Control of Crystal Growth in Bridgeman

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PergamonProg. C/ystalGro?/thand Charac t. Vo l 30 , pp . 217-236 . 1995

1995 Elsevier Science LtdPrinted in Great Bdtatn

0960-8974 /95 $29 .00

0960-8974(95)00006-2

C O N T R O L O F C R Y S T A L G R O W I "H IN B R I D G M A N F U R N A C E

C. Batur,* A. Srinivasan,* W. M. B. Duvalt and N. B. Singh¢

*Department of Mecha nical Engineering, University of Akron, Akron,OH 44325-3903, U.S.A.

1-Processing Scienc e and Te chno logy Branch , Materials Division, NA SA LE-WlS ResearchCenter, Cleveland, OH 44 135, U.S.A.

:l:Westinghouse Res earch and Dev elopm entCenter, 1310 Beulah Road,

PiMsburgh, PA 15235 , U.S .A.

A B S T R A C T

Co ntro l o f crys tal qual i ty dur ing crys tal g ro wth requires accu rate implem entat ion o f

thermal bou nda ry condit ions . W e identify th is problem as the furnace tem perature contro l

problem. Th e thermal bound ary conditions , in tu rn , d icta te the in ter face shape betw een the

so lid and the l iqu id reg ion o f the m ater ia l . Determ inat ion of the bounda ry condit ions for a

g iven des ired in ter face shape is considered as the materia l tem perature con tro l p roblem in

th is paper . W e out l ine the c ur ren t ef for ts for the so lu t ion of the furnace tem peraturecontro l and the materia l tem perature contro l p roblems. W e res tr ic t our rev iew to

Br idg m an grow th contro l techniques.

1. I N T R O D U C T I O N

De nsi ty o f crys tal defects , poly-crys tal lizat ion , hom ogenei ty densi ty of im pur i ty a toms and

non-un iform dis tr ibu t ion of a d opant material are v i ta l m easures def in ing the qual i ty of

gro w n crystals, Ge velbe r et al [ 15], [ 16]. Control of crystal quality w hile the crystal is

grow ing ins ide a furnace requires the m easuremen t of quali ty . A lternat ively , w e need in-

s itu m easurem ents that can be uniquely rela ted to qual ity. Fo r example, i f the crys tal is to

be u sed as an acou sto opt ical tunable f il ter, the aco ust ic and opt ical p roper t ies de f ine thequal i ty , therefore , som e mea surem ents re la ted to the acousto-opt ical p roper t ies should be

received in-si tu by the grow th co ntro l a lgor ithm.

Th ere are pract ical d if ficu l ties associated with the def in it ion and the m easure m ent of

quali ty . M ost of the t ime a set o f variab les def ines the quali ty and they are not a lw ays

access ib le dur ing growth . I f they can be mea sured after the grow th then the s ta tist ical

proces s con tro l techniques can be used to provide pract ical so lu tions to qual i ty contro l

problem. H ow eve r , th is is an off -l ine contro l methodo logy and the opt imizat ion is on ly

poss ib le a t the expense o f cos t ly tr ia ls. F rom the ma nufactur ing contro l po in t o f v iew, i t is

21 7



2 1 8 C . Ba t u r e t a l .

prac t ica l to ident i fy a var iable tha t wi l l provide a reasonable m easure o f qua l i ty whi le the

crysta l is growin g.

The shape o f so l id - liqu id in te r face is comm only accep ted a s an ind ica to r tha t c an be

re la ted to c rysta l qua li ty . Th e shape inf luences the c rysta ll ine perfec t ion and com posi t ion a l

hom oge nei ty . A crysta l tha t grow s wi th a f ia t in te rface has minimal therm al s t resses

becau se the the rma l cha nges a re in one d im ens ion on ly , Fu and W i lcox [ 13 ]. Du t t a e t a l

[10] ob serv ed tha t G aSb crysta ls gro w n wi th a fia t mel t -so l id in te rface exhib i ted very low

dis loca t ion dens i t ie s due to the reduc t ion o f the rma l s tr e sse s a t t he in t er face . The re fo re , i t

m akes p rac t i c a l sense to con t ro l the shape o f in t e r face by m an ipu la t ing the the rm a l

bo und ary con di t ion s surro und ing the in te rface . I f a cer ta in in te rface shape is desi red , there

is a corres po ndin g tem pera ture d is t r ibut ion inside the m ater ial tha t gen era tes th is spec i f ied

in te r face shape . T he t a sk o f the in t er face shape con t ro l l e r i s to e s t ab l ish th is t em pe ra tu re

d i s tr i tmt ion . We iden ti fy th i s p rob lem a s the m a ter ia l t empe ra tu re con t ro l p rob lem. The

required te m per a tur e d is t r ibut ion or , equiva lent ly , the desi red in te rface shape can b e

accom pl i shed by m an ipu la t ing the boun da ry cond i t ions a round the m a ter ia l . The se

bou nda ry cond i t ions are e s t ab l ished by the ~ rn ac e t empe ra tu re con tro l le r . The

tem pera tu re con t ro l p rob lem in the fu rnace is s imp le r than the t emp era tu re con t ro l

p rob lem ins ide the ma te ri al . W e iden t ify th is p rob lem a s the fu rnace t em pera tu re con t ro l

p rob lem.

F igu re 1 sho w s the func t iona l b locks o f the c rystal g row th con t ro l le r . I f t he in t e r face

. shape o r the in s ide ma te ri a l temp e ra tu re s can be measu red , i t is com pared w i th the de s i red

in te r face shape o r the de s i red ma te ri a l t empe ra tu re s and the re su l ting e r ro r s igna l i s sen t to

the ma te r i al t em pe ra tu re con t ro l le r . T he requ i red bounda ry cond i t ions o r equ iva len t ly the

fu rnace t emp era tu re s a re d e te rmined by the ma te r ia l t emp e ra tu re con t ro ll e r . The boun da ry

cond i t ions a re u sed a s the se t -po in t t empe ra tu re s fo r the fu rnace t em pera tu re con t ro l le r . I t

may be n o t i ced tha t the fu rnace t empera tu re con t ro l sys tem is on ly a m inor con t ro l loop

wi th in the f ram ew ork o f the in t e rface con t ro l sys tem. The ac tua l boun da ry cond i t ions

e s tab li shed by the fu rnace t empera tu re co n t ro l l e r de te rmine the t em pe ra tu re s in s ide the

ma te r i a l t h rou gh th e dynam ics o f the c rys ta l g row th mechan i sm.

I f the in t e r face shape can no t be measu red , i t is imposs ible to con t ro l the shape o fin te rface w i th a f eedback con t ro l system. We can on ly con t ro l the var iab les tha t w e can

mea su re o r e s t ima te . How eve r , u nde r ce r t a in c i rcumstances , it is poss ib le to r ecove r the

in te rface shape in fo rma t ion by pa rt ia l m easu remen t s . Fo r example , i f t he ou t s ide w a ll

su r face t em pera tu re s o f the c ruc ib le can be measu red , then , th roug h a m ode l o f the g ro w th

dynam ics, the sh ape inform at ion can be part ia lly recovere d . This inform at ion then can be

fed back to the m a te ri a l tem pe ra tu re con t ro l le r a s in the p rev ious ca se .

Th e b lock d iag ram o f F igu re 1 a lso h igh l igh t s the fac t tha t fo r a g iven des i red in te r face

shape o ne can de te rmine m ore than one t emp era tu re d is t r ibu t ion in s ide the m a te ri a l tha t

co r re s pon ds to the sam e shape . In o the r words , t he re may no t be a phys ica lly real iz ab letem pera tu re d i s t ribu t ion fo r a g iven in te r face shape . Th i s non-un ique ness p rob lem can be



C o n t r o l o f C r y s t a l G r o w t h

reso lved by in t rod uc in g a f i l te r tha t determines a phy s ica l l y rea li zab le tem pera ture

d i s t r i b u t i o n i n s i d e t h e m a t e r ia l f o r a g i v e n a r b i tr a r y d e s i r e d i n t e r f a c e s h a p e .

2 1 9

--•RE-FILTER

\ DESIRED\ INTERFACE\ SI-L~PE

DESIREDIV~TERtAL IEIV~. REQUIRED ACTUA LT r BOUND~J'~Y BOUN DARY

C O N D ~ C O ND IT IO N S, T z

I V l A T E R I A L rE r ES, r

R

, a / d~ SURFACE IF::MpI:RAIU~S

OR Iqlrd~AC~

MEASURED OR ESI11vIATEDMATE RIAL TEMPERATURES

E S T I M A T O R M E A ~ M E N T s Y s T E M- - -

Figure 1 . Crys ta l g row th con t ro l system.

Th e des i red in te r face shape i s spec if ied by the opera to r . The p re - f il te r de te rmines the phys ica l ly

real izab le tem pera tu re d is t ribu t ion ins ide the mate r ia l . The mate r ia l tem pera tu re co n t ro l le r

es tab l i shes the requ ired bo undary cond i t ions which , in tu rn , a re im plem ented by the fu rnac etem pera tu re con t ro l le r . The es t imator es timates the ac tua l m ate r ia l temp era tu res g iven par t ia l

me a s u r e m e n t s s u c h a s t h e a m p o u le ' s o u t s id e s u r f ac e t e mp e r a tu r e s o r t h e i n te r f ac e s h a p e

T h i s r e v i e w d o e s n o t a d d r e s s t h e c o n t r o l p r o b l e m s t h a t a re s p e c if ic to t h e m a t e ri a ls g r o w n

a n d i s l im i t e d t o f u r n a c e a n d m a t e ri a l t e m p e r a t u r e c o n t r o l p r o b l e m i n B r i d g m a n g r o w t h

t e c h n i q u e s . F o r a r e v i e w o f m a t e r ia l re l a te d g r o w t h p r o b l e m s , t h e r e a d e r i s r e f e r r e d t o

P e t r o s y a n [ 3 2 ]. F o r t h e c o n t r o l l e r d e s ig n i s su e s re l a te d t o t h e C z o c h r a l s k i p r o c e s s ,

G e v e l b e r a n d S t e p h a n o p o l u s [ 17 ] , [1 8 ] p r o v i d e a n e x c e l le n t d i s c u s s io n o n t h e m o d e l

b a s e d i n t e r f a c e c o n t r o l .

T h e o r g a n i z a t i o n o f t h e p a p e r i s a s f o ll o w s . C o n t r o l o f f u rn a c e t e m p e r a t u r e s , i .e ., t h e

m i n o r c o n t r o l l o o p o f F i g u r e 1 , i s d i s c u s s e d in S e c t i o n 2. B o t h m o d e l b a s e d a n d

c o n v e n t i o n a l P I D t e m p e r a t u r e c o n t r o l le r s a r e r e v i ew e d . T h e m a t e r i a l t e m p e r a t u r e c o n t r o l

p r o b l e m i s p r e s e n t e d i n S e c t i o n 3. T h e m e a s u r e m e n t o f i n te r f a ce s h a p e a n d t h e i n t e r fa c e

s h a p e c o n t r o l l e r d e s i g n a r e g i v e n h e r e f o r t h e v e r t ic a l B r i d g r n a n c o n f i g u r a t i o n . F i n a ll y ,

c o n c l u s i o n s a r e s t a t e d i n S e c t i o n 4.

2 . C O N T R O L O F F U R N A C E T E M P E R A T U R E S

2 . 1 D Y N A M I C M O D E L O F H E A T I N G Z O N E S

F o r a m u l t i - z o n e c r ys t al g r o w t h f u r n a ce , t h e d y n a m i c s b e t w e e n t h e z o n e t e m p e r a t u r e s a n d

t h e e n e r g y i n p u t t o h e a t i n g z o n e s c a n b e e x p r e s s e d b y a li n ea r m o d e l a s



2 2 0 C . Ba t u r e t a l .

y ( t ) = A l y ( t - l ) + A z y ( t - 2 ) + . . . + A . y ( t - n ) + B i u ( t - 1 ) + B 2 u ( t - 2 ) +

+ . . . + a . u ( t - n ) + e ( t ) + C t e ( t - 1 ) + . . . + C , e ( t - n ) 2.1.1

w h e r e y ( t ) e R p is a ( p ) d i m e n s i o n a l v e c t o r i n d ic a t in g z o n e t e m p e r a t u r e s f o r a ( p ) z o n e

f u r n a c e . S i m i l a rl y , t h e v e c t o r u ( t ) e R p d e n o t e s t h e h e a t e n e r g y to ( p ) h e a t i n g z o n e s .

S i n c e t h e h e a t e n e r g y f l o w i s p r o p o r t i o n a l t o t h e v o l t a g e a p p l i e d t o e l e c t r i c a l h e a t i n g

e l e m e n t , t h e t e r m u ( t ) c a n b e a l so c o n s i d e r e d a s t h e i n p u t v o l t a g e t o h e a t e r s . T h e

t e m p e r a t u r e c o n t r o l h a r d w a r e r e a d s th e t e m p e r a t u r e s y ( t ) a t t i m e ( t) , i m p l e m e n t s th e

f u r n a c e t e m p e r a t u r e c o n t r o l a l g o r it h m d e s c r ib e d b e l o w a n d g e n e r a t e s t h e c o n t r o l l e r o u t p u t

u ( t) . T h e u n c e r t a i n ty o f th i s l in e a r m o d e l is r e p r e s e n t e d b y t h e t e r m s a s s o c i a t e d w i t h

e ( t) . T h e t e r m e ( t ) i s a r a n d o m v e c t o r w i t h z e r o m e a n a n d f in i te v a r i an c e . T h e

u n c e r t a i n t y i s d u e t o s e v e r a l fa c t o r s s u c h a s t h e l i n e a r iz a t i o n o f t h e a c t u a l n o n - l i n e a rh e a t i n g d y n a m i c s , a f in i te n u m b e r o f c o e f fi c ie n t s i n t h e m o d e l , a n d o t h e r e x t e r n a l i n p u t s

s u c h a s i n f i lt r a t io n a n d r a d i a t i o n f lu x e s w h i c h a r e n o t e x p l i c i tl y c o n s i d e r e d i n t h e m o d e l .

I f th e r a d i a t i o n h e a t e x c h a n g e b e t w e e n t h e h e a t i n g z o n e s i s v e r y s t r o n g , t h e l in e a r m o d e l

r e p r e s e n t e d i n ( 2 . I . 1 ) w i l l b e a v e r y w e a k I oc '~ l r e p r e s e n t a t i o n o f t h e a c t u a l d y n a m i c s . T h e

c o e f f i c i e n t m a t r i c e s ( & ; i = l , 2 , . . . n ) , ( B i ; i = l , 2 , . . . n ) a n d ( C i ; i = l , 2 , . . . n ) , a r e a ll ( p x p )

m a t r i c e s t h a t n e e d t o b e d e t e r m i n e d e x p e r i m e n t a l ly f o r a g i v e n f u rn a c e . I f t h e h e a t i n g

z o n e s a r e w e l l i n s u l a t e d f r o m e a c h o t h e r , t h e r e i s in s i g n i fi c a n t t h e r m a l i n t e r a c t i o n b e t w e e n

t h e m . I n t h i s c a s e , t h e m a t r i x c o e f f i c i e n t s ( & , Bi, C i ) a r e s im p l i f ie d t o d i a g o n a l m a t r i c e s .

T h e p r o c e s s i d e n t i f ic a t i o n t e c h n i q u e s c a n b e u s e d t o e s t i m a t e t h e c o e f f i c i e n t m a t r i c e s ,L j u n g [ 2 7 ], S o d e r s t r o m a n d S t o i c a [ 4 0] . I n t h e s e te c h n i q u e s , th e h e a t e r i n p u t s i g n a ls u ( t )

a r e c h o s e n s u c h t h a t t h e z o n e t e m p e r a t u r e s y ( t ) a re d i s tu r b e d s li g h tl y a r o u n d t h e i r n o r m a l

o p e r a t i n g t e m p e r a t u r e s . I n o r d e r t o o b t a in c o n s i s t e n t e s ti m a t e s o f t h e m a t r ix c o e f f i c ie n t s ,

t h e i n p u t s i g n a l s u ( t ) a r e r a n d o m s e q u e n c e s . T h e r e s u l t in g i n p u t o u t p u t d a t a ( u ( t ) , y ( t );

t = l , 2 , . . ) a r e s u b s t i t u t e d i n t o ( 2 . 1 . 1 ) a n d t h e m a t r i x c o e f f i c i e n t s ( A i , B i , C i ) a r e e s t i m a t e d

b y t h e l e a s t s q u a r e s m i n i m i z a t io n te c h n i q u e . T h e p e r f o r m a n c e i n d e x o f t h e l e a st s q u a r e s

a l g o r i t h m i s t h e m i n i m i z a t i o n o f t h e s u m o f s q u a r e s o f t h e r e s i d u a l s , i .e . ,

N

^ 2 ( t ) + ( t )= ~ . e ~ . . . p

t= l (2 .1 .2 )

w h e r e N i s t h e n u m b e r o f s am p l i n g p o i n ts d u r i n g w h i c h th e i d e n t i fi c a ti o n d a t a

( u ( t ), y ( t ); t = l , 2 , . . N ) i s c o l l e c te d a n d e , ( t ) i s t h e it h c o m p o n e n t o f t h e r e s i d u a l v e c t o r

~ ( t ) , d e f i n e d f r o m (2 . 1 .1 ) as

e ( t ) = y ( t ) - . 4 , y ( t -

1 ) - . . . - ~ 4 . y ( t- n ) - B , y ( t -

1 ) - . . . - B . y ( t- n ) - C ~ ( t -

1 ) - . . .- ( ~ . ~ ( t - n )

( 2 . 1 . 3 )



Cont ro l o f C rys ta l Grow th 221

2 .2 M O D E L B A S E D T E M P E R A T U R E C O N T R O L L E R S

If the crucib le is t rans lated axially , as in the c ase of mo st Br idgm an furnaces , the ax ial

tem pe ratu re distribution inside the furnace is kept constant. T here fore, the set-po int

tem peratu res for the furnace tem perature contro l sys tem should also remain constan t . Thisi s known as the tempera tu re r egu la tion p rob lem. In the case o f E lec t ro Dynam ic Grad ien t

(ED G) b ased furnaces , such as the Mellen furnace of Parsey and Thiel [31] , R ajendran and

M ellen [34], the axial temp eratu re distribution is translated wh ile the cru cible remains

constan t . T he crys tal g ro wth rate is determined by the speed of the m oving tempera ture

gradien t . In th is case, the set-poin t temperatures for the tem perature co ntro l sys tem

chan ge with respect to t ime. This is a lso the case for Rapid Therm al Process ing (RTP )

w her e a specif ic tem perature- t ime profi le has to be implem ented with in a few se conds ,

El la [ 11 ] Th e c ontro l ler des ign problem in th is case is mu ch m ore chal lenging and i t is

know~ as the se rvo p rob lem in con t ro l s. T he g row th r a te fo r mos t c ry s ta ls in Br idgman

furnaces is very s low, typ ical ly in the range o f a few ram/hour. In contras t , m ost

tem pera ture contro l lers w ork on the sampling per iod ranging f rom 0 .5 to 10 seconds .

The refore , the required changes in the set-pqin t tem peratures for the E D G type furnaces

are no t very fast for a typ ical tem perature contro l ler des ign problem. This ph eno m enon

just i f ies the use o f the regulator type contro l ler des ign for bo th the constan t thermal

gradien t and the var iab le thermal grad ien t (ED G) type furnaces. Ho we ver , for fas t

grow ing crys tals , the furnace temp erature contro l ler des ign should be based on the servo

con trolle r design principles. In the next sections, w e will review the con trolle r design

techniqu es for the furnace temp erature contro l p roblem.

2 .2 .1 S IN G L E I N P U T S IN G L E O U T P U T T E M P E R A T U R E C O N T R O L L E R S

This is the c ase w here the re is ins ignif ican t in teract ions betw een heat ing zon es an d the

contro l ler des ign can be per forme d separately for each zone. From (2 .1 .1) , dy nam ics of

each zone can be w r i tt en as

y(t) = aly (t-1) +a2y(t-2)+...+ a.y(t-n)+ blu(t-1 )+ b2u(t-2)+

+. . .+ b .u( t-n)+ e( t ) +cle ( t -1)+ . . .+ c~e(t -n) 2.2.1.1

w here u( t) , y ( t ) and e( t ) are n ow scalar variables represent ing the vol tag e to h eat ing

e lemen t , the tempera tu re in the hea t ing zone and the uncer ta in ty te rm o f the m odel ,

respect ively .

The furnace tem perature contro l ler can be des igned by minimizing the w eighted o utput

error variance, i .e. ,

I = E { [ r ( t + 1 ) - y ( t + 1)]2 + ~ , [ u ( t ) - u ( t - 1)]: } 2.2.1 .2

wh ere (E ) is the e xpectat ion operator , r ( t ) is the set-poin t tem perature for the heat ing

zone, y ( t ) is the zone tem perature and (~ ,) is a posit ive w eight parameter . The bas ic



222 C. Batur e t a l .

The t e rms w i t h ( ^ ) i nd i ca te t he e s t i ma t ed pa r ame t e r s o f t he ma t r i x coe f f ic i en t s. U n l es s t he

ma t r ix coe f f ic ients (Ci; i= l ,2 , . .n) a re a ll zero , the minim izat ion of (2 .1 .2 ) i s a non- l ine ar

mi n i mi za t i on p rob l em an d can be pe r fo rmed by i t e ra t ive t echn i ques on l y .

F i gu re 2 show s a t yp i ca l i npu t ou t pu t i den t if ica t ion da t a f o r an e igh t zone t r anspa ren t

crys ta l gro w th fu rna ce f rom Ba tur e t a l [3] and Sr in ivasan e t ai [36]. Th e input s ignal is a

P s e u d o R a n d o m B i n a r y S i g n al [ 40 ] t h a t d is tu r b s th e f u r n a c e t e m p e r a t u r e s a r o u n d t h e i r

norm a l o pe ra t i ng po i n t f o r co ns i s ten t i den t if i ca t ion o f ma t r i x coe f f ic i en ts .

300

25 0

ii irt"

t--~; 200

ii i

uJI.-a 150Z<

I-

o..

Z 100

LUZON

5O

00

FURNACE INPUT OUTPUT IDENTIFICATION DATA

f I I I I I

T Y P I C A L P R B S I N P U T S IG N A L T O H E A T E R

I I I I I I

200 400 600 800 1000 1200 1400

SAMPLES

Figure 2. Experimental input output data fo r identification of furnace zone dynamics

Plots sh ow typical iden tification data obtained from a transparent crystal growth furnace . All eight

zones are simultaneously perturbed around 225 C. For clarity, only input to zone one is shown.

Th e othe r zone inputs are simply the shifted version o f this pseudo random b inary input. D ata a re

sampled in eve ry 4 seconds.





224

e ( t) = r ( t ) - y ( t )

C. Batur et aL

( 2 . 3 . 2 )

T h e t e m p e r a t u r e c o n t r o l l e r re a d s t h e z o n e t e m p e r a t u r e y ( t ) a t t i m e ( t) , d e t e r m i n e s t h ec o n t r o l l e r o u t p u t u ( t ) f r o m ( 2 . 3 .1 ) a n d s e n d s it t o t h e h e a t in g e l e m e n t . T h e s a m e p r o c e s s

i s r e p e a t e d a g a i n a f t e r a s a m p l i n g p e r i o d ( T ) . F o r m o s t i n d u s t r ia l c o n t r o l l e r s , t h e

d e r i v a t i v e e f f e c t , i .e . , t h e l a s t t e r m o f ( 2 . 3 . 1 ) i s i m p l e m e n t e d wi t h a l o w p a s s f i l t e r s i n c e

t h e d e r i v a t iv e t e r m a m p l i fi e s t h e h i g h f r e q u e n c y n o i s e th a t m a y e x i s t in t h e t e m p e r a t u r e

c o n t r o l l o o p a n d c a u s e s s a t u r a ti o n i n t h e a m p l if ie r s. D e t e r m i n a t i o n o f t h e c o n t r o l l e r

p a r a m e t e r s K p , Ti a n d T d f o r a g i v e n fu r n a c e is k n o w n a s t h e c o n t r o l l e r t u n i n g . A v a s t

m a j o r i ty o f P I D c o n t r o l l e rs a r e t u n e d m a n u a l ly b y c o n t r o l e n g i n e e rs , b a s e d o n t h e i r p a s t

e x p e r i e n c e s a n d h e u r i s t i c r u l e s , A s t r o m e t a l [ 2] . S o m e o f t h e s e h e u r i s t i c r u l e s a r e a l s o

c a p t u r e d b y c o m m e r c i a l e x p e r t t u n i n g s y s t em s s u c h as F o x b o r o E X A C T c o n t r o l l e r, e .g . ,

C a U a g h a n e t a l [ 6 ].

I f a p a r t ia l m o d e l c a n b e c o n s t r u c t e d f o r t h e h e a t i n g z o n e d y n a m i c s , t h e c o n t r o l l e r t u n i n g

b e c o m e s l e s s h e ur is t ic . A c o m m o n l y u s e d s i m p l e m o d e l i s a fi rs t o r d e r s y s t e m w i t h d e a d

t ime , i . e . ,

y ( t ) = a y ( t - 1) + b u ( t - d ) ( 2 . 3 . 3 )

w h e r e u ( t ) i s t h e v o l t a g e t o h e a t in g e l e m e n t a t ti m e ( t ), y ( t ) is th e z o n e t e m p e r a t u r e a n d

( d ) i n d i c a t e s th e t i m e d e l a y b e t w e e n i n p u t a n d o u t p u t . A n e x p e r i m e n t a l i d e n ti f ic a t io n o f

t h e m o d e l p a r a m e t e r s ( a , b , a n d d ) c a n b e p e r f o r m e d b y e x c i ti n g t h e h e a t i n g z o n e b y as t e p c h a n g e o r a r a n d o m c h a n g e i n t h e v o l t a g e u ( t ), s e e , f o r e x a m p l e , A s t r o m a n d

W i t t e n m a r k [ 1 . O n c e th e m o d e l p a r a m e t e r s a r e e s t im a t e d , t h e c o n t r o l l e r p a r a m e t e r s c a n

b e d e t e r m i n e d b y t h e w e l l k n o w n Z i e g l e r N i c h o l s ru l e s [4 3 ]. S o m e c o m m e r c i a l

t e m p e r a t u r e c o n t r o l le r s c a n i m p l e m e n t a m o d e l b a s e d s e l f- t u n in g a l g o r it h m i n o r d e r t o

d e t e r m i n e th e c o e f f ic i e n ts o f t h e P I D c o n t ro l le r . T h e s e c o n t r o l l e rs d o n o n e e d a n o p e r a t o r

t o t u n e t h e c o n t r o l l e r .

F o l l o w i n g t h e w o r k o f A s t r o m e t al [2 ] a n d K a y a a n d T i tu s [ 2 5 ], s o m e c u r r e n t i nd u s t ri a l

t e m p e r a t u r e c o n t r o l le r s a n d t h e ir a u to - t u n i n g p r o p e r t ie s c a n b e s u m m a r i z e d a s i n T a b l e 1 .

Control ler ManufacturerE X A C T F o x b o r o

U D C 6 0 0 0 H o n e y w e l l

S L P C Y o k o g a w a

C L C O 4 B a i le y C o n t r o ls

Tuning T echniqueI d e n ti f ie s z o n e d y n a m i c s b y t h e r e s p o n s e o f

h e a t i n g z o n e t o d i s tu r b a n c e s. U s e s Z i e g l e r - N i c h o l s

t y p e r u l e s t o t u n e t h e c o n t r o l l e r .

I d e n ti fi e s t h e z o n e d y n a m i c s b y a s t e p c h a n g e i n

v o l t a g e u . H e u r i s ti c a n d Z i e g l e r N i c h o l s t y p e r u l e s

a r e u s e d t o t u n e t h e P I D c o n t r o ll e r .

M o d e l o f t y p e (2 . 1 . 3 ) is i d e n ti f ie d b y i n t r o d u c i n g

a s t ep c h a n g e i n p r o c e s s i n p u t u . P I D p a r a m e t e r s a r e

d e t e r m i n e d w i t h t h e m o d e l p a ra m e t e r s . E x a c t

e q u a t i o n s a r e n o t p u b l i s h e d .

I d e n ti f ie s P r o c e s s M o d e l b y S t e p R e s p o n s e .



DPR 900 F i sher Con t ro ls

S I P A R T S i emen s

TO S D I C To s h i b a

S Y S M A C O m r o n

FU JI PY X-4 Fuji Elect r ic

Control of Crystal Growth

Identi fies P rocess M odel Freq uenc y Response.

Identi fies P rocess M odel by S tep Resp onse.

Identi fies P rocess M odel by R ando m Input u( t ) .

Implem ents fuzzy cont ro l wi th use r def ined ru les

and membership functions.

Implem ents fuzzy co nt ro l w i th f ixed ru les .

T a b l e 1 . C u r r e n t T e m p e r a t u r e C o n t r o ll e rs a n d T u n i n g P r o p e r t i e s

225

Th e S YS M AC cont ro l ler of Om ron i s a fuzzy logic cont ro l ler [39] w i thout sel f- tuning

proper ty . I t implem ents fuzzy cont ro l ru les such as" i f t h e t e m p e r a t u r e e r r o r i s Smal l ,

a n d t h e c h a n g e i n te m p e r a t u r e e r r o r i s M e d i u m t h e n a p p l y S m a l l P o s i t iv e h e a t e r

c o n t r o l vol tag e" . Th e quali fiers Sm all , M edium, etc . are def ined through thei r

m em bership functions. Fo r a detai led analysis o f fuzzy control lers, see, for example, Ba tur

and Kaspar i an [4] . The SYS MA C con t ro ll e r can accom mo date up to 128 ru les. The

fuzzy tem pera ture cont ro l ler of Fuj i [14] work s on the set -point fo l lowing er ror wh ich

happ ens fo l lowing disturbances or cha nges in the set-points. I t im plem ents a f ixed set o f

fuzzy c ont ro l ru les .

3 . C O N T R O L O F M A T E R I A L T E M P E R A T U R E

Te m pera ture and conve ct ive f low dis tr ibution ins ide the material determ ine the shape o f

the solid-liquid interface. This distribution, in turn, is dic tate d by the furn ac e axialtem pe ratu re profi le, p art icularly nea r the sol id-liquid interface. Additionally, th e a m po ule 's

t ranslat ional veloci ty , the hea t losses from the s ide wal ls o f the am poule, and pressu re in

the app aratus af fect the material tem perature d is tr ibution and consequ ent ly the shape of

in ter face. Since the heat losses f rom the s ide wai l s are not con t ro llable param eters dur ing

grow th, w e w i ll consider the axial furnace temperature d is t r ibut ion and th e t ranslat ional

rate as the m ain proc ess variables tha t affect the shape o f the sol id-liquid interfac e during

growth .

Taghav i and Duv al [41 ] analy tically determined the requi red furnace tem pera ture prof i le

in order to obtain a f iat in ter face in the s teady s tate . Th e resul t s are obtained u nde r thesimpl ify ing assump t ions that the thermophysical proper t ies of the me l t and crystal are

equal and inde pende nt of tempe rature. Fur thermore, co nvec t ive f lows inside the m el t are

assum ed negligible. Th eir results indicated that a f iat interface requires a rath er cha llenging

axial temp era ture distr ibution w hich includes a discontinuity at the interface. Ne verth eless,

th is w ork i s the f i rs t ser ious at tempt to so lve the material tem perature cont ro l problem in

the steady state.

Da ntzig and Tor torel l i [8] , and Dantzig [9] have s tudied the ef fects of furnace axial

tem pera ture d is t ribut ion on the shape of the solid- liquid inter face. Th ey posed th e

problem as an op t imizat ion problem w here the fo l lowing perform ance index is minimizedwi th respect to zone tem peratures Tz, i .e. ,



2 2 6 C . Ba t u r e t a l .

I(T~) = (T~ - T) r (Tr - T ) (3.1)

w here Tr(x ,y) is the des ired reference temp erature that w e wa nt to es tab lish at a g iven

loca tion (x,y) inside the material. T(x,y) is the actual material tem per atur e distr ibution

estab l ished at the sam e locat ion . This d is tr ibu tion is gene rated by the zon e tem perature s

Tz , dynam ics o f the furnace and the mater ia l a t hand . Dantzig and co-wo rkers ob tained

the tem peratu re d is tr ibu tion T(x ,y) by s imulating the thermal dynam ics of the m ater ia l by a

f in i te e lem ent model .

Th e c onst i tu t ive equat ions descr ib ing the dynam ics of the in ter face are g iven by the

foll ow ing partial differential equations, see e.g., Jasinski and Witt [21],

OTp , c , = V . ( k , V T )

(3.2)

,gTpsc, ~ = V . (k , VT )

( 3 . 3 )

k a T k a T = p L y, - g - , - g

(3 .4)

w he re (T) is the material tem peratu re, (p) is the density, (c) is the specif ic he at and (k ) isthe c ond uc tive heat transfer coefficient. T he subscripts (1) and (s) refer to liquid and solid,

respect ively and (n) ind icates the uni t normal vecto r to sur face. The equat ion (3 .4) is

know n as the S te fan ' s cond i t ion wh ich descr ibes the hea t f low ba lance a t the in te r face .

He re, (L) is the la ten t heat of the materia l and (v) is the crys tal g row th veloci ty . The

gro w th dynam ics can b e c haracter ized by s imulating (3 .2) - (3 .4) by f ini te e lem ent , f in i te

d if ference or with n on- l inear e lectr ical analogue s imulators [19] , [22] un der p roper

bounda ry cond i tions .

A_~er spatial disc retization by the f inite elemen t techniqu e, the m aterial tem pe rat ure 's

dynam ic model can be wr i t ten in an abstract form as ,

M ( T ) T +K (T)T = T~ ( 3 . 5 )

wh ere (M ) is the tem perature dep enden t thermal mass matrix, (K) is the thermal s t if fness

matr ix and (T) is the tem perature ins ide the materia l. Th e material tem peratu re contro l

p rob lem beco me s the de te rmina tion o f the fu rnace zone tempera tu res Tz such tha t the

perform ance index (3 .1) is minimum. The problem can be considered as the constrained

minim izat ion problom w here the constrains are g iven by (3.5) . Alternat ively , one ma y v iew

the problem as in inverse problem in heat t ransfer wh ere the bou ndary con dit ions (Tz) is to

be determ ined such that a des ired temp erature d ist ribu t ion (Tr ) o r , equivalen tly, a

desire d interfa ce shape can be established inside the material.



Control of Crystal Growth 227

T h e s o l u t io n t o i n v e r se p r o b l e m i s f o u n d b y t h e f o ll o w i n g s te p s:

1 . T h e r e f e r e n c e t e m p e r a t u r e d i s t r ib u t i o n ( T , ) i s s p e c i f ie d f o r t h e t i m e i n s t a n t ( t) .

2 . F o r a g i v e n s e t o f z o n e t e m p e r a t u r e s ( T ~ ) , t h e m a t e r ia l d y n a m i c s i s s i m u l a t e d b y ( 3 . 5 )

u n t i l a s t e a d y s t a t e m a t e r i a l t e m p e r a t u r e d i s tr i b u t io n T ( x , y ) i s f o u n d .

3 . A n e w s e t o f z o n e t e m p e r a t u r e ( T ~ ) i s d e t e r m i n e d b y t h e g r a d i e n t d e s c e n t a l g o r it h m ,

i .e . ,

= ( o t a ) - a ,

( 3 . 6 )

w h e r e ( e t ) is o p t i m i z e d b y a l i n e s e a r c h a l g o r i th m .

4 . S t e p s ( 2 ) - ( 3 ) a r e r e p e a t e d u n t i l c o n v e r g e n c e i s a c h i e v e d . A t t h e e n d o f t h i s st e p , t h e

m a t e r i a l t e m p e r a t u r e ( T ) i s t h e l e a s t sq u a r e s a p p r o x i m a t i o n t o t h e d e s i r e d m a t e ri a lt e m p e r a t u r e ( T r ).

5 . T o g r o w t h e c r y s t al , a n e w r e f e r e n c e t e m p e r a t u r e s ( T , ) i s s p e c if i ed a t t i m e 0 + 8 0 i n

s u c h a w a y t h a t i t c o r r e s p o n d s t o t h e d e s i r e d s h a p e o f t h e i n te r f a c e a t t im e ( t + r t ) .

D a n t z i g a n d c o - w o r k e r s s o l v e d t o m a t er ia l t e m p e r a t u r e c o n t r o l p r o b l e m f o r t h e s t e a d y

s t a t e c o n d i t i o n s , i . e . , t h e o p t i m i z a t i o n i s p e r f o r m e d o n c e t h e t e m p e r a t u r e s , s i m u l a t e d b y

( 3 . 5 ) , r e a c h e d t h e i r s t e a d y s t a t e v a l u e s . T h e y a ls o a s s u m e d t h a t t h e r e i s n o a v a i l a b l e

m e a s u r e m e n t f o r t h e m a t e r ia l t e m p e r a t u r e o r t h e i n t e rf a c e s h a pe .

i f t h e t e m p e r a t u r e d y n a m i c s r e p r e se n t e d b y ( 3 . 5 ) i s a t ru e r e p r e s e n t a t io n f o r t h e m a t e ri a l

a n d t h e f u r n a c e , a n d i f t h is d y n a m i c s d o e s n o t c h a n g e i n ti m e , t h e n t h e i r s o l u t i o n i s

o p t i m u m i n a l e a s t s q u a r e s s e n s e . Ho w e v e r , i f t h e m o d e l i s n o t e x a c t , a s i t i s e x p e c t e d f o r

m o s t a p p l i c a ti o n s , w e n e e d s o m e f e e d b a c k s ig n a l t o m a k e c o r r e c t i o n s i n o r d e r t o

c o m p e n s a t e f o r t h e m o d e l u n c e r t a i n t i e s . S r i n i v a s a n e t a l [ 3 7 ] c o n s i d e r e d t h e s i t u a t i o n

w h e r e s o m e f e e d b a c k i n f o r m a t i o n m a y b e a v a il ab le . F o r e x a m p l e , it m a y b e p o s s i b le t o

m e a s u r e t h e o u t s i d e s u r f a ce t e m p e r a t u r e s o f t h e a m p o u l e . A l te r n a ti v e ly , t h r o u g h i m a g e

p r o c e s s i n g t e c h n i q u e s , o n e c a n d e t e r m i n e t h e s h a p e o f th e i n t e r fa c e , t h e r e fo r e , t h e m a t e r ia l

t e m p e r a t u r e s a t i n t e r fa c e p o i n ts . I n d e e d , t h e i n t e rf a c e s h a p e c a n b e q u a n t i f ie d f o r

t r a n s p a r e n t f u r n a c e s , s e e , f o r e x a m p l e , B a t u r e t a l [ 3 ] , K a s p a r i a n e t a l [ 2 4 ], P o t t s a n dW i l c o x [ 3 3 ], C h a n g a n d W i l c o x [7 ] , N e u g e b a u e r a n d W i l c o x [ 30 ] a n d L a n e t a l [2 6 ] . E v e n

f o r n o n - t r a n s p a r e n t f u r n a c e s , X - r a y i m a g i n g c a n l o c a t e th e i n t e r f a c e , . e . g ., F r i p p e t a l [ 1 2 ],

H u b e r t e t a l [ 2 0 ], K a k i m o t o e t a l [ 2 3 ], a n d W a r g o a n d W i t t [4 2 ] a s in th e c a s e o f

C z o c h r a l s k i m e t h o d . A n o t h e r i n - s it u m e a s u r e m e n t to o l t o l o c a t e th e i n t e r f a c e is t o u s e a n

E d d y c u r r e n t p r o b e a s d e m o n s t r a t e d b y S t e fa n i e t a l [ 3 8] a n d R o s e n e t a l [ 3 5 ].

T h e m a t e r i a l t e m p e r a t u r e c o n t r o l t e c h n i q u e p r o p o s e d b y S r i n iv a s a n e t al [ 3 7 ] i s

i m p l e m e n t e d t h r o u g h t h e f o l lo w i n g s t ep s .

1 . F o r e a c h s a m p l i n g p e r i o d , t h e m a t e r i a l t e m p e r a t u r e d y n a m i c s i s r e p r e s e n t e d b y a f in i t e

e l e m e n t m o d e l a s i n (3 . 5 ).2 . T h e f u r n a c e z o n e t e m p e r a t u r e s e t -p o i n ts a r e d e t e r m i n e d b y



2 2 8

= +

C. Batur e t a l .

(3.7)

w he re (]1) is the estim ated tempe rature distr ibution inside the material, (Kf) is the gain o f

the contro l le r and (TQ is the b ias s ignal that m akes sure that

T = T , (3 .8 )

in the ste ady state. If all tem peratu res inside the material can be me asu red (an ideal case) ,

then ]h = T. Ho we ver , i f on ly part ia l me asureme nts such as the amp oule ' s ou ts ide sur face

tem perature s or the in ter face shape is avai lab le , then (T ) is on ly an es t imate o f unk now n

mater ia l tempera tures (T). A consis ten t est imate of (T) can be determ ined by apply ing

Kalm an es t imation techniques to the dynam ic mod el (3 .5) , p rovided that the re is asuff ic ien t n um ber of tem perature measurem ents , M aciejowski [28].

The f eedback con t ro l ga in (Kf) is determined by minimizing the per forman ce index of the

Linea r Quadrat ic R egulator , i .e. ,

I(T~) = (7~ - T) r Q(T~ - T) + (T ,)r R(T~)(3 .9)

w her e (Q) and (R) are the user specif ied weigh t matr ices emphasizing on the cost o f

contro l act ion .

As in the case o f Da ntzig and co -work ers , the in ter face is t rans lated by changing the

re fe rence tem pera tu re (T , ) acco rd ing to a g iven des i red g rowth r a te fo r the c rys ta l. The

am poule is assum ed to be in a f ixed posi t ion inside the furnace. F igure 3 show s the

required furnace zo ne tem peratures for the grow th o f f la t in terface in the case of s imulated

L e a d B r o m id e g r o w th .

The cho ice o f r e f e rence tempera tu re (Tr ) fo r the mate rial t em pera tu re con t ro lle r i s

determ ined by the d es ired in terface shape. Ho we ver , for a g iven des ired in ter face shape

one can specify m ore than one set o f reference temp eratures and they al l cor resp ond to thesam e shape. Therefore , addit ional constrain ts are need ed to uniquely determine the

referenc e temp erature . On e obvious constrain is the physical real izab il it iy o f the req uested

tem peratu re d is t ribu t ion ins ide the materia l. One can only specify a set o f reference

tem perature s (T ,) that can be realizab le by the furnace tem perature d is t r ibu tion see, for

example, Srinivasan et al [37]. Oth er constrains such as therm ally indu ced stress limits

arou nd th e interfa ce and inside the crystal can also be used to specify the re feren ce

tem peratures , as d iscussed by Gevelber and Stephanopolus [15] in the case o f Czoc hralsk i

g r o w th .

Th e crystal qual i ty contro l p roblem becom es much m ore d if ficu l t i f one also conside rsthe inf luence of convec t ion in the melt. In fact , as s imulated by C hang and B row n [44] and

M urray et a l [29], the comp osi t ional un iformity depend s on the conve ct ive f lows and the



Control of Crystal Growth 229

shape of the m elt-sol id interface. In all three tech niques pre sented in this sect ion, i .e. ,

Ta gha vi an d Duv al [41], Da ntzig and Tortorelli [8], and Srinivasan e t al [37], it is

assum ed that the con duct ion is the main heat t ransfer phenom enon inside the ampo ule.

The refore, thei r resul ts ma y not be valid for large Rayleigh or Pec klet number. From the

cont ro l e ngineer ing point of v iew, the m ain di ff iculty is the lack o f mea surem ents o f the

var iables to be cont ro l led . A s in the case o f temperatures , i f the o uts ide surface

tem per ature s or the interfac e shape is measurable, then i t is possible to est ima te the inside

mater ial temp eratures . How eve r , the measurem ent of the flow veloci t ies and pat terns

inside the m el t is prohibit ive. Therefore, the const ruct ion of the feedba ck cont ro l laws to

m anipula te the con vec tive f lows to a desirable pat tern is also prohibi tive at this stage.

C O N C L U S I O N

An ideal contro l sytem should control the quality of crystal during grow th. Th is defini t ion

implies in-si tu m easurem ent of quali ty . H ow ever , the qual i ty can m ost ly be determ ined

off-line by a co m bination o f variables such as den si ty of crystal defects, poly-

crystal lization, dens i ty o f imp uri ty atom s, uniformity of dop ing material , etc. T hese

variables are not no rm ally available to the control ler due to m eas urem ent diff icul ties.

Th e sha pe of the sol id-liquid interface can be related to crystal quali ty since i t influences

the crystal line perfect ion and composi t ional hom ogenei ty . The refore, i t i s pract ical to

cont ro l the shape by m anipulating the thermal boundary condi t ions surrounding the

in ter face. For a g iven in ter face shape, there i s a correspon ding tem perature d is tr ibutioninside the m aterial that ge ner ates this specified interface shape. T he task o f the in terface

shap e con trol ler is to establish this tempe rature distr ibution by set t ing up the ap prop riate

bou nd ary cond it ions, i .e. the tem pera ture distribution inside the furnace .

W e su rveyed the cu r ren t con t ro l des ign methodo log ies fo r t he con t ro l o f zone

tem peratu res ins ide the furnace. M ost industr ial tem perature cont ro l lers can h andle the

tem pera ture cont ro l problem easi ly i f there i s negl ig ib le in teract ion am ong heat ing zones ,

s ince the zon e dynam ics are general ly s low. I f the therma l in teract ions are s t rong, a m ul t i-

input mu l t i-output m odel based con t ro l ler i s needed.

The m ater ial tem perature cont ro l problem can be solved wi th or wi thout the feedback.

Open loop solu t ions not requi ring feedback temp erature measu reme nts can determ ine the

optim um fu rnace tem pera ture distribution for a desired interface shape. Th e optima li ty is

in a least square s sense and the solut ion only applies to stea dy state condit ions. If som e

m easure m ents such as the outs ide ampou le surface temp eratures and/or the in ter face

shape a re available then a Linea r Quadrat ic Reg ulator can be designed to ge nerate the

opt imu m fu rnace tem perature d is tr ibution for any time instant. These tec hnique s are

successful ly appl ied to s imulated crystal grow th and remain to be d em onst rated on real

crystal grow th systems.



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2 3 2 C . Ba t u r e t a l ,

13. Fu, T . W . , W i lcox, W. R. , " Inf luence of Insula t ion on S tabi l i ty o f Inter face Shap e

and P o s i t ion i n t he V er ti ca l Br idgm an S tockbarge r Tecn ique" , Journa l o f Crys t a l Grow th ,48, pp: 416-424, 1980.

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15. Geve lbe r , M . A . , S t ephanopoulus , G . , W argo , J . " Dyn am ics and Con t ro l o f t he

Czoc hra l sk i P rocess" , Journa l o f Crys t a l Grow th , 91 . pp : 199 , 1988 .

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C r y s t al G r o w t h P r o c e s s ", J o u r n a l o f D y n a m i c S y s te m s M e s u r e m e n t s a n d C o n t ro l , M a r c h

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L iqu id In t e r f ace S h ape by X - ray r ad iography Dur ing S i li con S ing l e Crys t a l Grow th" ,

Journa l of Cry s ta l G row th, 91, pp: 509-514, 1988.

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of S t e r eo Im aging fo r Recog ni t ion o f Crys t a l S ur face S hapes", Journa l o f Crys t a l Growth ,

Vol . , 141 455-464, 1994.



Control of Crystal Growth

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233

26 . Lan , C. W . , Y an g, D . T . , T ing , C . C . , Chert , F . C . , "A T r a n s p a re n t M u l t iz o n e F u r n a c e

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110, pp: 713-723, 1991.

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S tockb arge r T echn ique" , Journa l o f Crys t a l Grow th , 89 , pp : 143-154 , 1988 .

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M ater ia l s" , Journal of Cry s ta l Gro wth, 85, pp: 130-135, 1987.

35 . Rosen , G . J . , Car l son , F . M . , W i l cox , W . R . , W al lace, J . P . " M on i to r in g C dTeS ol id if i ca ti on by an Ed dy C ur ren t Techn ique" , N in th Amer ican Co nfe rence o n C rys t a l

Grow th , Augu s t 1 -6 , 1993 , pp : 16 8

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Con fe rence , S ess ion o f M ic roe lec t ron ics and Thermal P rocess Con t ro l, June 21 , 1995 .



2 3 4 C , B a tu r et aL

38. S tefani, J . A. , T ien, J . K. , Cho e, K. S . , W al lace , J . P . "M ul t i f r eq ue nc y Edd y Cu rrent

Diag nos t i c s o f Axia l and Rad ia l Thermal P rofi le s Dur ing S i li con Crys t a l Grow th" , Journa l

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224, 1992.

43 . Z i eg l e r, J . G ., N icho l s , N B . , " Op t imu m,S e t t ings fo r Automat i c C ont ro l l e r s" , T rans ,AS ME, 64 , pp .759-768 , 1942 .

44 . Chang , C . J ., B row n R . A . , , " R ad i a l S egrega t i on Induced by Na tura l Conv ec t i on and

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pp :343-364 , 1983 .



Cont ro l o f C rys ta l Grow th 235

C. Batur received B.S.M.E and M .S. de grees from the Technical Univers i ty of Is tanbul ,

Tu rkey in 1970 and 1971 respectively. H is Ph. D deg ree is from the Un iversity o f

Leices ter , England, 1976. H e is presently o n the facul ty of De partme nt o f Mechanical

Eng ineering at the Univers i ty of Akron, Akron, OH, U SA. Dr. B atur published

extens ively in proce ss control, co m pute r imaging and neural-fuzzy systems. C urrently he

w ork s on th e co ntrol o f interface shape dt~ring crystal grow th and the structural para me ter

contro l in polym er process ing machines .

Arvind Srin ivasan received the B. Tech degree in Mechanical E ngineering from th e Indian

Inst i tu te o f Technology, Madras , India in 1989, an M.S. deg ree in mechanical enginee ring

in 1991 and a M.S. degre e in electr ical engineering in 1993 from the Un ivers i ty of A kron.

H e obtained h is Ph.D. degr ee in mechanical engineering in 1 994. H e s tudied the contro l o f

in terface shape during crys tal growth. Be twe en 1991 and 1994 he co nducted research on

prob lem s related to b oth tem perature con trol and solid-liquid interface contro l in crystal

grow th furnaces. T his research was supported by the N AS A Lewis Research Center ,

Clev eland Ohio. Currently he work s for InterBol Inc., Canton, O hio. Dr. Srinivasan 's

resea rch interests span the areas o f m odeling and contro l of distributed systems, system

theory , multivariable robust com rol, neural netw ork and fuzzy logic.



236 C. Batur e t a / .

W al t e r M. D uva l r ece ived h i s P h .D in M echan i ca l Eng inee r ing f rom R ensse l ae r

P o ly t ech n i c Ins t i t u t e i n 1984 . H i s re sea rch a rea was in t he two-p hase f l ow, cond ensa t i on

and evap ora t i on o f a mu l t icom ponen t f lu id i ns ide a hor i zon t a l tube .H e i s a resea rch sc ient i s t in the m ater ia ls divis ion, proces s ing sc ienc e and techn olo gy

b r a n c h , a t N A S A R e s e a r c h C e n t e r. H e j o i n e d N A S A L e w i s i n 1 98 5 t o p i o n e e r th e n e w l y -

fo rmed r esea rch a r ea i n t he m a te ri a ls d i~ s ion o n m ic rograv it y e ff ec ts on c rys t a l g ro w th

t r anspor t phenom ena . H i s r e sea rch in t e r es ts inc lude con t ro l sys t ems fo r op t imiza t i on o f

c r y s ta l g r o w t h p r o c e s se s , e x p e r im e n t s a n d c o m p u t a t io n s o f c r y s ta l g r o w t h p h e n o m e n a ,

hydro dyn am ic i ns tab i li ty , phase ch ange phenomena, and chao t i c dynam ics o f d i s sipa t ivesys t ems .

N . B . S i n g h j o i n e d t h e W e s t in g h o u s e S c ie n c e a n d T e c h n o l o g y C e n t e r i n 1 98 4 a R e r

resea rch an d t each ing exper i ence o f more t ha t 15 yea r s in t he a r ea o f so li d i fi ca ti on and

c r y s ta l g r o w t h . H e o b t a i n e d M s c a n d P h D d e g r e e s f r o m t h e G o r a k h p u r U n i v e r s i t y ( U P ) ,

I n d ia , a n d w a s a f a c u l ty m e m b e r i n t h e C h e m i s t ry d e p a rt m e n t o f T i l ak D h a r i P o s t -

Grad ua t e Col l ege un t i l June 1979 , wh en be j o ined Rensse l ae r P o ly t echn i c Ins t i t u te , T roy ,

N ew Yo rk . A t W es tm_ghouse he i s t he p rogram man ager fo r c rys t a l g row th . D r . S ingh has

pub l i shed ex t ens ive ly i n the a r e o f c rys t a l g row th and cha rac t e r iza t i on and i s fe l l ow o f

A S M i n te r n a ti o n a l. H e h a s b e e n i n v o lv e d in th e o r g a n i z in g a n d p r o g r a m c o m m i tt e e s o f

m any na t iona l and i n t e rna t i ona l confe rences and wo rkshops . Dr . S ingh i s an ac t ive

m e m b e r o f A A C G ( e le c t ed ex e c u ti v e c o m m i tt e e m e m b e r ) , A S M , T M S , A I A A , a n d

S igm a Xi S c i en ti fi c Resea rch S oc i e ty , and he i s t he founde r o f P i tt sburgh Ch ap te r o f

C r y s t a l G r o w e r s .

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Control of Crystal Growth in Bridgeman