THERMODYNAMICS OF EQUILIBRIUM AND NON-EQUILIBRIUM CRYSTALLIZATION OF Ge AND Si

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THERMODYNAMICS OF EQUILIBRIUM AND NON-EQUILIBRIUM CRYSTALLIZATION OF Ge

AND Si

D. Turnbull

To cite this version:

D. Turnbull. THERMODYNAMICS OF EQUILIBRIUM AND NON-EQUILIBRIUM CRYSTAL- LIZATION OF Ge AND Si. Journal de Physique Colloques, 1982, 43 (C1), pp.C1-259-C1-269.

�10.1051/jphyscol:1982136�. �jpa-00221793�

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JOURNAL DE P H Y S I Q U E

CoZZoque CI, suppldment au nOIO, Tome 43, octobre 1982 page C1-259

THERMODYNAMICS OF E Q U I L I B R I U M AND NON-EQUILIBRIUM CRYSTALLIZATION OF Ge AND S i

D. Turnbull

Division o f Applied Sciences, Hamard University, Pierce HaZZ, MA 02138 Cambridge, U . S. A.

Resume.- A l a p r e s s i o n d'une atmosphgre, l ' c t a t c r i s t a l l i n , c-sc, d e G e o u S i s e l i q u e f i e thermodynamiquement a l a temperature T e t donne un l i q u i d e metal-

cR

l i q u e ( R m ) . Un & a t amorphe semiconducteur, a-sc, e s t forme p a r d i v e r s e s tech- niques de d e p o s i t i o n , bombardement i o n i q u e du c r i s t a l ou d e S i , en trempant ultra-rapidementtm. Les r e s u l t a t s obtenus pour l e s c h a l e u r s d e c r i s t a l l i s a t i i o n

des e t a t s a-sc d e Ge e t S i s e r o n t p r e s e n t e s e t compares 2 ceux obtenus pour Rm + c-sc. A l ' a i d e de c e s c h a l e u r s , e t en u t i l i s a n t l e peu d e donnees connues pour l e s c h a l e u r s c a p a c i t i v e s e t un modgle de l ' e n t r o p i e i3O°K de a-sc, l ' e n e r - g i e l i b r e d e a-sc a e t 6 c a l c u l B e r e l a t i v e m e n t B c-sc. L ' a n a l y s e i n d i q u e qu'une t r a n s f o r m a t i o n dans l e regime m e t a s t a b l e e s t admise comme d t a n t d e pre- mier o r d r e pour a-sc + Rm B T * ^0,8TcR pour Ge e t 0,9 TcR pour S i .

aR

La c o m p a t i b i l i t 6 d e c e s c a l c u l s avec l e s connaissances d e c i n e t i q u e de r e c r i s - t a l l i s a t i o n de c-sc qui c o n d u i t i c o u v r i r c-sc de couches endommagees e t Z former a-sc ii p a r t i r d e Rm p a r trempe u l t r a - r a p i d e e s t d i s c u t e e .

Abstract.- A t one atmosphere p r e s s u r e t h e c r y s t a l l i n e s t a t e , c-sc, of Ge o r S i m e l t s thermodynamically a t temperature T to a m e t a l l i c l i q u i d (Ern). An

cR

amorphous semiconducting s t a t e , a-sc, i s formed i n v a r i o u s d e p o s i t i o n proces- s e s , by i o n bombardment of t h e c r y s t a l o r , of S i , by u l t r a - r a p i d quenching of Rm. Measurements of t h e h e a t s of c r y s t a l l i z a t i o n of t h e a-sc s t a t e s of Ge and S i w i l l be reviewed and compared w i t h t h o s e of Rm + c-sc. Using t h e s e h e a t s , i n c o n j u n c t i o n w i t h l i m i t e d h e a t c a p a c i t y d a t a and a model c a l c u l a t i o n o f t h e O°K e n t r o p y of a-sc t h e f r e e energy of a-sc r e l a t i v e t o t h e c-sc s t a t e h a s been computed. The a n a l y s i s i n d i c a t e s a t r a n s f o r m a t i o n i n t h e m e t a s t a b l e regime-assumed, b u t n o t proven, t o be f i r s t o r d e r o f a-sc ⁺Rm a t T ". 0.8 TcR f o r Ge and

". 0.9 TcQ f o r S i . aR

The c o n s i s t e n c y of t h e s e c a l c u l a t i o n s w i t h experience on t h e k i n e t i c s o f regrowth of c-sc i n t o damaged o v e r l a y s on c-sc and on t h e formation o f a - s c from Rm i n u l t r a - r a p i d quenching i s d i s c u s s e d .

1. I n t r o d u c t i o n . - The condensed phases which t a k e p a r t i n t r a n s f o r m a t i o n s o c c u r r i n g i n t h e high energy f l u e n c e p r o c e s s i n g of S i ( o r Ge) a t low p r e s s u r e a r e t h e c r y s t a l l i n e ( c - s c ) , l i q u i d m e t a l l i c (Em) and amorphous semiconducting ( a - s c ) . I n t h e s e t r a n s f o r m a t i o n s , i m p u r i t y c o n c e n t r a t i o n s f a r beyond t h o s e a t thermodynamic e q u i l i b r i u m may become i n c o r p o r a t e d o r "trapped" i n t h e growing phase. The thermodynamic i n t e r r e l a t i o n of t h e c-sc and Rm phases i s f a i r l y w e l l c h a r a c t e r i z e d b u t t h a t of a-sc t o c-sc o r Rm i s s t i l l q u i t e u n c e r t a i n .

The e x i s t i n g thermodynamic and k i n e t i c o b s e r v a t i o n s e s t a b l i s h t h a t a-sc i s l e s s s t a b l e t h a n c-sc a t a l l temperatures from O°K t o w e l l above t h a t , TcR, a t which c-sc and Rm c o e x i s t a t e q u i l i b r i u m . Also, it seems c l e a r from t h e s e o b s e r v a t i o n s , a s

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1982136

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C1-260 JOURNAL DE PHYSIQUE

w e l l a s from t h e m e l t i n g behavior of c-sc, t h a t t h e Rm phase i s more s t a b l e t h a n a-sc from T > TcR t o temperatures w e l l below TcR, and s o i n t h e range of Rm mesta- s t a b i l i t y , r e l a t i v e t o c-sc. However, t h e measured e n t h a l p i e s , AH^^, of a-sc-c-sc i n d i c a t e t h a t a-sc i s more s t a b l e t h a n Rm a t T = O°K. It f o l l o w s t h a t , i f

c r y s t a l l i z a t i o n i s avoided, t h e r e should be a temperature, Tag, between O°K and TcR a t which a thermodynamic t r a n s i t i o n I?m++a-sc would occur [1,21. Whether t h i s t r a n s i t i o n would be thermodynamically continuous o r d i s c o n t i n u o u s i s n o t e s t a b l i s h e d though, i n view of t h e a t t e n d a n t change i n c o o r d i n a t i o n number, i t i s p l a u s i b l e t h a t it would b e d i s c o n t i n u o u s . There i s a l s o t h e problem of how unique o r thermodykun- i c a l l y d e f i n e d t h e a-sc s t a t e would be a t T

cR'

2. S t r u c t u r a l R e l a x a t i o n and Thermal P r o p e r t i e s of t h e a-sc S t a t e . - A s p r e p a r e d , a-sc specimens a r e i n c o n f i g u r a t i o n a l l y f r o z e n s t a t e s and s o , depending on t h e c o n d i t i o n s of t h e i r p r e p a r a t i o n , t h e y e x h i b i t sope specimen t o specimen d i s p e r s i o n of e n t h a l p i e s .

The e n t h a l p y of c r y s t a l l i z a t i o n , AH^^, of a-sc may c o n s i s t of c o n t r i b u t i o n s ( a ) from p o i n t (e.g. d a n g l i n g bonds) and extended i m p e r f e c t i o n s (e.g. i n t e r n a l boundaries) i n t h e amorphous s t r u c t u r e and (b) due t o d i s t o r t i o n s of t h e bond a n g l e s from t h e i r i d e a l t e t r a h e d r a l v a l u e s . Upon a n n e a l i n g some i r r e v e r s i b l e d e c r e a s e s i n energy p r i o r t o c r y s t a l l i z a t i o n may occur owing t o a n n i h i l a t i o n of p o i n t d e f e c t s and o t h e r s t r u c t u r a l r e l a x a t i o n p r o c e s s e s . The temperature of t r a n s i t i o n between a-sc and Rm should b e h i g h e r t h e g r e a t e r is t h e d e g r e e of r e l a x a t i o n of a-sc. There may e x i s t a f u l l y r e l a x e d 131 semiconducting f l u i d s t a t e of a-sc which would be i n e q u i l i b r i u m w i t h Rm a t some c h a r a c t e r i s t i c maximum Tag. Such a s t a t e might be s i m i l a r

s t r u c t u r a l l y and dynamically t o t h e l i q u i d s t a t e of f u s e d s i l i c a . I f t h e CRN model i s a p p r o p r i a t e f o r i t s s t r u c t u r e i t s excess enthalpy over t h a t of c-sc should be due p r i m a r i l y t o t h e bond d i s t o r t i o n s b u t w i t h some c o n t r i b u t i o n from p o i n t d e f e c t s p r e s e n t i n thermal e q u i l i b r i u m .

Assuming t h a t s t r u c t u r a l r e l a x a t i o n occurs by a s i n g l e t h e r m a l l y a c t i v a t e d p r o c e s s , i t s time c o n s t a n t , T, may be expressed a s :

T = T~ exp ~ Q , , ~ / R T )

where Q i s t h e a c t i v a t i o n energy f o r r e l a x a t i o n . When a-sc i s h e a t e d a t some r e 1

c o n s t a n t r a t e , ?, i t s s t r u c t u r a l r e l a x a t i o n should be v i r t u a l l y completed, i f c r y s t a l l i z a t i o n does n o t i n t e r v e n e , a t some temperature, Tx. By a crude a n a l y s i s , assuming Q >> RT T may b e r e l a t e d t o + a s follows:

r e 1 x x

-

^Qs+^T(T^{) .} ^{( 1 )}

R

It seems r e a s o n a b l e t o suppose t h a t t h e upper l i m i t i n g v a l u e of T would correspond t o t h e time c o n s t a n t , 'TD, f o r t h e d i f f u s i v e jump of a network atom i n t h e c-sc c r y s t a l which may be expressed a s :

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where A i s t h e n e a r e s t neighbor spacing and D i s t h e s e l f - d i f f u s i v i t y i n c-sc.

From t h e s e l f - d i f f u s i v i t y and eq. ( 1 ) upper l i m i t i n g v a l u e s of Tx f o r s p e c i f i e d h e a t i n g r a t e s can be e v a l u a t e d numerically. These may t h e n be compared w i t h t h e temperatures, Tkc, a t which t h e c r y s t a l l i z a t i o n r a t e of a-sc r e a c h e s i t s maximum d u r i n g measurement of t h e c r y s t a l l i z a t i o n enthalpy a t corresponding $. They should a l s o l i e i n t h e temperature range of t h e g l a s s t r a n s i t i o n temperatures e s t i m a t e d e a r l i e r [31.

The Arrhenius c o n s t a n t s , D and QD i n the equation D = D exp [-QD/RTI

which d e s c r i b e t h e experimental s e l f - d i f f u s i v i t i e s i n t h e c-sc phase of Ge and S i a r e l i s t e d i n Table 1. Also shown i n t h e Table a r e t h e computed d i f f u s i v i t i e s a t TcQ and t h e upper l i m i t i n g v a l u e of T s c a l e d w i t h TcQ(i.e.

X - T ~ / T ~ ~ ) c a l c u l a t e d

T r x - from e q u a t i o n ( l ) , with T = r f o r t h e h e a t i n g r a t e s C! = 1 and 100°/min.

D'

Table 1: Arrhenius c o n s t a n t s f o r s e l f d i f f u s i o n i n c-sc ~e~ and sib and reduced temperatures f o r complete r e l a x a t i o n of a-sc s t a t e s a t 2 h e a t i n g r a t e s c a l c u l a t e d therefrom.

Element Q D ( T )

2 0°-l zcR ^{T r x}

(cm s e c ) ( e . v . ) (cm s e c - l ) ? = 1°/min ? = 100°/min

Ge 10 3.0 6 X 1 0 - l ~ 0.61 0.68

a . LETAW A.H., PORTNOY W.M. and SLIFKIN L . , Phys. Rev.

102 (1956) 636;

b. GRFIELD J.M. and MASTERS B.J., J. Appl. Phys. 38

(1967) 3148.

There have been a number of measurements, beginning w i t h t h a t of Chen and Turnbull [41, of t h e enthalpy of c r y s t a l l i z a t i o n of t h e a-sc s t a t e of Ge p r e p a r e d by a v a r i e t y of t e c h n i q u e s (vapor-,electro-deposition, e t c . ) . Excepting f o r a s i n g l e measurement of Rudee [51 t h e temperature r a n g e s and e n t h a l p i e s of c r y s t a l l i z a t i o n obtained i n t h e s e v e r a l i n v e s t i g a t i o n s [4,5,6,71 a r e i n f a i r agreement. The most r e c e n t d e t e r m i n a t i o n by Fan and Andersen [81 on s p u t t e r d e p o s i t e d f i l m s , and of Donovan e t a l . [91, on f i l m s amorphized by i o n damage and c r y s t a l l i z e d by e p i t a x i a l regrowth of t h e undamaged c r y s t a l a l s o g i v e AH and Tkc v a l u e s i n g o d agreement

a c

w i t h t h e e a r l i e r d e t e r m i n a t i o n s . These f a t t e r r e s u l t s and t h o s e of Chen and Turnbull a r e summarized i n Table 2. Also shown a r e t h e approximate h e a t i n g r a t e s i n t h e experiments and t h e c a l c u l a t e d T v a l u e s corresponding t o them.

r x

We n o t e t h a t t h e r e i s g o d agreement on ^N0.305 f o r t h e r a t i o of t h e c r y s t a l l i z a t i o n e n t h a l p y of a-sc Ge t o t h a t of t h e Rm phase a t e q u i l i b r i u m . The e s t i m a t e d tempera- t u r e s , T ~ a t complete r e l a x a t i o n a r e somewhat l a r g e r t h a n t h o s e , Trkc, when t h e ~ , c r y s t a l l i z a t i o n r a t e s a r e maximum a t t h e s p e c i f i e d $. However, s i n c e t h e d i f f e r - ences T r x - Trkcr a r e n o t l a r g e it seems l i k e l y t h a t much of t h e r e l a x a t i o n may have occurred b e f o r e t h e c r y s t a l l i z a t i o n o n s e t .

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C1-262 JOURNAL DE PHYSIQUE

Table 2: Enthalpy and temperature of c r y s t a l l i z a t i o n of a-sc s t a t e s Germanium

hcR = 0.386e.v. TcL = 1233OK

Method of

p r e p a r a t i o n 5 (O/min) -

&cR Vapor-or e l e c t r o -

d e p o s i t i o n 80 0.308tO. 015 0.63 0.68 141

S p u t t e r d e p o s i t i o n 10 0.303+_0.025 0.63 0.645 [81

iw i o n damage 40 0.315+0.025 0.58 0.66 191

S i l i c o n

AH^^ ⁼^0.528e.v. ^TcL⁼^1693OK

S p u t t e r d e p o s i t i o n 10 0.197+0.02 0.63 0.69 [81

S p u t t e r d e p o s i t i o n S i Ge

1-x x

Xe,Ar Ion Damage 40 0.221**+0.01 0.555 0.71 [91

x+O e x t r a p o l a t i o n

* * average of 20 d e t e r m i n a t i o n s

Recently two s e t s of measurements of t h e e n t h a l p y of c r y s t a l l i z a t i o n of a-sc S i have been r e p o r t e d [8,91. The r e s u l t s a r e a l s o summarized i n Table 2. Those of Fan and Andersen [81 on s p u t t e r d e p o s i t e d f i l m s s c a t t e r e d q u i t e widely and we n o t e t h a t t h e i r average AH f o r f i l m s with no Ge admixture l i e s w e l l below t h e e x t r a p o l a t i o n

a c

of AH of t h e i r a-sc SilexGex films t o x=O. The r e s u l t s of Donovan e t a l . 191cx-1 a-sc a c

f i l m s formed by A o r Xe i o n damage s c a t t e r e d much l e s s and gave a n average AH,^

.-.

which l i e s between t h e Fan and Andersen d i r e c t and e x t r a p o l a t e d v a l u e s . Donovan e t a l . deduced t h e r a t e of regrowth, a - s c + c - s c , and i t s temperature dependence, u ( T ) , from t h e r a t e s of h e a t r e l e a s e measured on some of t h e i r specimens. T h e i r u(T) was i n f a i r agreement w i t h t h a t o b t a i n e d by Czepregi e t a l . [ l o ] from Rutherford Back S c a t t e r i n g measurements on f i l m s produced by i o n damage.

The reduced v a l u e , 0.22, of t h e c r y s t a l l i z a t i o n enthalpy of a-sc S i lies w e l l below t h a t , 0.31, of a-sc Ge. This l a c k of s c a l i n g with AH^^ was unexpected b u t t h e c u r r e n t s t a t e of t h e microscopic t h e o r y f o r cohesion i s h a r d l y adequate f o r assessment of i t s s i g n i f i c a n c e .

The temperatures T and Trkc from t h e v a r i o u s h e a t i n g r a t e s used i n t h e r x

c r y s t a l l i z a t i o n s were c a l c u l a t e d f o r S i i n t h e same way a s t h o s e f o r Ge. We n o t e t h a t T f o r i o n amorphized f i l m s of both S i and Ge, a r e c o n s i d e r a b l y below t h o s e

r k c

of s p u t t e r o r vapor d e p o s i t e d a-sc f i l m s s o t h a t t h e displacement of T from T r x r k c i s c o n s i d e r a b l y h i g h e r . T h i s behavior probably r e f l e c t s t h a t i o n amorphized f i l m s a r e more p u r e , s o t h a t t h e i m p u r i t y d r a g on c r y s t a l growth i s l e s s , t h a n a r e a-sc f i l m s prepared by t h e o t h e r t e c h n i q u e s .

TO c a l c u l a t e t h e temperature, Tag, of t h e m e t a s t a b l e a-sc Rm e q u i l i b r i u m we need f o r t h e a-sc, b e s i d e s i t s e n t h a l p y of c r y s t a l l i z a t i o n , t h e temperature dependence

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of i t s e n t h a l p y , H(T), and i t s entropy r e l a t i v e t o c-sc a t some temperature. Chen and Turnbull [41 measured H(T) of a-sc G e from 300°K t o Tkc. Bagley and Chen [ I ] and Spaepen and t h e a u t h o r [21, independently, used t h e s e r e s u l t s t o e s t i m a t e T

a % Both c a l c u l a t i o n s were based on p l a u s i b l e , b u t f a r from r i g o r o u s , e x t r a p o l a t i o n s of H(T) from Tkc t o TaR and on model assessments of t h e entropy d i f f e r e n c e between t h e a-sc and c-sc s t a t e s a t O°K, Asac(0). I n t h e a n a l y s i s of Spaepen and t h e w r i t e r

[ l l ASac(Ol was equated t o t h e upper l i m i t i n g c o n f i g u r a t i o n a l entropy of t h e 4-coordinated random metwork, a s computed by Spaepen [ I l l f o l l o w i n g B e l l and Dean's procedure [121 f o r SiOZ. Both c a l c u l a t i o n s gave T (=T /.T 0.8, a temperature

r a t aR cR

w e l l above Trkc and Trx a t t h e h e a t i n g r a t e s used i n t h e c r y s t a l l i z a t i o n s t u d i e s , f o r t h e s c a l e d temperature of t h e e q u i l i b r i u m , assumed t o b e f i r s t o r d e r .

Following t h e procedure of Spaepen and t h e w r i t e r [Z], Donovan e t a l . [91 e s t i m a t e d from t h e i r t h e r m a l d a t a that TraRN0.9 f o r a-sc S i . They assumed t h a t t h e h e a t capacity-temperature r e l a t i o n C (T) f o r a-sc S i s c a l e s , w i t h TcR, a s t h a t f o r a-sc

P

Ge. The h i g h e r T f o r S i , r e l a t i v e t o Ge, r e f l e c t s t h e lower s c a l e d AHac of S i . r a t

The major concern w i t h t h e s e T c a l c u l a t i o n s i s on t h e v a l i d i t y of assuming t h a t r a t

t h e m e t a s t a b l e a - s c + c R t r a n s i t i o n i s thermodynamically d i s c o n t i n u o u s . A thermodynamically continuous t r a n s i t i o n would b e a t t e n d e d by l a r g e i n c r e a s e s i n C (TI of

P a-sc i n t h e t r a n s i t i o n range which, because of r a p i d c r y s t a l l i z t i o n , is h a r d l y a c c e s s i b l e t o measurement. However t h e assumption t h a t t h e t r a n s i t i o n i s

d i s c o n t i n u o u s i s supported by c e r t a i n o b s e r v a t i o n s on t h e c r y s t a l l i z a t i o n k i n e t i c s of t h e Rm and a-sc phases. I n p a r t i c u l a r , Rm Ge h a s been undercooled t o

temperatures a s low a s 0.8TcR [131 with no measurable c r y s t a l n u c l e a t i o n and when s o undercooled it seemed q u i t e f l u i d , and i t s c r y s t a l l i z a t i o n , when i n i t i a t e d , occurred much more r a p i d l y t h a n would have t h a t of a similar mass of a-sc. Also, t h e o b s e r v a t i o n s of ~ o k o r o w s k i e t a l . [I41 t o b e d i s c u s s e d l a t e r , t h a t e p i t a x i a l regrowth of S i c-sc i n t o a-sc o v e r l a y s p e r s i s t s on i t s Arrhenius course from low temperature t o T approaching TcR i n d i c a t e t h a t no continuous t r a n s i t i o n occurred.

3 . Survey of E p i t a x i a l Regrowth K i n e t i c s . - The f o r m a l t h e o r y f o r t h e k i n e t i c s and morphology of t h e movement of crystal-amorphous i n t e r f a c e s h a s been reviewed comprehensively i n o t h e r p u b l i c a t i o n s [ I S ] . A b r i e f summary and c e r t a i n a s p e c t s of t h e t h e o r y a r e p r e s e n t e d h e r e .

The movement occurs by a two s t e p sequence:

(1) atomic rearrangement of t h e i n t e r f a c e and

( 2 ) t r a n s p o r t of t h e h e a t of t r a n s i t i o n and i m p u r i t y t o o r from t h e i n t e r f a c e . The v e l o c i t y u of a p l a n a r i n t e r f a c e i n a p u r e system i s r e l a t e d t o t h e k i n e t i c c o n s t a n t , ui, of t h e i n t e r f a c i a l rearrangement p r o c e s s by t h e expression:

-AS (T -T )

m m i

u - u , 1 1 - e x p [ RTi I )

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JOURNAL DE PHYSIQUE

where u = fXki i

k . = i n t e r f a c i a l rearrangement frequency f = f r a c t i o n of growth s i t e s on i n t e r f a c e

= i n t e r f a c e displacement/rearrangement T . = i n t e r f a c e temperature

T = e q u i l i b r i u m temperature m

AS = entropy of t r a n s f o r m a t i o n t o amorphous s t a t e . m

I n t h e l i n e a r k i n e t i c regime t h i s e q u a t i o n reduces t o :

u a l s o may b e r e l a t e d t o t h e r a t e of h e a t t r a n s p o r t by a n e x p r e s s i o n having the

a p p l i c a b l e where t h e temperature i s uniform i n t h e e x t e r n a l phase (e.g. o v e r l a y on C-SC) where

K = t h e r m a l c o n d u c t i v i t y of phase through which h e a t i s e x t r a c t e d 8 = gram atomic volume; assumed e q u a l in t h e 2 phases

AH = gram atomic e n t h a l p y of t r a n s i t i o n m

(grad T)i = thermal g r a d i e n t a t t h e i n t e r f a c e

A t s t e a d y s t a t e u i s c o n s t a n t and t h e d e p a r t u r e , Ti, of t h e i n t e r f a c e temperature from e q u i l i b r i u m becomes, i n t h e l i n e a r k i n e t i c regime

-K (grad T) RTi - T m - T i = * T i z ( ni

) (

^{(Asm) Tm}

)

Provided t h e i n t e r f a c e i s moving, i t s temperature must l i e between T and t h e m

temperature of t h e h e a t source ( i n m e l t i n g ) o r s i n k ( i n c r y s t a l l i z a t i o n ) . Such displacements of t h e i n t e r f a c e t e m p e r a t u r e s from e q u i l i b r i u m a r e , of c o u r s e , r e q u i r e d f o r t h e d e p o s i t i o n of m e t a s t a b l e s t a t e s (e.g. s u p e r s a t u r a t e d s o l u t i o n s o r amorphous phases) by t h e moving f r o n t .

When t h e i n t e r f a c e p r o c e s s i s slow compared w i t h thermal t r a n s p o r t

[ i . e . gr grad T) i>>ui] Ti approaches ( s e e F i g . 1 ) t h e s o u r c e o r s i n k temperature and t h e i n t e r f a c e motion i s s a i d t o be " i n t e r f a c e l i m i t e d " .

The t h e r m a l l y a c t i v a t e d s o l i d state e p i t a x i a l regrowth of c-sc i n t o amorphous a-sc o v e r l a y s i s , i n t h i s s e n s e , i n t e r f a c e l i m i t e d even when t h e thermal g r a d i e n t i s minute. A t t h e o p p o s i t e l i m i t [u.>?-K(grad T ) . 1 T. i s l i t t l e d i s p l a c e d from

1 1

e q u i l i b r i u m and i n t e r f a c e motion i s s a i d t o be " d i f f u s i o n l i m i t e d " . With a p p l i e d g r a d i e n t s of t h e u s u a l magnitude, t h e growth of m e t a l and c-sc c r y s t a l s i n t o t h e i r pure undercooled m e t a l m e l t s i s i n t h i s d i f f u s i o n l i m i t e d regime. The u l t i m a t e l i m i t on t h e speed of t h e i n t e r f a c e p r o c e s s would be imposed by t h e frequency of c o l l i s i o n s of atoms from t h e m e l t o n t o t h e i n t e r f a c e [16,171, i n which c a s e u

i

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would correspond t o t h e sound speed, u , i n t h e l i q u i d 1151. A n a n a l y s i s of C o r i e l l and t h e a u t h o r 1181 i n d i c a t e d t h a t t h e growth speed of N i d e n d r i t e s i n h i g h l y undercooled molten N i [I91 may be s o l i m i t e d .

F i g . 1 ^:schematic r e p r e s e n t a t i o n of p o s i t i o n s of c r y s t a l - m e l t i n t e r f a c e temperatures, T., r e l a t i v e t o e q u i l i b r i u m t e m p e r a t u r e s , TcR andt Tal, and h e a t s h k teraperature, T , i n undercooling, o r h e a t s o u r c e temperature, T+, i n s u p e r h e a t i n g .

I n high energy f l u e n c e p r o c e s s i n g , t h e thermal g r a d i e n t s may b e s o extreme t h a t t h e r e w i l l b e l a r g e displacements of T from e q u i l i b r i u m even i f t h e i n t e r f a c e

i

p r o c e s s e s a r e v e r y r a p i d . It i s under t h e s e c o n d i t i o n s t h a t e x t e n s i v e i m p u r i t y t r a p p i n g s o r m e t a s t a b l e phase d e p o s i t i o n may occur. Large i n t e r f a c i a l under- c o o l i n g s c a n b e s u s t a i n e d d u r i n g growth only provided the frequency of homophase c r y s t a l n u c l e a t i o n remains v e r y low. A c t u a l l y , t h e e x i s t i n g e x p e r i e n c e i n d i c a t e s t h a t t h e s e f r e q u e n c i e s do n o t reach measurable l e v e l s i n l i q u i d m e t a l s , u n l e s s t h e reduced undercooling ~r =(?) exceeds, a t l e a s t , 0.20 t o 0.25 I201 .

S i m i l a r l y , s u b s t a n t i a l s u p e r h e a t i n g of t h e i n t e r f a c i a l r e g i o n must a t t e n d u l t r a - r a p i d heterogeneous m e l t i n g which o c c u r s because of t h e c o n s i d e r a b l e r e s i s t a n c e of c r y s t a l i n t e r i o r s t o homophase n u c l e a t i o n of t h e melt.

4. ImpurityTrapping.- The thermodynamic c o n d i t i o n s f o r t h e non-equilibrium

i n c o r p o r a t i o n of i m p u r i t i e s i n c r y s t a l s d u r i n g t h e i r growth were s e t f o r t h by Baker and Cahn [211. I m p u r i t i e s a r e s a i d t o be trapped when, f o l l o w i n g t h e p a s s a g e of t h e c r y s t a l l i z a t i o n f r o n t , t h e y a r e lodged i n t h e c r y s t a l z i t a chemical p o t e n t i a l l e v e l h i g h e r t h a n t h a t i n t h e m e l t a t t h e i n t e r f a c e . Trapping i s thermodynamically p e r m i t t e d when t h e t o t a l f r e e energy of t h e system i s decreased by passage of t h e f r o n t .

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C1-266 JOURNAL DE PHYSIQUE

Suppose t h a t t h e motion of t h e c r y s t a l - m e l t f r o n t i n a d i l u t e b i n a r y melt of B i n A is " d i f f u s i o n l e s s " ; i . e . t h e r e i s no i m p u r i t y r e d i s t r i b u t i o n accompanying t h e growth. The thermodynamic c o n d i t i o n f o r such motion i s t h e n :

AGc = (1-xB) AD +x Au < 0

A B E

where AGc i s t h e o v e r a l l f r e e energy of c r y s t a l l i z a t i o n

xB = atom f r a c t i o n of t h e i m p u r i t y B; e q u a l i n melt and c r y s t a l AuB = The d i f f e r e n c e between t h e chemical p o t e n t i a l of B, a t x g , a t

t h e s p e c i f i e d temperature, f n t h e c r y s t a l and t h e melt.

ApA = The d i f f e r e n c e between t h e chemical p o t e n t i a l of t h e major c o n s t i t u e n t A, a t x and T, i n t h e c r y s t a l m e l t .

B

When A!+, > 0 t h i s c o n d i t i o n i s s t i l l f u l f i l l e d i f AuA i s s u f f i c i e n t l y n e g a t i v e . The c o n d i t i o n AG = 0 d e f i n e s a temperature-composition r e l a t i o n T (X ) , l y i n g

0 B

between t h e l i q u i d u s and s o l i d u s r e l a t i o n s , below which t h e c r y s t a l growth can b e d i f f u s i o n l e s s . For d i l u t e s o l u t i o n s t h e chemical p o t e n t i a l changes i n d i f f u s i o n - l e s s growth should be, approximately,

AV, A S ~ ( ? ~ - T ~ ) = A S ~ A T A i ApB = R T ~ (l/ke) R ~ where :

AS: = entropy of c r y s t a l l i z a t i o n of p u r e A TR ⁼l i q u i d u s temperature a t composition x

B

ke ⁼e q u i l i b r i u m d i s t r i b u t i o n c o e f f i c i e n t of B between c r y s t a l and melt a t i n t e r f a c e temperature T

i

The d e p r e s s i o n of t h e i n t e r f a c e temperature below t h e l i q u i d u s must t h e n f u l f i l l t h e c o n d i t i o n :

The maximum amounts of v a r i o u s i m p u r i t y r e p o r t e d by White e t a l . [221 t o be t r a p p e d i n S i d u r i n g r a p i d e p i t a x i a l regrowth would r e q u i r e i n t e r f a c i a l undercoolings ranging, depending on t h e i m p u r i t y , from 10 t o 70°c. A r e c e n t a n a l y s i s of t h e k i n e t i c s of impurity t r a p p i n g was p r e s e n t e d by Aziz 1231.

5. a - s c t f R m T r a n s i t i o n . - We have noted t h a t t h i s t r a n s i t i o n should occur i n t h e temperature range where c-sc i s t h e most s t a b l e phase. T h e r e f o r e , i t s exposure r e q u i r e s t h a t c r y s t a l l i z a t i o n be bypassed i n t h e h e a t i n g of a-sc o r t h e deep under- c o o l i n g of Rm. Such exposure would be favored by u l t r a - r a p i d quenching of Rm o r h e a t i n g of a-sc. I t a l s o r e q u i r e s t h a t t h e frequency of homophase n u c l e a t i o n of c-sc i n a-sc and Rm be r e l a t i v e l y low a t t h e t r a n s i t i o n temperature and t h a t t h e number d e n s i t y of e x t r a n e o u s c r y s t a l l i z a t i o n c e n t e r s b e k e p t minimal. ^P.L. Liu e t a l . [241 and Tsu e t a l . [25] discovered t h a t t h i n amorphous o v e r l a y s were formed on t h e s u r f a c e of i n i t i a l l y p e r f e c t S i c r y s t a l s i r r a d i a t e d under c e r t a i n c o n d i t i o n s by picosecond l a s e r p u l s e s . The d i f f r a c t i o n and o p t i c a l r e f l e c t i v i t y behavior of

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t h e s e o v e r l a y s proved t o be t h e same a s t h a t of a-sc formed by o t h e r t e c h n i q u e s . J . M . Liu e t a l . 1261 showed t h a t a-sc a l s o formed when p r e c a u t i o n s were t a k e n t o i n s u r e t h a t t h e s i l i c o n s u r f a c e s were oxide f r e e d u r i n g i r r a d i a t i o n . h hey supposed

124,26J t h a t it d e r i v e d from Rm which had been undercooled t o temperatures w e l l below Tat d u r i n g t h e extreme quench.

Spaepen and t h e a u t h o r [I51 proposed t h a t i f T1 l i e s s u f f i c i e n t l y below Tat a-sc w i l l be formed i n p r e f e r e n c e t o c-sc a t t h e c-sc-Rm i n t e r f a c e . Thus, beyond some c r i t i c a l i n t e r f a c i a l undercooling a-sc r a t h e r t h a n c-sc would be d e p o s i t e d by t h e advancing i n t e r f a c e . Such a t r a n s i t i o n i n t h e mode of regrowth r e q u i r e s t h a t t h e p r o d u c t f k i , between t h e growth s i t e f r a c t i o n and t h e i n t e r f a c i a l rearrangement frequency, be much h i g h e r f o r a-sc ^fRm t h a n f o r c-sc f Rm regrowth. Indeed, we e x p e c t t h a t f f o r t h e a-sc-Em i n t e r f a c e should l i e w e l l above t h a t f o r low index f a c e s i n c o n t a c t with Rm. The o b s e r v a t i o n s of Ohdomari e t a l . [271 t h a t a - s c , r a t h e r t h a n c-sc, d e p o s i t s from t h e vapor on c l e a n c-sc s u r f a c e s i n d i c a t e t h a t i n t h i s p r o c e s s a l s o t h e i n t e r f a c e attachment r a t e i s h i g h e r on a-sc t h a n on c-sc.

Recently C u l l i s e t a l . [281 r e p o r t e d formation of a-sc on c-sc S i i n h i g h energy nanosecond p u l s i n g where t h e c a l c u l a t e d regrowth speed exceeded some c r i t i c a l valve, u*, c h a r a c t e r i s t i c of t h e s u r f a c e o r i e n t a t i o n . u* was 15-20 meters/sec. on (100) f a c e s and somewhat l e s s t h a n t h i s v a l u e on (111) f a c e s . I n t h e model of Spaepen and t h e a u t h o r 1151, t h e i n t e r f a c i a l undercooling a t which t h e t r a n s i t i o n from c r y s t a l t o amorphous regrowth occurs should be a t l e a s t a s l a r g e a s t h e displacement of T from T ^@200-250° according t o t h e e s t i m a t e of Donovan e t a l . t91.

aR cR'

Taking 250° a s t h e i n t e r f a c i a l undercooling a t t h e growth t r a n s i t i o n we o b t a i n , from eq. (41, ui .-. 35 meters/sec. on (100) which i s a b o u t a f a c t o r of l o 2 ^lower

t h a n t h e speed of sound i n Ilm S i . This r e s u l t i m p l i e s t h a t t h e p r o d u c t of t h e growth s i t e f a c t o r , f , and a c t i v a t i o n f a c t o r , e -Qu'RTi, where Q i s an a c t i v a t i o n

u

energy f o r growth, N - lo-' on (100) a t 1450°K; t h u s , f 9 0.01 and QU 4 0.57 e.v.

Kurz [29] e s t i m a t e s t h a t regrowth speeds of 50 t o 60 meters/sec., a t l e a s t , were reached by a-sc i n t h e picosecond l a s e r p u l s i n g experiments. I t appears t h a t t h e a c t i v a t i o n energy f o r t h i s regrowth should be determined p r i m a r i l y by t h e change i n l o c a l o r d e r a t t h e i n t e r f a c e and s o should d i f f e r l i t t l e from t h a t f o r regrowth t o c-sc. I f s o , it would f o l l o w t h a t t h e growth s i t e f r a c t i o n on (100) would be no more t h a n 10-I of t h a t a t t h e a-sc-Rm i n t e r f a c e s o t h a t Q 0.3 e.v.

u

The evidence on whether t h e r e v e r s e t r a n s i t i o n a - s c + Rm o c c u r s i n S i a t temperatures w e l l below TcR s t i l l appears c o n t r a d i c t o r y . B a e r i e t a l . [301 i n f e r r e d from t h e i r thermal a n a l y s i s and m i c r o s t r u c t u r a l o b s e r v a t i o n s t h a t an a-sc f i l m on c-sc S i d i d , i n f a c t , melt a t t e m p e r a t u r e s s e v e r a l hundred degrees below TcR d u r i n g pulsed e-beam h e a t i n g . Recently C u l l i s e t a l . [311 observed a c e l l u l a r s e g r e g a t i o n p a t t e r n of I n w i t h i n an amorphous a-sc l a y e r regrown on S i a f t e r 2.5 nanosecond l a s e r i r r a d i a t i o n of an I n i o n amorphized a-sc l a y e r . This s e g r e g a t i o n was explained by supposing t h a t t h e o r i g i n a l l a y e r melted t o h i g h l y undercooled Rm.

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C1-268 JOURNAL DE PHYSIQUE

F u r t h e r i n d i r e c t evidence f o r t h e occurrence of an a-sc+Rm t r a n s i t i o n i s provided by t h e "explosive" c r y s t a l l i z a t i o n of t h i n f i l m s of a-sc Ge i n which t h e

c r y s t a l l i z a t i o n f r o n t moves a t speeds i n t h e meters/sec. range. Gilmer and Leamy [32] p r e s e n t e d a model f o r t h i s p r o c e s s according t o which t h e r a p i d growth i s s u s t a i n e d through a t h i n l a y e r of Rm formed between t h e a-sc and c-sc by l i b e r a t i o n of t h e h e a t of c r y s t a l l i z a t i o n . To t r i g g e r t h e r a p i d growth, it i s n e c e s s a r y o n l y t h a t Ti r e a c h a temperature s l i g h t l y exceeding T

aR'

I n c o n t r a s t , Knapp and Picraux 1331 and Kokorowski e t a l . [I41 have p r e s e n t e d evidence t h a t a-sc l a y e r s on c-Si p e r s i s t , a t l e a s t f o r s h o r t p e r i o d s , t o temperatures w e l l above t h e c a l c u l a t e d TaR. I n p a r t i c u l a r , a s a l r e a d y noted, Kokorowski e t a l . r e p o r t e d t h a t t h e c-sc r a t e s of regrowth i n a - s c f o l l o w , w i t h i n experimental e r r o r , a s i n g l y a c t i v a t e d Arrhenius c o u r s e from low t e m p e r a t u r e s t o TcR. However, s i n c e t h e regrowth r a t e mustga t o z e r o a t t h e a - s c + - s c e q u i l i b r i u m t h i s behavior i n d i c a t e s , provided t h e t e m p e r a t u r e s were c a l c u l a t e d c o r r e c t l y , t h a t the e q u i l i b r i u m temperature, T a c must l i e f a r above TcR w i t h t h e c o r o l l a r y t h a t t h e thermodynamic Tat must b e w e l l below TcR 131. The a u t h o r s u g g e s t e d [31 t h a t t h e a p p a r e n t s u p e r h e a t i n g i n t h e s e experiments might r e f l e c t t h e r e s i s t a n c e of t h e random network (cRN) s t r u c t u r e t o im n u c l e a t i o n . An e s t i m a t e based on simple nucleation t h e o r y , i n c o n j u n c t i o n with t h e undercooling behavior of Rm, i n d i c a t e d t h a t a super- h e a t i n g of o r d e r 20% of TaR may be r e q u i r e d f o r measurable n u c l e a t i o n w i t h i n an i d e a l CRN s t r u c t u r e . Thus, a t s m a l l s u p e r h e a t i n g Rm wouId only n u c l e a t e a t t h e ex- t e r n a l s u r f a c e o r a t i n t e r n a l i m p e r f e c t i o n s , e . g . v o i d s , and t h e r e could b e ,

depending on t h e method of specimen p r e p a r a t i o n and s u r f a c e t r e a t m e n t , widespecimen t o specimen v a r i a t i o n s i n t h e k i n e t i c s and morphology of i t s formation.

Acknowledgement: This r e s e a r c h was supported i n p a r t by a N a t i o n a l Science Foundation g r a n t NSF-DMR80-20247. The a u t h o r h a s b e n e f i t e d from d i s c u s s i o n w i t h E.P. Donovan, J . M . Poate and F. Spaepen.

References.

1. BAGLEY B.G. and CHEN H.S., Am. I n s t . Phys. Conf. Proc. 50 (1979) 97.

2. SPAEPEN F. and TURNBULL D., Am. I n s t . Phys. Conf. Proc. 50 (1979) 50.

3. TURNBULL D . , Mats. Res. Soc. Sym. Proc.,Ed. by P i c r a u x S.T. and Choyke W . J . North Holland, Amsterdam (1982) 103-108.

4. CNEN H.S. and TURNBULL D . , J . Appl. Phys., 40 (1969) 4214.

5. RUDEE M.L., Thin S o l i d Films, 3 (1972) 207,

6. LYTLE F.W., SAYERS D.E. and EIKUM A.K.. 3. Non-Cryst. S o l i d s , 13 (1973) 68.

7. TEMKIN R.J. and PAUL W., Proc. F i f t h I n t e r n a t i o n a l (Garmisch) Conference on Amorphous Semiconductors, ~ d . by Stuke 3. and Brenig W., Taylor and F r a n c i s

(1974) 1193-1200.

8 . FlPN J.C.C. and ANDERSEN H . , J. Appl. Phys. 2 (1981) 4003.

9. DONOVAN E.P., SPAEPEN F., TURNBULL D., POATE J . M . and JACOBSON D.C., submitted t o Phys. Rev. L e t t e r s .

10. CZEPREGI L., KENNEDY E.F., MAYER J . W . and SIGMON T.W., J. Appl. PhyS. 49

(1978) 3906.

11. SPAEPEN F., P h i l . Mag. 30 (1974) 417.

12. BELL R. J. and DEAN P., Phys. Chem. GkiSS 9 (1968) 125.

13. TURNBULL D. and CECH R.E., J. Appl. Phys. 1 (1950) 804.

14. KOKOROWSKI S.A., OLSEN G.L. and HESS L.D., J. Appl. Phys. 2 (1982) 921.

15. SPAEPEN F. arid TURNBULL D . , "Laser Annealing of S o l i d s " , Ed. by P o a t e J . M . and Mayer J . W . , i n p r e s s , Academic P r e s s , NY (1982).

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TURNBULL D., J. de Physique 35, (1974) C-4.1-9.

TURNBULL D. and BAGLEY B.G., " T r e a t i s e on S o l i d S t a t e Chemistry", Ed. by Hannay N.B., Plenum P r e s s , 5 (1975) 513-554.

CORIELL S.C. and TURNBULL D., Acta M e t a l l u r g i c a , i n p r e s s .

WALKER J . L . , Presented i n B. Chalmers, " P r i n c i p l e s of S o l i d i f i c a t i o n s " , Chapter 4, Wiley, NY (1964).

TURNBULL D . , Contemporary Phys. 10 (1969) 873.

BAKER J.C. and CAHN J . W . , " S o l i d i f i c a t i o n " , Am. Soc. Metals, Metal Park, Ohio (1971) 23-58.

WHITE C.W., WILSON S.R., APPLETON B.R. and YOUNG F.W., J. Appl. PhyS. 51

(1980) 738.

A Z I Z M.J., 3. Appl. Phys. 53 (1982) 1158.

LIU P.L., YEN R., BWlEMBERGEN N., and HODGSON R.T., Appl. PhyS. L e t t . 34

(1979) 864.

TSU R., HODGSON R.T. , TAN T.Y. and BAGLIN J.E. , Phys. Rev. L e t t . 42 (1979) 1356.

LIU J . M . , YEN R., DONOVAN E.P., BLOEMBERGEN N. and HODGSON R.T., ~ p p l . Phys.

L e t t . 38 (1981) 617.

OHDOMARI I., KAKUMU M., SUGAHARA H . , BORI M. and SAITO T., J. Appl. Phys- 52 (1981) 6617.

C U L ~ S A.G., WEBBER H. C., CHEW N.G. , POATE J . M . and BAERI P. , PhyS . Rev. L e t t . 49 (1982) 219.

KUR;~-H., P r i v a t e ~ o m u n i c a t i o n , s e e a l s o LIU J.M., YEN R. ,KURZ H. and BLOEMBERGEN N., Appl. PhyS. L e t t . 39 (1981) 755.

BAERI P., FOTI G., POATE J.M. and CULLIS A.G., Phys. Rev. L e t t . 5 (1980) 2036.

CULLIS A.G., WEBBER H.C. and CHEW N.G., Appl. Phys. Rev. L e t t . % (1982) 998.

G I W R G.H. and LEAMY H . J . , "Laser and E l e c t r o n Beam P r o c e s s i n g of M a t e r i a l s " , Ed. by White C.W. and Peercy P.S., Academic P r e s s , NY (1980) 227-233.

KNAPP J . A . and PICRAUX S.T., Appl. Phys. L e t t . 38 (1981) 873.

H.F. MATARE.- You showed t h a t t h e energy o f c r y s t a l l i z a t i o n i s lower i n t h e c a s e of s i l i c o n t h a n i n t h e c a s e of germanium, i s t h e r e an e x p l a n a t i o n i n view o f t h e o t h e r p r o p e r t i e s which a r e q u i t e s i m i l a r ? With t h e h i g h e r m e l t i n g p o i n t o f s i l i c o n one would e x p e c t t h e o p p o s i t e .

D. TURNBULL.- The v a l u e s shown were e n e r g i e s s c a l e d t o t h o s e o f c r y s t a l l i z a t i o n . I n a c t u a l magnitude t h e energy o f c r y s t a l l i z a t i o n of t h e amorphous phase o f s i l i c o n i s approximately e q u a l t o t h a t of germanium. We have no e x p l a n a t i o n f o r why t h e s c a l i n g c o r r e l a t i o n does n o t work i n t h i s c a s e . However t h e r e seems t o b e no w e l l founded microscopic t h e o r y f o r why t h e s c a l i n g r e l a t i o n s do work f o r such o t h e r p r o p e r t i e s a s t h e s h e a r moduli and s e l f d i f f u s i o n c o e f f i c i e n t s .

R. BARTON.- Are you o f t h e o p i n i o n t h a t most amorphous f i l m s encountered i n p r a c t i c e r e p r e s e n t t h e m e t a s t a b l e s t a t e o r some o t h e r "frozen i n " u n s t a b l e s t a t e ? And what i s t h e o r i g i n o f your f r e e energy curve f o r a-Si ?

D. TURNBULL.- My view i s t h a t most amorphous S i o r Ge f i l m s can be d e s c r i b e d a s

"frozen" from a m e t a s t a b l e random network s t r u c t u r e i n which v a r i o u s i m p e r f e c t i o n s ( v o i d s , e t c ) have been i n c o r p o r a t e d . The method of c a l c u l a t i o n o f t h e f r e e energy of t h e amorphous r e l a t i v e t o t h e c r y s t a l l i n e s t a t e o f S i is d e s c r i b e d i n t h e w r i t t e n t e x t of t h e paper.