

OJooo 88 -

0.00044 -/\/

f_10O 0~ ~ . ~-

-35-

010 6

304-

228-

152-

76~

-36-

oo

0

0

cla

1 CV

CIO .-

0

00

0 CD

-37-

RE-ERENCES

1. I).u fll.ird, P. [1974]: 'A modified Newton method for the solution of ill,'ol ditjoned systems of rionlinear equations with applications to multiplesnlmotinj', Huir. Math. vol 22, pp 289-315.

o. DoLan, E.P., Miller, Y.Y.H., Schilders, W.H.A. [1980]: 'Uniform numericai

methods for problems with initial and boundary layers', Boole Press,Dub iL n.

3. Harkowich, P.A. [1984]: 'A singular perturbation analysis of the

fundoinental semiconductor equations', SIAM J. Appl. Math.

4. Harkowich, P.A., Ringhofer, C.A. [1984]: A singularly perturbed boundaryvalue problem modelling a semiconductor device.

5. Hlarkowich, P.A., Rinqhofer, et. al. [1983]: 'A singular perturbationapproach for the fundimental semiconductor equations', IEEE Trans. onLlectr. Uev., vol LU-3D, pp 1165-1181

6. ;iock, P.S. [I. 3]: 'Analysis of rratherrtlcal models of semiconductordevices', Hoole Press, Dublin.

7. Hinqhofer, C.A. [19841 a): 'On collocation schemes for singularlyperturbed two point boundary value problems'.

. irmncofer, C.A. [1964J b): 'An asymptotic analysis of a transientp-ii-junction model', submitted to SIAM J. Appl. MaLh.

9. Wl(erhcrr, S. [1984]: 'Analysis and simulation of semiconductor devices',S pr i ,njer.

10. Sze, S.M.. [1969]: 'Physics of semiconductor devices', Wiley -ti terscience.

II. ,an Hoosbroeck, W. [1950]: 'Theory of flows of electrons and holes in; f.r;4kii,,m irid other semiconductors', Bell. Syst. Tech. I., vol 29, pp56 o0- G 7.

SECURITY CLASSIFICATION OF THIS PAGE (When Data EntereNU

REPORT DOCUMENTATION PAGE READ CSORTIN GoBEFORE COMPLETING FORM

1REPORT NUMBER 2. GOVT ACCESSIONNO3.RCPET$AALGUMR:, ~~2801L l: ', L

( 4. TITLE (and Subtitle) S. TYPE OF REPORT & PERIOD COVEREDSummary Report - no specific

A SECOND ORDER DIFFERENCE SCHEME FOR TRANSIENT reporting period

SEMIC.jNDUCTOR DEVICE SIMULATION 6. PERFORMING ORG. REPORT NUMBER

7. AUTHOR(a) S. CONTRACT OR GRANT NUMBER(@)

Christian A. Ringhofer DAAG9-80-C-0041

9. PERFORMING ORGANIZATION NAME AND ADDRESS 10. PROGRAM ELEMENT, PROJECT. TASKMathematics Research Center, University of AREA A WORK UNIT NUMBERS

Work Unit Number 3 -610 Walnut Street Wisconsin Numerical Analysis andMadison, Wisconsin 53706 Scientific Computing

11. CONTROLLI.MG OFFICE NAME AND ADDRESS 12. REPORT DATE

U. S. Army Research Office March 1985P.O. Box 12211 IS. NUMBER OF PAGES

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16. DISTRIBUTION STATEMENT (of thle Report)

Approved for public release; distribution unlimited.

17. DISTRIBUTION STATEMENT (of the abotroct entered In Block 20, if different from Report)

IS. SUPPLEMENTARY NOTES

19. KEY WORDS (Continue on revere de If neceeary aid Identify by block number)

Semiconductors, Singular Perturbations, Finite Differences

20. ABSTRACT (Continue on reveree aide If neceeeary end Identify by block number)

A second order scheme for the solution of the transient fundamental

semicondiu'2tor device equations is presented which does not suffer from

timestep restrictions due to the stiffness of the analytical problem. The

second order accuracy as well as the stability properties are demonstrated

on the simulati-n of a p-n-junction diode.

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7-85

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