MODELING THE EBIC MEASUREMENTS OF DIFFUSION LENGTHS AND THE RECOMBINATION CONTRAST AT EXTENDED DEFECTS

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MODELING THE EBIC MEASUREMENTS OF DIFFUSION LENGTHS AND THE

RECOMBINATION CONTRAST AT EXTENDED DEFECTS

C. Donolato

To cite this version:

C. Donolato. MODELING THE EBIC MEASUREMENTS OF DIFFUSION LENGTHS AND THE

RECOMBINATION CONTRAST AT EXTENDED DEFECTS. Journal de Physique Colloques, 1989,

50 (C6), pp.C6-57-C6-64. �10.1051/jphyscol:1989605�. �jpa-00229636�

R E W E DE PHYSIQUE

APPLIQUEE

Colloque C6, Supplbment au n06, Tome 24, Juin 1989

MODELING THE EBIC MEASUREMENTS OF DIFFUSION LENGTHS AND THE RECOMBINATION CONTRAST AT EXTENDED DEFECTS

C. DONOLATO

C N R , I s t i t u t o Lamel, V i a C a s t a g n o l i 1, I-40126 Bologna, I t a l y

Resume

-

h e d e s c r i p t i o n a l t e r n a t i v e , basee s u r l e concept de p r o b a b i l i t e d e e c t e , e s t proposee pour 1 'i n t e r p r e t a t i o n d'experiences typiaues

r e a l isees par l a technique du courant i n d u i t p a r f a i s c e a u d ' e l e c t r o n s (EBIC) associee au microscope e l e c t r o n i q u e

a

balayafle. On montre que l a d i s t r i b u t i o n

9

( r ) de c e t t e p r o b a b i l i t e dans 1 ' e c h a n t i l l o n o b e i t

a

une equation de d i f f u s i o n homogene e t que l e courant i n d u i t r e s u l t e de l ' e x c i t a t i o n des p r o p r i e t 6 s de l ' e c h a n t i l l o n par l a f o n c t i o n de qeneration du faisceau

e l e c t r o n i q u e . L ' o b j e c t i f des experiences ESIC e s t de determiner l e s paramstres e s s e n t i e l s c a r a c t e r i sant 1 e semi conducteur ou 1 e defaut, d o n t depend.

D i f f e r e n t e s m6thodes d'acces 2 c e t t e i n f o r m a t i o n sont discutees e t quelques nouvelles p o s s i b i l i t 6 s sont presentees.

A b s t r a c t

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An a l t e r n a t i v e d e s c r i p t i o n i s proposed o f t y p i c a l experiments t h a t a r e performed w i t h t h e e l e c t r o n beam induced c u r r e n t (EBIC) technique o f t h e scanning e l e c t r o n microscope, on t h e b a s i s of t h e n o t i o n o f c h a r g e - c o l l e c t i o n p r o b a b i l i t y . I t i s shown t h a t t h e d i s t r i b u t i o n p ( r ) o f t h i s p r o b a b i l i t y i n t h e specimen obeys a homogeneous d i f f u s i o n equation, and t h e induced c u r r e n t r e s u l t s from probing t h i s sample p r o p e r t y w i t h t h e generation f u n c t i o n o f t h e e l e c t r o n beam.

EBIC experiments aim a t determining t h e e s s e n t i a l semiconductor o r d e f e c t parameters upon which i s dependent. D i f f e r e n t methods of r e c o v e r i n g t h i s i n f o r m a t i o n a r e discussed and some new p o s s i b i l i t i e s a r e presented,

1

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INTRODUCTION

The e l e c t r o n beam induced c u r r e n t (EBIC) technique o f t h e scanning e l e c t r o n microscope (SEM) has been used e x t e n s i v e l y t o c h a r a c t e r i z e semiconductor m a t e r i a l s and devices; review a r t i - c l e s on t h e p r i n c i p l e s and a p p l i c a t i o n s o f EBIC are a v a i l a b l e 11-31.

This paper aims a t demonstrating t h a t d i f f e r e n t EBIC experiments can be given a common pheno- menological d e s c r i p t i o n u s i n g t h e n o t i o n o f charge c o l 7 e c t i o n p r o b a b i l i t y , i.e. t h e proba- b i l i t y t h a t an i n j e c t e d m i n o r i t y c a r r i e r w i l l be c o l l e c t e d by t h e j u n c t i o n c o n t a c t and thus c o n t r i b u t e s t o t h e induced c u r r e n t . The concept o f one-dimensional charge c o l l e c t i o n proba- b i l i t y has been used by Possin and K i r k p a t r i c k /4/ t o analyze q u a n t i t a t i v e l y EBIC measure- ments on p l a n a r p-n j u n c t i o n s . Here t h i s n o t i o n w i l l be g e n e r a l i z e d t o t h r e e dimensions and moreover, i t w i l l be shown t h a t t h e c h a r g e - c o l l e c t i o n p r o b a b i l i t y d i s t r i b u t i o n i n a g i v e n s t r u c t u r e can be c a l c u l a t e d d i r e c t l y by s o l v i n g a homogeneous d i f f u s i o n equation, even i n t h e presence o f defects. The induced c u r r e n t can then be described as t h e r e s u l t o f probing t h i s l o c a l p r o p e r t y o f a device w i t h t h e generation volume o f t h e SEM e l e c t r o n beam.

As observed i n / 4 / , t h e c h a r g e - c o l l e c t i o n p r o b a b i l i t y d i s t r i b u t i o n c o n t a i n s a l l i n f o r m a t i o n about t h e motion o f m i n o r i t y c a r r i e r s t h a t can be obtained by t h e EBIC technique. Therefore c h a r a c t e r i z i n g q u a n t i t a t i v e l y a semiconductor s t r u c t u r e by EBIC means r e c o v e r i n g t h i s d i s t r i - b u t i o n ( o r the e s s e n t i a l parameters upon which i t i s dependent), under t h e assumption t h a t t h e probe f u n c t i o n o f t h e e l e c t r o n beam (i.e. t h e s p a t i a l d i s t r i b u t i o n o f t h e generation) i s known. Typical methods used t o perform t h i s r e c o n s t r u c t i o n and new p o s s i b i l i t i e s a r e discus- sed, making reference t o s p e c i f i c experiments f o r c l a r i t y o f t h e discussion.

2

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THE CHARGE COLLECTION PROBABILITY

The usual way of d e s c r i b i n g an EBIC experiment f o l l o w s t h e sequence o f p h y s i c a l processes t h a t occur i n t h e semiconductor sample as a r e s u l t o f t h e e l e c t r o n bombardment, i . e . :

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1989605

F i g . 1

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The two e q u i v a l e n t d e s c r i p t i o n s o f charge c o l l e c t i o n . ( a ) : through t h e d e n s i t y p ( 1 ) o f beam-injected m i n o r i t y c a r r i e r s ; (b): by t h e charge c o l l e c t i o n p r o b a b i l i t y cp

(1).

a) t h e generation o f e l e c t r o n - h o l e p a i r s by t h e e l e c t r o n beam b ) t h e t r a n s p o r t o f beam-generated m i n o r i t y c a r r i e r s

c ) the c o l l e c t i o n o f m i n o r i t y c a r r i e r s by t h e j u n c t i o n .

The process a ) i s described by i n t r o d u c i n g t h e generation f u n c t i o n g ( 1 ) s - I 1

,

^which

g i v e s t h e generation r a t e o f p a i r s p e r u n i t volume a t t h e p o s i t i o n

1

i n t h e sample. A number o f a n a l y t i c a l approximations o f t h i s f u n c t i o n have been used, t o simp1 i f y t h e a n a l y t i c a l treatment o f t h e problem b) /I/, t h e s i m p l e s t generation model being t h a t o f t h e p o i n t source; Monte-Carlo s i m u l a t i o n s o f g(1) have a l s o been employed /5,6/.

The generation o f c a r r i e r s w i t h s p a t i a l r a t e g ( 1 ) b r i n g s about a steady-state c o n c e n t r a t i o n p ( 1 ) [ cm - 3 ] o f excess m i n o r i t y c a r r i e r s . The two f u n c t i o n s a r e connected by t h e d i f f u s i o n equation:

where D and z a r e t h e d i f f u s i o n c o e f f i c i e n t and l i f e t i m e o f m i n o r i t y c a r r i e r s , r e s p e c t i v e - l y ; z may depend on 1 i f the sample i s n o t homogeneous o r c o n t a i n s l o c a l i z e d defects. To s p e c i f y u n i q u e l y p f o r ,a given g, we must add t o E q . ( l ) s u i t a b l e boundary c o n d i t i o n s . For instance, i n t h e i d e a l i z e d case i l l u s t r a t e d i n F i g . l a ( s e m i - i n f i n i t e semiconductor, w i t h t h e surface as c o l l e c t o r ) , t h e boundary c o n d i t i o n s are:

Once t h e s o l u t i o n o f Eq.(l) w i t h ( 2 ) has been determined ( s t e p b ) ) , t h e c o l l e c t e d p a r t i c l e c u r r e n t I [ s - l ] i s c a l c u l a t e d by i n t e g r a t i n g over t h e c o l l e c t i o n plane z=O t h e d e n s i t y o f t h e d i f f u s i o n c u r r e n t ( s t e p c ) ) :

+co tal

The r e s u l t i n g expression f o r I i s /7/:

CO

I = (exp(-z/L) g,(z) dz

0

where

L

= ( D z ) i s t h e m i n o r i t y - c a r r i e r d i f f u s i o n length, and

f

J J - 0 0 - 0 0

i s t h e depth d i s t r i b u t i o n o f t h e generation. Equation ( 4 ) shows t h a t I can be expressed through two one-variable f u n c t i o n s : exp(-z/L), which i s r e l a t e d t o a specimen p r o p e r t y ( L ) , and g ( z ) which describes t h e e x t e r n a l generation. Therefore, one m i g h t wonder whether i t was r e a l 1 4 necessary t o c a l c u l a t e f i r s t the three-dimensional d i s t r i b u t i o n p ( r ) t o o b t a i n Eq.(4).

A c t u a l l y t h e c a l c u l a t i o n o f p ( 1 ) can be avoided by i n t r o d u c i n g t h e f i n c t i o n cp(r) which represents t h e charge c o l l e c t i o n p r o b a b i l i t y a t t h e p o i n t

r.

On t h e b a s i s o f a r e c i p r o c i t y theorem f o r charge c o l l e c t i o n /8/, i t has been shown t h a t

~ ( 1 ) obeys t h e homogeneous v e r s i o n o f E q . ( l ) :

w i t h t h e boundary c o n d i t i o n s

Equations (6), ( 7 ) i n v o l v e o n l y device p r o p e r t i e s and n o t those o f t h e generation, and a l s o t a k e i n t o account t h e symmetry o f t h e sample. I n f a c t , i t i s easy t o see t h a t t h e s o l u t i o n t o Eqs. (6), ( 7 ) i s t h e simple one-variable f u n c t i o n

Y ( Z ) = exp(-z/L) ( 8 )

The independence o f cp o f x, y i s a consequence o f t h e t r a n s l a t i o n a l symmetry o f t h e s t r u c - t u r e o f Fig.la along these d i r e c t i o n s . The induced c u r r e n t I i s then obtained by m u l t i p l y i n g t h e generation r a t e a t a depth z by t h e corresponding charge c o l l e c t i o n p r o b a b i l i t y and i n t e g r a t i n g over z; t h i s leads again t o Eq. ( 4 ) . The presence o f a d e p l e t i o n l a y e r , where a l l generated c a r r i e r s a r e assumed t o be c o l l e c t e d , can be taken i n t o account simply by speci- f y i n g t h a t cp = 1 w i t h i n t h e depleted region. I n t h e more general case o f a specimen where

cp = ~ ( r ) , Eq. ( 4 ) becomes:

I ( & ) =

I

^v cp(r) g ( r - 1 ) dV = ^{0 (9)}

where t

=

( x ,y ) i s t h e v e c t o r d e s c r i b i n g t h e beam p o s i t i o n i n t h e xy plane and t h e symbol

*

denoTes t h & t8o-dimensional convolution.

The a l t e r n a t i v e p i c t u r e o f charge c o l l e c t i o n f o l l o w i n g from Eq.(9) i s represented schemati- c a l l y i n F i g . l b and i n c l u d e s t h e two f o l l o w i n g steps o f t h e modeling:

a l ) t h e d i s t r i b u t i o n o f t h e charge c o l l e c t i o n p r o b a b i l i t y i n t h e specimen b l ) t h e sampling o f t h i s d i s t r i b u t i o n w i t h t h e beam probe.

This f o r m u l a t i o n corresponds c l o s e l y t o t h e theory o f image formation i n scanning microscopes /9/; i t may s i m p l i f y both t h e a n a l y t i c a l c a l c u l a t i o n s , as j u s t shown i n a p a r t i c u l a r case, and t h e numerical treatment o f t h e charge c o l l e c t i o n problem i n c o n f i g u r a t i o n s o f 1 ow symme- t r y . I n f a c t , t h e homogeneous Eq.(4) i s w e l l s u i t e d t o numerical s o l u t i o n ; once cp(r) has been c a l c u l a t e d over a s u i t a b l e g r i d , t h e value o f I f o r d i f f e r e n t g ( 1 ) ( e i t h e r known a n a l y - t i c a l l y o r through Monte C a r l o s i m u l a t i o n ) can be computed by u s i n g t h e d i s c r e t i z e d v e r s i o n o f Eq. (9).

3

-

THE CHARGE COLLECTION EFFICIENCY

Having o u t l i n e d how t h e beam-induced c u r r e n t can be c o n v e n i e n t l y computed, l e t us consider t h e i n v e r s e problem i.e. how from EBIC measurements, through proper modeling, i n f o r m a t i o n can be gained on e l e c t r i c a l p r o p e r t i e s o f t h e m a t e r i a l / d e v i c e under i n v e s t i g a t i o n .

Here t h e simple one-dimensional case o f a Schottky diode i r r a d i a t e d w i t h a s t a t i o n a r y e l e c - t r o n beam w i l l be examined (Fig.2). From Eq. (9), w i t h

t

= 0, and w r i t i n g more e x p l i c i t l y g,(z).= g .h(z,E), where g [s-']is t h e t o t a l generation r a t e and h(z,E) i s t h e normalized one-d~men!?~onal generation ?unction, we have

(10) 0

I t i s convenient t o i n t r o d u c e t h e charge c o l l e c t i o n e f f i c i e n c y o f t h e device q ( E ) = I ( E ) / g o /2,4/, i.e. t h e f r a c t i o n o f t h e i n j e c t e d c a r r i e r t h a t i s c o l l e c t e d and gives r i s e t o t h e

electron

I ^beam

Schottky I

Fig.2

-

Schematic r e p r e s e n t a t i o n o f charge c o l l e c t i o n e f f i c i e n c y measurements on a Schottky diode.

induced c u r r e n t . A measurement o f q(E) f o r d i f f e r e n t values o f E y i e l d s t h e data from which i n f o r m a t i o n about 9 ( z ) i s obtained /4,11,12/.

I t i s convenient t o r e w r i t e Eq. ( l o ) u s i n g i n p l a c e o f ,E t h e primary e l e c t r o n range R, which f o r a given m a t e r i a l i s a known f u n c t i o n o f E. Thus q and 9 w i l l be expressed as f u n c t i o n s o f v a r i a b l e s w i t h t h e same p h y s i c a l dimension ( t h a t o f a l e n g t h ) ; moreover, t h i s s u b s t i t u t i o n a l l o w s e x p l o i t i n g t h e e m p i r i c a l r e s u l t

/ l o /

t h a t h(z,E(R)) i s o n l y dependent upon z/R and n o t on z and R separately. Hence Eq.(lO) becomes /4/:

where A(z/R) i s a u n i v e r s a l f u n c t i o n independent o f t h e beam energy

/lo/.

^{I f} we were a b l e t o probe t h e device w i t h a p o i n t source o f c a r r i e r s , t h e measured c o l l e c t i o n e f f i c i e n c y would equal t h e c o l l e c t i o n p r o b a b i l i t y a t t h e source p o s i t i o n . Since e x p e r i m e n t a l l y we can o n l y probe t h e sample w i t h an extended source, we measure a smeared v e r s i o n o f q ( z ) , i.e. q(R).

The method most f r e q u e n t l y used t o recover i n f o r m a t i o n about (p ( i n p a r t i c u l a r t o determine t h e value o f t h e b u l k d i f f u s i o n l e n g t h L o f t h e semiconductor) from q(E) data assumes t h e form o f 9 t o be known a p r i o r i i n terms o f a small number o f device parameters; t h e values o f these parameters a r e then found by f i t t i n g t h e t h e o r e t i c a l expressions obtainend from E q . ( l l ) t o experiment /4,11,12/. Typical parameters f o r a Schottky diode a r e t h e thickness h o f t h e m e t a l l i z a t i o n , t h e w i d t h W o f t h e d e p l e t i o n l a y e r and L. For t h e purpose o f determining L, however, i t i s n o t necessary t o f i t t h e whole q(R) p r o f i l e , b u t o n l y i t s h i g h energy p a r t , which i s more s e n s i t i v e t o t h e value o f L /2,11/. An approximate expression f o r q(R) v a l i d

i n t h e l i m i t o f R >>h

+

W has been proposed r e c e n t l y 1131:

where q o i s t h e c o l l e c t i o n e f f i c i e n c y o f an i d e a l Schottky b a r r i e r w i t h h=W=O.

The method o f the model f u n c t i o n i s u s e f u l as l o n g as t h e form assumed f o r 9 i s known t o describe adequately t h e device and t h e number o f parameters i n v o l v e d i s small.

There a r e r e l e v a n t p r a c t i c a l cases, however, where these c o n d i t i o n s a r e n o t met; t h i s hap- pens, f o r instance, w i t h t h e Schottky s t r u c t u r e o f Fig.2 i f l a y e r s w i t h enhanced recombina- t i o n (and g e n e r a l l y unknown w i d t h and p o s i t i o n ) a r e present i n t h e device. I n t h i s case a d i r e c t method o f r e c o v e r i n g 9 from q(R) w i t h o u t any assumption about t h e form o f appears d e s i r a b l e . This i s p o s s i b l e i n p r i n c i p l e , s i n c e E q . ( l l ) i s an i n t e g r a l equation which has a unique s o l u t i o n f o r a given q(R) and a known A / 4 / . When A i s a polynomial o f t h e v a r i a - b l e

5

= z/R, as i n t h e approximation by Everhart and H o f f 1101, an e x p l i c i t s o l u t i o n f o r

E B I C C1

E L E C T R O N B E A M

S C H O T T K Y

X

Fig.3

-

Schematic i l l u s t r a t i o n o f c o l l e c t i o n e f f i c i e n c y measurements on bevels f o r depth p r o f i l i n g o f t h e d i f f u s i o n l e n g t h . The a c t u a l value o f a i s about 3'.

(p o f Eq. (11) can be found and has t h e form /14/:

where q' and q " a r e t h e f i r s t and second d e r i v a t i v e o f q(R), r e s p e c t i v e l y . The a p p l i c a t i o n o f Eq.(13) t o p r a c t i c a l cases, however, r e q u i r e s e v a l u a t i n g r e l i a b l y q ' and q " from a c t u a l n o i s y experimental data, and has n o t y e t been attempted.

Another s i t u a t i o n where i t i s d i f f i c u l t t o g i v e a p r i o r i t h e form o f y occurs when L i s depth-dependent, i . e . when L = L ( z ) , f o r i n s t a n c e as r e s u l t o f g e t t e r i n g treatments. The r e c o n s t r u c t i o n o f t h e r e l e v a n t f u n c t i o n L ( z ) i s made easier, i f t h e experimental arrangement o f Fig.3 i s used, i.e. i f q i s measured a t a f i x e d beam energy a t d i f f e r e n t p o s i t i o n s on a Schottky diode f a b r i c a t e d on t h e sample a f t e r angle beveling. This makes i t p o s s i b l e t o probe t h e semiconductor a t d i f f e r e n t depths w i t h a f i x e d generation volume, whereas i n energy-de- pendent measurements o f q d i f f e r e n t depths a r e reached by changing t h e e x t e n s i o n o f t h e generation region. Sampling a v a r i a b l e property, as L ( z ) , w i t h a f i x e d probe i s expected t o y i e l d b e t t e r chance o f r e c o v e r i n g d i r e c t l y t h i s property.

I n f a c t , i f q ( z ) i s t h e value o f t h e c o l l e c t i o n e f f i c i e n c y when t h e beam i s a t a d i s t a n c e x = z / s i n a from t h e bevel edge ( a i s t h e bevel angle), i t i s p o s s i b l e t o g i v e a simple r e l a t i o n between L ( z ) and q ( z ) i n t h e i d e a l case o f n e g l i g i b l e thickness o f t h e d e p l e t i o n l a y e r /15/:

where

a being t h e average generation depth. The presence o f t h e d e r i v a t i v e i n Eq.(14) sharpens up t h e smooth experimental q ( z ) p r o f i l e t o y i e l d L ( z ) (Fig.4). The advantage o f performing t h e measurements a t a f i x e d beam energy i s a l s o r e f l e c t e d i n t h e form o f Eq:(14), which i n v o l v e s t h e f i r s t d e r i v a t i v e o f q only, and n o t t h e second one as i n Eq.(13).

The modeling o f experiments t h a t i n v o l v e a two-dimensional d i s t r i b u t i o n o f (p (e.g. t h e a n a l y s i s o f EBIC scans obtained on a cross-section o f a p-n j u n c t i o n ) i s o b v i o u s l y more d i f f i c u l t and f u r t h e r approximations must be i n t r o d u c e d t o l i m i t t h e complexity o f t h e discussion. Some methods and problems o f t h e two-dimensionalcase a r e discussed i n /16/.

4

-

THE EBIC CONTRAST OF DEFECTS

Equation ( 6 ) f o r t h e c h a r g e - c o l l e c t i o n probabi 1 i t y a l s o holds i f z i s position-dependent, and t h e r e f o r e can be used t o describe charge c o l l e c t i o n i n t h e presence o f semiconductor defects. By r e p r e s e n t i n g a d e f e c t as a r e g i o n F where t h e l i f e t i m e z' (supposed constant

Fig.4

-

Computer simulated c o l l e c t i o n e f f i c i e n c y p r o f i l e q ( z ) f o r a h y p o t h e t i c a l d i f f u s i o n l e n g t h p r o f i l e (continuous 1 in e ) . The r e c o n s t r u c t e d d i f f u s i o n l e n g t h p r o f i l e ( d o t s ) has been c a l c u l a t e d u s i n g Eqs. (14), (15).

w i t h i n F) i s s m a l l e r than t h e b u l k l i f e t i m e z /7/, we o b t a i n from Eq.(6)

where Y = ( l / r l '

-

^{1 / z )} and e ( r ) = 1 i n s i d e F and vanishes elsewhere; t h e boundary c o n d i t i o n s a r e s t i l l those o f Eq.(7).

The f a c t o r Y [ s - I ] represents t h e recombination s t r e n g t h o f t h e d e f e c t . Equation (16) can be solved approximately by t r e a t i n g t h e term c o n t a i n i n g y as a p e r t u r b a t i o n and w r i t i n g y =

= IP, + 9,

,

where yo i s t h e s o l u t i o n t o Eq. (16) i n absence o f t h e d e f e c t (see Eq. ( 8 ) ) , and 9, i s a ( s m a l l ) d e v i a t i o n from 9, produced by t h e presence o f t h e d e f e c t ; t h e approximate s o l u t i o n i s /16/:

~ ( 1 ) =

'PO+

IP, = exp(-z/L)

-

^y,/ exp(-zl/L) G(1.1' ) dV1 (17)

F

where G i s t h e Green's f u n c t i o n o f Eq.(6) t h a t vanishes a t t h e surface. The second term i n Eq.(17) represents t h e decrease o f t h e charge c o l l e c t i o n p r o b a b i l i t y i n t h e device due t o t h e presence o f t h e defect. T h i s produces a decrease o f t h e EBIC c u r r e n t , which can be c a l c u l a t e d from Eq. (9) t o be

I * =

lm

^g

^*

^{9, dz (18)}

The corresponding image c o n t r a s t i s given by

where 10 i s t h e background c u r r e n t and has t h e expression a l r e a d y given i n Eq.(4). I t i s n o t d i f f i c u l t t o check t h a t Eq.(19) reproduces t h e expression d e r i v e d p r e v i o u s l y /7/ s t a r t i n g from t h e s o l u t i o n o f t h e d i f f u s i o n equation f o r p ( r ) .

The two approaches t o t h e d e f e c t c o n t r a s t c a l c u l a t i o n correspond t o d i f f e r e n t b u t e q u i v a l e n t d e s c r i p t i o n s o f charge c o l l e c t i o n , as i n Sec.2. According t o t h e former model /7/, t h e presence o f a d e f e c t m o d i f i e s t h e d i s t r i b u t i o n p ( r ) o f t h e beam-injected m i n o r i t y c a r r i e r s ( i n a way t h a t i s dependent on t h e beam-defect d i s t a n c e ) and hence t h e induced c u r r e n t through a r e d u c t i o n o f t h e g r a d i e n t o f p a t t h e c h a r g e - c o l l e c t i n g surface. I n t h e present

approach, a d e f e c t reduces i n i t s surroundings t h e p r o b a b i l i t y t h a t a generated c a r r i e r reaches t h e j u n c t i o n ( i . e . (p ); t h e induced c u r r e n t i s t h e r e s u l t o f sampling (p w i t h t h e generation volume, and t h e r e f o r e i s lower when t h e beam i s near t h e d e f e c t .

The f o r m u l a t i o n o f t h e problem i n terms o f charge c o l l e c t i o n p r o b a b i l i t y does n o t s i m p l i f y , i n general, t h e expression o f t h e image c o n t r a s t , s i n c e t h e presence o f a d e f e c t breaks t h e symmetry o f t h e o r i g i n a l device. For d e f e c t s o f s p e c i a l shape, however, t h e defect-device c o n f i g u r a t i o n may r e t a i n some symmetry and t h e i n t r o d u c t i o n o f (p w i l l a l l o w some s i m p l i f i c a - t i o n o f t h e treatment.

Eq.(19) has been g e n e r a l l y used t o determine y from EBIC measurements by computing i*m,,, i.e. t h e maximum c o n t r a s t , and comparing t h i s value t o t h e experimental one. There i s , how- ever, a f u r t h e r c h a r a c t e r i s t i c p r o p e r t y o f i * ( t ) = i " ( x ,y ) t h a t can be used advantageously f o r t h e same purpose, i.e., i t s i n t e g r a l over th< imageopl%ne:

- w -00

I f i* i s p l o t t e d on t h e z a x i s as a f u n c t i o n o f x

,

y t h e q u a n t i t y V [cm 2 ] represents a volume, which describes tf?e o v e r a l l i n f l u e n c e o f tRe d @ f e c t on t h e image c o n t r a s t . Substi- t u t i n g Eq. (19) i n t o Eq. (20) and t a k i n g i n t o account t h a t G ( z , r t ) = G(x-x'

,

^{y - y '}

,

^z,zl), s i n c e t h e s t r u c t u r e o f Fig.1, i n absence o f t h e d e f e c t , has t r a n s l a t i o n a l i n v a r i a n c e along x,y, we o b t a i n :

where

GI

i s t h e one-dimensional Green's f u n c t i o n r e s u l t i n g from t h e i n t e g r a t i o n o f G w i t h r e s p e c t t o x,y. The e v a l u a t i o n o f t h e i n t e g r a l i n l a r g e brackets r e q u i r e s a c t u a l l y an i n t e - g r a t i o n w i t h r e s p e c t t o z ' only, since t h e f u n c t i o n t o be i n t e g r a t e d over F depends o n l y on t h i s v a r i a b l e . Equation (21) shows t h a t , f o r a given d e f e c t geometry, V can be c a l c u l a t e d i n terms o f t h e one-variable f u n c t i o n s h and GI, whereas t h e e v a l u a t i o n o f i*,,, would i n v o l v e t h e t h r e e - v a r i a b l e f u n c t i o n s g and 6.

The e v a l u a t i o n o f V from an experimental image does n o t r e q u i r e much a d d i t i o n a l e f f o r t , i f a d i g i t a l a c q u i s i t i o n system i s connected t o t h e SEM, and g r e a t l y simp1 i f i e s t h e c a l c u l a t i o n s r e q u i r e d t o recover y from EBIC c o n t r a s t data. An example o f t h e use o f an i n t e g r a l p r o p e r t y o f t h e EBIC image ( t h e area o f t h e c o n t r a s t p r o f i l e ) t o e v a l u a t e t h e recombination s t r e n g t h o f d i s l o c a t i o n s has been r e p o r t e d i n /17/.

To i l l u s t r a t e another u s e f u l p r o p e r t y o f V, l e t us consider a s t r a i g h t s e m i - i n f i n i t e d i s l o c a - t i o n t i l t e d by an angle a from t h e normal t o t h e sample surface ( t h e z a x i s ) . I n t h i s case dV' = ( a /cos a )dzl, a being t h e cross-section o f t h e recombination c y l i n d e r associated w i t h t h e d i s l o c a t i o n ; t h e p r o d u c t yd = y o [cm2 s-']can be c a l l e d l i n e recombination v e l o c i t y o f t h e d i s l o c a t i o n . Hence Eq. (21) becomes

e x p ( - z t / L ) GJz,zl) dz'

which gives, i n p a r t i c u l a r , t h e r e l a t i o n

between t h e value o f V o f a s t r a i g h t d i s l o c a t i o n t i l t e d by a from t h e

z

a x i s and t h a t o f a perpendicular d i s l o c a t i o n w i t h t h e same yd

.

Therefore, f o r t h i s d e f e c t , a d e v i a t i o n from t h e normal incidence changes t h e maximum c o n t r a s t and lowers t h e symmetry o f t h e EBIC image, b u t leaves t h e product V - c o s a unaffected. This p r o p e r t y can be u s e f u l t o compare t h e a c t i v i t y o f approximately s t r a i g h t d i s l o c a t i o n s w i t h d i f f e r e n t , b u t known, i n c l i n a t i o n angles w i t h r e s p e c t t o t h e surface.

5

-

DISCUSSION AND CONCLUSIONS

The d e s c r i p t i o n o f steady-state EBIC experiments i n terms o f c h a r g e - c o l l e c t i o n p r o b a b i l i t y o f f e r s some formal and computational advantages, b u t r e l i e s on t h e same p h y s i c a l approximat- i o n s used p r e v i o u s l y /7/. Thus, f o r instance, t h e model assumes low i n j e c t i o n c o n d i t i o n s , where ~ ( 1 ) can be c a l c u l a t e d independently o f g ( r ) . Injection-dependent e f f e c t s ( b o t h i n t h e b u l k /18/ and a t d e f e c t s /19,20/) can be described by t a k i n g i n t o account t h a t t h e p r o b e

f u n c t i o n g ( 1 ) can modify t h e p r e - e x i s t e n t d i s t r i b u t i o n o f t h e c h a r g e - c o l l e c t i o n p r o b a b i l i t y , so t h a t t h e c a l c u l a t i o n steps a l ) and b l ) o f Sec.2 would no l o n g e r be independent.

The present d i s c u s s i o n o f t h e d e f e c t c o n t r a s t uses t h e f i r s t o r d e r ( l i n e a r ) approximation o f 9 i n t h e presence o f a d e f e c t . Higher oder approximations t o the c o n t r a s t have been develo- ped by Pasemann /21/ f o r a s t r a i g h t d i s l o c a t i o n p a r a l l e l t o t h e surface, and an exact calcu- l a t i o n has been g i v e n f o r a plane g r a i n boundary perpendicular t o t h e sample s u r f a c e /22/.

These treatments, w h i l e a l l o w i n g a more accurate d e t e r m i n a t i o n o f t h e s t r e n g t h o f h i g h l y recombining defects, produce a n o n - l i n e a r c o n t r a s t f u n c t i o n (i.e. i* i s n o t p r o p o r t i o n a l t o

y ) and t h e r e f o r e t h e advantages r e l a t e d t o l i n e a r i t y a r e l o s t . The s u b j e c t o f EBIC c o n t r a s t modeling has been reviewed r e c e n t l y by Jakubowicz /23/.

F u r t h e r extensions o f t h e l i n e a r model have a l s o been considered. Thus Joy and Pimentel /24/

and Sieber /25/ have described recombination a t a d i s l o c a t i o n p a r t i a l l y l y i n g i n t h e deple- t i o n region, where c a r r i e r motion i s n o t l o n g e r p u r e l y d i f f u s i v e , by a t t r i b u t i n g t o t h e d i s l o c a t i o n an e f f e c t i v e r a d i u s (and hence a recombination s t r e n g t h yd ) dependent on t h e depth p o s i t i o n ( o r , e q u i v a l e n t l y , on t h e e l e c t r i c f i e l d ) i n t h e d e p l e t i o n l a y e r . Wilshaw and Booker /20/ have spe,cified i n yd

,

which has been i n t r o d u c e d i n Sec.4 on a p u r e l y phenomeno- l o g i c a l basis, i t s dependence on t h e temperature and i n j e c t i o n l e v e l by modeling a d d i t i o n a l l y t h e u n d e r l y i n g microscopical recombination process.

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