THERMAL MODELING OF CW LASER-CRYSTALLIZATION OF SOI

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THERMAL MODELING OF CW LASER-CRYSTALLIZATION OF SOI

J. Hode, J. Joly

To cite this version:

J. Hode, J. Joly. THERMAL MODELING OF CW LASER-CRYSTALLIZATION OF SOI. Journal

de Physique Colloques, 1983, 44 (C5), pp.C5-343-C5-350. �10.1051/jphyscol:1983551�. �jpa-00223137�

THERMAL MODELING OF

CW

LASER-CRYSTALLIZATION OF SO1

J . M . Hode and J . P . J o l y

L. E.T. I., Commissariat a' Z 'Energie Atomique, 85X, 38041 GrenobZe Cedex, France

Rgsumk - Dans l e c a d r e d'une gtude de r e c r i s t a l l i s a t i o n de s i l i c i u m s u r i s o - l a n t p a r f a i s c e a u l a s e r c o n t i n u balayk nous prgsentons d e s modsles thermiques a n a l y t i q u e s prenant en compte l e s d i f f Q r e r ~ t s paramstres physiques ( c o n d u c t i v i t 6 thermique, r g f l e c t i v i t k , c h a l e u r l a t e n t e ) . L ' i n f l u e n c e de c e s paramstres e s t 6valuke e t comnparge aux r g s u l t a t s expkrimentaux.

A b s t r a c t - We p r e s e n t a n a l y t i c a l thermal models f o r CW l a s e r - c r y s t a l l . i z a t i o n o f SOI, t a k i n g account of v a r i o u s p h y s i c a l parameters ( t h e r m a l c o n d u c t i v i t y , r e f l e c t i v i t y , l a t e n t h e a t )

.

The i n f l u e n c e of t h e s e parameters i s e v a l u a t e d and compared t o experiment.

1NTRODUCTION : Many thermal models of CW l a s e r a n n e a l i n g o r CW l a s e r c r y s t a l l i z a t i o n have been a l r e a d y given (1-10). For good understanding of what happens d u r i n g c r y s - t a l l i z a t i o n it i s n e c e s s a r y t o develop an e l a b o r a t e model t a k i n g account of t h e i n f l u e n c e of each p h y s i c a l parameter. A s it was impossible t o t r e a t t h e e n t i r e problem, we developed s e v e r a l models t o p o i n t o u t t h r e e d i f f e r e n t e f f e c t s :

.

^A s t e a d y - s t a t e one t o show t h e i n f l u e n c e of t h e change o f r e f l e c t i v i t y a t t h e m e l t i n g p o i n t .

.

Two n o n - s t a t i o n a r y s o l u t i o n s u s i n g G r e e n ' s f u n c t i o n f o r a t h r e e - l a y e r s t r u c t u r e t o t a k e account of scan speed and l a t e n t heat.

I n o u r c a l c u l a t i o n s , we used an e l l i p t i c a l beam w i t h a Gaussian i n t e n s i t y p r o f i l e I ( x , y ) = I. exp -

[[$]'+ kwl2]

⁰

^{= L}

^{) .}

r@w2

We a l s o used t h e v a r i a b l e s and t h e normalized v a r i a b l e s i n d i c a t e d i n Fig. 1 . INFLUENCE OF THE SILICON REFLECTIVITY : It i s w e l l known t h a t s i l i c o n r e f l e c t i v i t y s h o r ~ s an a b r u p t t r a n s i t i o n a t t h e m e l t i n g p o i n t (from a s o l i d s t a t e v a l u e o f Rs =

44

% t o a l i q u i d v a l u e o f R 1 = 72 ,%). Designating P a s t h e power r e q u i r e d t o o b t a i n a l i n e a r temperature r i s e OM corresponding t o t h e m e l t i n g temperature we o b t a i n OM = ~ ( l - ~ s ) / ( 2 f i k w ) ( 2 , 3 ) . I f we

i n c r e a s e P from AP a l i t t l e d i s k of r a d i u s AR w i l l melt and t h e r e f l e c t i v i t y w i l l be R 1 on t h i s d i s k . According t o appendix I1 t h e t e m p e r a t u r e a t t h e edge o f t h i s d i s k w i l l be a t t h e f i r s t o r d e r i n AR and AP :

The t e m p e r a t u r e a t t h e c e n t e r of t h e beam will b e P ( 1 - R s ) I + -

---[

^{2 f i k w P 1 - R s w}

d~rection

From ( 1 ) and ( 2 ) we o b t a i n

&

=

2

R 1 - RS dR ₍₄₎ P 1 - R s w and from ( 1 ) and ( 3 ) we o b t a i n t h e t e m p e r a t u r e v a r i a t i o n at t h e c e n t e r o f t h e beam

F i g . 1 : our parameters

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983551

C5-344 JOURNAL DE PHYSIQUE

We can s e e t h a t when m e l t i n g o c c u r s , t h e temperature a t t h e c e n t e r of t h e bean tends t o d e c r e a s e with i n c r e a s i n g power. This i n c o n s i s t e n c y i s c l e a r l y demonstrated by t h e t h e r m a l p r o f i l e p r e s e n t e d i n f i g ( 2 , a ) . So we concluded t h a t it must e x i s t a t r a n s i t i o n r e g i o n a t t h e edge of t h e molten zone where t h e r e f l e c t i v i t y v a r i e s c o n t i n u o u s l y from R 1 t o Rs and where t h e temperature i s OM ( f i g . 2 , b ) . This r e g i o n has been d e s c r i b e d by HAWKINS and BIEGELSEN ( 1 1 ) a s composed of s o l i d l a m e l l a e i n l i q u i d s i l i c o n and i s c l e a r l y v i s i b l e on t h e micrographs of samples a f t e r SECCO ETCHING ( 12) ( f i g . 3 ) . To e v a l u a t e t h i s r e g i o n (width R2) and t h e f u l l y melted r e g i o n (width R ~ ) , we chose a polynomia.1 form o f t h e r e f l e c t i v i t y -

" i R '

^{+ $}

Rs + ( R 1 - R S )

R2 - R 1

Which g i v e s Rs f o r R = Rg and R 1 f o r R = R1. For each v a l u e of R 1 , we must choose p, $ and R t o a s s u r e a f l a t t e m p e r a t u r e p r o f i l e f o r R , ,( R

<

^R2, by minimizing

2

2

(ad? /aR) dR with r e s p e c t t o p, $ and R2 ( + i s t h e normalized temperature

$ = O/ Omax, Omax = P( 1-Rs) / ( 2 f i k w ) ) . This computer work can be done i n r e l a t i v e l y s h o r t CPU t i m e s because t h e s u r f a c e t e m p e r a t u r e d i s t r i b u t i o n corresponding t o such a r e f l e c t i v i t y i s reduced t o a s i n g l e i n t e g r a l on a f i n i t e i n t e r v a l a s demonstrated i n appendix I. These c a l c u l a t i o n s were done f o r s e v e r a l v a l u e s of R 1 and Rs l e a d i n g t o d i f f e r e n t v a l u e s of ( R 1 - R S ) / (1- R S ) which i s t h e important parameter of t h e problem. Then i t has been p o s s i b l e t o c a l c u l a t e t h e width o f t h e f u l l y melted zone v e r s u s t h e beam power f o r d i f f e r e n t v a l u e s of t h e r a d i u s and d i f f e r e n t caping l a y e r s . The r e s u l t i s shown i n f i g .

4

and shows good agreement w i t h experiment. (N.B. : A l l c a l c u l a t i o n s were done and compared t o experiment f o r Poly S i d e p o s i t e d on bulk s i l i c o n and a r e e a s i l y t r a n s p o s e d t o a SO1 s t r u c t u r e using t h e method exposed i n t h e next p a r a g r a p h ) .

DYNAMIC CASE FOR A THREE LAYER STRUCTURE : We d e r i v e d t h e Green's f u n c t i o n t o t h e h e a t e q u a t i o n f o r a t h r e e l a y e r s t r u c t u r e and apply t o a scanning Gaussian e l l i p t i - c a l beam. The r e s u l t s a r e p r e s e n t e d i n Appendix 111 and l e a d t o t y p i c a l t h e r m a l p r o f i l e s ( f i g . 5 ) and i s o t h e r m a l l i n e s ( f i g . 6 ) . The a n a l y t i c a l i n t e g r a t i o n presen- t e d i n appendix 111 h a s been c a r r i e d a s f a r a s p o s s i b l e so it i s n e c e s s a r y t o perform a numerical i n t e g r a t i o n t o c a l c u l a t e t h e temperature. This i n t e g r a t i o n r e - q u i r e s l o n g CPU t i m e s even on a powerful computer s i n c e t h e a n a l y t i c a l i n t e g r a t i o n l e a d s t o a double i n t e g r a l . So, we p r e s e n t now an approximate model assuming a one-

SCAN SPEED : 80 CM/s Scan

BEAM RADIUS : 50 PM ^direction

I

~ i g . 6 : I s o t h e r m a l l i n e s o b t a i n e d by t h e t h r e e - l a y e r model proposed i n Anp. I11

( . 5 Poly S i on 1 . 5

u

t h e r m a l oxide )

scan

beam radius : z5pm scan vcloslty: Bocrnls prcheatlng ;500'C

100 50 0 50 100

F i g . 7 : Isothermal lires o b t a i n e d w i t h t h e sim- p l i f i e d model

(same c o n d i t i o n s a s i n P i g . 6 )

Beam power ( W ) 0

F -

: F u l l y melted width v e r s u s beam power

Fig. 5 : Typical temperature d i s t r i b u t i o n a l o n g t h e x ax i s

( - 5 v p o l y S i on 1.5

v

t h e r m a l o x i d e ) x(pm)

INCOMING POWER : 11 W

I

C5-346 JOURNAL DE PHYSIQUE

dimensional h e a t flow i n t h e i n s u l a t i n g l a y e r and i n t h e upper s i l i c o n l a y e r . The t e m p e r a t u r e d i s t r i b u t i o n i n t h e b u l k s i l i c o n i s given by t h e model proposed by NISSIM et

(4)

and temperature i n t h e upper l a y e r s i s o b t a i n e d by i n t e g r a t i n g t h e h e a t flow. Isothermal l i n e s c a l c u l a t e d w i t h t h i s model a r e p r e s e n t e d i n F i g . 7 and can b e compared t o t h o s e p r e s e n t e d i n F i g . 6 . We can s e e a r e l a t i v e e r r o r of about 6 % f o r a beam v e l o c i t y of 80 cm/s. It can be noted t h a t t h i s e r r o r d e c r e a s e s w i t h d e c r e a s i n g v e l o c i t y . I n t h i s c a s e , it i s p o s s i b l e t o apply t h e i n v e r s e

Kirchhoff t r a n s f o r m ( 1 3 ) t o t h e b u l k s i l i c o n temperature i n o r d e r t o o b t a i n t h e a c t u a l temperature r i s e , s i n c e t h e d i f f u s i v i t y of s i l i c o n i s r e l a t i v e l y c o n s t a n t a t high t e m p e r a t u r e s . S i n c e t h e S i 0 2 t h e r m a l c o n d u c t i v i t y have l i t t l e temperature dependence t h e temperature r i s e i n t h e St02 i s o b t a i n e d by i n t e g r a t i n g t h e h e a t flow. The a c t u a l t e m p e r a t u r e r i s e i n t h e upper s i l i c o n l a y e r i s a l s o o b t a i n e d by i n t e g r a t i n g t h e h e a t flow b u t we must t a k e account of t h e temperature dependence o f t h e t h e r m a l c o n d u c t i v i t y of s i l i c o n . F i n a l l y , t h e temperature r i s e w i l l b e g i v e n by :

P ^{2 2 2}

1 X o ) d 0 - ( a l - 2 ) e

-

^{x2/w2 e - ~}

/p

JI,,

Y , a l l I T ~ W

INFLUENCE OF THE LATENT HEAT : As r e p o r t e d i n Appendix I11 we a p p l i e d t h e Green's f u n c t i o n c a l c u l a t e d i n t h e same appendix t o a moving s t r a i g h t c r y s t a l l i z a t i o n f r o n t . An example of t h e t e m p e r a t u r e r i s e due t o t h e l a t e n t h e a t i s p r e s e n t e d i n F i g . 8

.

It shows t h a t t h e e f f e c t of l a t e n t heat i s v e r y important and w i l l a f f e c t s t r o n g l y t h e shape of t h e t e m p e r a t u r e p r o f i l e n e a r t h e c r y s t a l l i z a t i o n f r o n t . The high v a l u e of t h e t e m p e r a t u r e r i s e w i l l cause a

g r e a t s u p e r c o o l i n g which w i l l be r e s p o n s i b l e f o r i n t e r f a c e i n s t a b i l i t y and d e n d r i t i c growth a s r e p o r t e d p r e v i o u s l y ( 1 4 ) . It must be noted t h a t t h e s o l u t i o n proposed i n Appendix I11 i s a l s o reduced t o a double i n t e g r a l and t h a t t h e numerical i n t e g r a t i o n i s v e r y l o n g . But we c a n ' t apply t h e method proposed i n t h e l a s t paragraph because t h e s i z e o f t h e h e a t source i s v e r y s h o r t s o t h a t t h e h e a t flow i s three-dimensional i n t h e i n s l u a t i n g l a y e r . P r e s e n t e f f o r t s c o n c e n t r a t e on t r e a t i n g t h e c a s e of a curved c r y s t a l l i z a - t i o n f r o n t and on adding it t o t h e t e m p e r a t u r e d i s t r i b u t i o n produced by t h e beam ( p r e s e n t e d i n t h e l a s t p a r a g r a p h ) .

CONCLUSION : We have p r e s e n t e d a n a l y t i c a l models i n c o r p o r a t i n g e a c h p h y s i c a l parameter of CW l a s e r c r y s t a l l i z a t i o n of SOI. (tempera- t u r e dependent t h e r m a l c o n d u c t i v i t y and s u r f a c e r e f l e c t i v i t y , l a t e n t h e a t ) . They a r e a v e r y powerful t o o l f o r good comprehension of l a s e r - i n d u c e d c y r s t a l l i z a t i o n .

8 : Temperature r i s e due t o l a t e n t

!

* .

5 p Poly S i on 1.5 p thermal o x i d e )

I front '46th : 50pm

ssen veloclly: 80 cmls

We a r e going t o d e s c r i b e a new way t o c a l c u l a t e t h e temperature d i s t r i b u t i o n under t h e same hypotheses a s Lax ( 2 ) . ( o n l y i n case of s u r f a c e a b s o r p t i o n , i . e . W = m ) 1 . Surface temperature ( Z = 0 ) : Lax's formula can be w r i t t e n a s f o l l o w s :

f m T ( R ) = JSE

k

lrn

^{J o ( h R)} F (A) dh I ,w i t F A = JO(ARI f ( ~ ) R ~ R (I,*) I

lfrn&

^{iARsin0 + e-iARsin@}

Using t h e r e l a t i o n s h i p J o ( XR) =

-

2 r

1

^{dO (1,3)}

We can w r i t e ( F ( X ) i s an even f u n c t i o n )

A

T = - Iow 2.rrk

i,' doi'--iX~sin'

^{F(A) dA I F} d e s i g n a t e s t h e F o u r i e r t r a n s f o r m of

-00

F, we o b t a i n T =

low

A F (RsinO) do (I,&)

A

Let U s now c a l c u l a t e F. Using eq. ( I , 2 ) and (1,3) we f i n d :

'0

A

S e t t i n g u = y c o t g o we o b t a i n F ( y ) = 2 f

(4-1

du ( i , 5 ) Two i n t e r e s t i n g c a s e s :

A 2 2

.

@aussian beam : f ( R ) = e-" s o t h a t F(.) = 2e y 2

1:

^{e-. du = fie-y}

.

Truncated Gaussian beam :

f i

f ( ~ ) = 0 i f R > a F ( y ) = 0 i f y > a

-R 2 . A 2

f ( ~ ) = e 1 f R < a F ( ~ ) = &e-' e r f ( d a v ) i f y < a

2. ~ u l k temperatune : Lax's formula i s : T =

-

O w o R F( 1 ) k 0

C5-348 JOURNAL DE PHYSIQUE

-/'Iz ^F(A)

^A

S e t t i n g G ( X ) = e e q ( 1 , b ) g i v e s T =

-

~ ( R s i n O ) dO

The f o u r i e r t r a n s f o r m of

.-IX1

^{i s 2 Z} s o t h a t 2 2

z

^{+ Y}

Avery l o n g i n t e g r a t i o n shows t h a t

= 1 coshB cosa

-

^{) y2}

+ z2

^{o h}2 - o s 2 a

[za

^- s i n h 2B With cosh

a

^{= ( A + B )} /2 and cosa = ( A - R) / 2

2

and B =

z2 +

^y2

A =

Jz2 ^{+ I R}

^{- y,}

F i n a l l y , we f i n d T ( R , Z ) =

-

1ow Ilk

APPENDIX I1 :

We apply t h e r e s u l t of Appendix I t o t h e c a s e o f an a b r u p t change of r e f l e c t i v i t y f o r R = a

.

( R 1 i f R < a , Rs i f R > a ) .

The r e f l e c t i v i t y can be w r i t t e n Rs + ( R 1 - R S ) Y ( a - R ) where Y i s t h e h e a v i s i d e f u n c t i o n .

The h e a t source i s given by f ( R ) 5

( z )

w i t h f ( R ) = ( ~ - R S ) I ~ e-R

[

1 Rl-Rs ~^{1 -Rs} (-R a

1

According t o appendix I , we f i n d T(R,Z = 0 ) :

T ( R , O ) = 2& kw

( w i t h B=

5

TT i f R

:

^{a and B = Arcsin}

[z]

i f ~ > a ) P ( I - R S )

Two i n t e r e s t i n g v a l u e s : T(0,O) = ---

2 6 kw

[

^{R s}

2fi kv 1 - Rs

1

APPENDIX I11 :

Green's function for a thkee-layer structure

-

applications

We applied the technique described by Burgener and Reedy ( 9 ) to the S O 1 structure (~ig. 9 ) . Using the same notations as those we derived from the Green's function (after Laplace transform).

-

G , = - 4 1 nD,

\r[e-il

^{IZ-Z'I +}

^{~ ~ ' 1}

⁺ (clrll/n2) ^{2 tanh} +

^(n,

tanh ^a,)

^- (n2

a2)

C O S ~

(n,

z) r l

-

^(an, /

n2)2]

e

-n(

^OQS~('I., z') JO

(513) - 5

cosh

(n,

a,) D

I

-

^{2a cosh ('1, z')}

G 3 =

-

' I 1

1

+

zanh

(n2

a2) With D = 1 +

a

tanh

(n,

a, ) ^'19

'1 2

-

^{+ tanh ('1 a2)}

2

1. Temperature distribution

induced-?~-an_-e11i~t_ic~L~?earn

The integrations are identical to those presented by Burgener and Reedy ( 9 ) We obtain :

With HI =

C5-350 JOURNAL DE PHYSIQUE

as21 exp

t%

( * 1 + ~ 2 - ~ ) ] 1

=

2

^~1 f o ~ h ( ~ ~ , ) cosh(Q2b-2)

a +

tanh (%A2) Q2

1 t.

*

t a n h ( R , ~ ~ )

~1 = 1

+ @-

t a n h ( Q l ~ , )

n2

Q2

a +

tanh (%A2)

Q2

The h e a t source i s Q = Lv 6 ( x '

-

^{v t ' ) i f} z ' <a, and L. <y' <w Q = 0 anywhere e l s e

A f t e r i n t e g r a t i o n w e o b t a i n :

+03 71/2

K . e-iSXsinO sin( FcosO)

R e k J

]

^{cos (SY cos0) dO}

"

^{7% +} iV,sinO ScosO

With K1 = 1 - cash ( R I Z )

D'cosh ( 0 1 ~ , ) K Z = H z s i n h ( R , A,) K 3 = H 3 s i n h ( 0 1 lbl)

' ( 1 , "1 A , = 3 V , = K

REFERENCES w

4

( 1 ) H.E. CLINE add T.R. ANTHONY,J.A.P., Bol. 48, No 9 (3895) 1977 (2) M.

LAX,

J.A.P., Vol. 48, N O 9 (3919) I977

( 3 ) M. 1SS(, Appl. Phys. L e t t . , 33 ( 7 8 6 ) 1978

( 4 ) Y . I . EISSIM, A. LEITOILA, R.B. GOLD, J-F. GIBBONS, J.A.P., Vol. 51, No 1 (274) I980

( 5 ) J . M . HODE ,DEA Report 1980

( 6 ) S.A. KOKOROWSKI, G.L. HOLSON, L.D. MESS, Laser and E l e c t r o n Beam S o l i d I n t e r - a c t i o n s and M a t e r i a l s P r o c e s s i n g (North Holland, New-York, 1981) P. I39 ( 7 ) J.E. N O D Y and R. H. ITENDEL, J.A.P., Vol. 53, No 6 (4364) 1982

( 8 ) I . D . COLDER and R. SUE, J.A.P.

,

Vol. 53, No 11 (7545) 1982 ( 9 ) M.L. BURGETTER and R.E. REEDY, J.A.P., Vol. 53, NO 6 (4357) 1982

(10) C . I . DROWLEY, C.HU, T . I . KAMINS, Electrochemical S o c i e t y S p r i n g Meeting (Montreal, 1982) Extended A b s t r a c t No 145 p. 234

( 1 1 ) kl.G. HAWKINS, D.K. BIEGEGSEN, Appl. Phys. L e t t . , Vol. 42, No 4 1983 ( 1 2 ) SECCO drARAGONA, J . Electrochem. Soc., Vol. I 1 9 (948) I972

( 13) H.S. CARSLAW and J. C. JAEGER, Conduction of h e a t i n s o l i d s 2nd ed. (Oxford U n i v e r s i t y , London, 1949) P. 89

(14) J . M . HODE, J.P. JOLY, P . JEtTCH Paper p r e s e n t e d a t the ~ l e c t r o c h e m i c a l S o c i e t y Spring Meeting ( ~ o n t r e a l , 1982)