DIFFUSION OF ELECTRONS AND HOLES IN DOPED SEMICONDUCTORS AT HIGH LATTICE AND ELECTRONIC TEMPERATURES

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DIFFUSION OF ELECTRONS AND HOLES IN DOPED SEMICONDUCTORS AT HIGH LATTICE

AND ELECTRONIC TEMPERATURES

M. Wautelet

To cite this version:

M. Wautelet. DIFFUSION OF ELECTRONS AND HOLES IN DOPED SEMICONDUCTORS AT

HIGH LATTICE AND ELECTRONIC TEMPERATURES. Journal de Physique Colloques, 1981, 42

(C7), pp.C7-131-C7-136. �10.1051/jphyscol:1981714�. �jpa-00221650�

JOURNAL DE PHYSIQUE

Colloque C7, supplément au n°10

Tome 42, octobre 1981 page C7-131

DIFFUSION OF ELECTRONS AND HOLES I N DOPED SEMICONDUCTORS AT HIGH LATTICE AND ELECTRONIC TEMPERATURES

M. Wautelet

Faculte des Sciences, Universite de I 'Etat, B-7000 Moris, Belgium

Résumé. - I l e s t montré comment l e s p r o p r i é t é s de diffusion des électrons e t des trous sont modifiées lorsque leur d e n s i t é e t l a température du réseau sont élevées.

En p a r t i c u l i e r , i l en r é s u l t e que l e t r a n s f e r t d ' é n e r g i e des électrons au réseau dépend fortement des conditions expérimentales : l e chauffage e s t plus efficace à basse qu'à haute puissance. La c r é a t i o n de défauts par l ' i r r a d i a t i o n laser peut conduire à l ' e x i s t e n c e d'un é t a t métastable de longue durée de v i e près de l a s u r - face i r r a d i é e du semiconducteur.

Abstract. - I t i s shown how the diffusion p r o p e r t i e s of e l e c t r o n s and holes are modified when t h e i r density and the l a t t i c e temperature are high. In p a r t i c u l a r , i t

a r i s e s t h a t the t r a n s f e r of energy from the e l e c t r o n s to the l a t t i c e depends strong- ly on experimental conditions : heating i s more e f f i c i e n t a t low than a t high power.

Laser-induced defects may r e s u l t in the existence of a long-lived metastable s t a t e near the i r r a d i a t e d surface of the semiconductor.

1. I n t r o d u c t i o n . - Under conditions of intense o p t i c a l pumping or e l e c t r i c a l i n j e c - t i o n , i t i s possible to obtain s o l i d s i n highly excited s t a t e s , i . e . very far from equilibrium. Relaxation to the fundamental s t a t e then occurs v i a processes which a r e not always well understood ( 1 ) . I t i s generally considered t h a t the physical mechanisms involved in these r e l a x a t i o n processes may be described s t a r t i n g from a knowledge of the quasi-equilibrium p r o p e r t i e s of the system under study, plus some non-linear phenomena. In t h i s work, a d i f f e r e n t approach i s taken : i t i s assumed t h a t conditions are established in which the temperature of e l e c t r i c a l c a r r i e r s , T , i s larger than the l a t t i c e temperature, T . The "instantaneous" physical p r o - p e r t i e s of the solid are then studied with the use of non-equilibrium thermodyna- mics. This point of view already allowed to discuss some p r o p e r t i e s of i n t r i n s i c

semiconductors, l i k e band gaps (2) and diffusion of e l e c t r i c c a r r i e r s (3-5) and heat ( 5 ) . These r e s u l t s are summarised and the reasoning i s extended to doped semi- conductors in Section 2. Diffusion of e l e c t r o n s and holes under the above condi- t i o n s a r e discussed in Section 3 .

2. Thermodynamic a 1 parameters.- As a s t a r t i n g point, l e t us consider t h a t T i s defined from the number of excited electron-hole p a i r s , n, v i a the law of mass action (2) . When T ?* T

, i t can be shown t h a t the semiconductor band gap i s not

e,n L

unequivocally defined (2) : the o p t i c a l "no phonon" band gap ( o p t i c a l gap) i s no longer equal t o the thermal, or chemical p o t e n t i a l , band gap. Indeed, the o p t i c a l

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981714

C7- 132 JOURNAL DE PHYSIQUE

gap i s t h e i n c r e a s e i n i n t e r n a l energy upon j n c r e a s e of t h e c a r r i e r d e n s i t y by one e l e c t r o n and one h o l e under c o n d i t i o n s of c o n s t a n t entropy and volume, w h i l e t h e thermal one i s t h e i n c r e a s e i n f r e e energy a t c o n s t a n t temperature and p r e s s u r e ( 6 ) . Both a r e e q u a l under normal e q u i l i b r i u m c o n d i t i o n s . When T P TL, o p t i c a l proper-

e , n t i e s and c a r r i e r d i f f u s i o n do n o t measure t h e same q u a n t i t y .

The o p t i c a l gap depends on t h e c a r r i e r c o n c e n t r a t i o n , n, and TL following (3) :

where (- bnl 13) and (- aEVib(TL, n ) ) a r e t h e exchange-correlation and electron-pho- non c o n t r i b u t i o n s r e s p e c t i v e l y . cr and b a r e c o n s t a n t a t high n. Evib(TL, n) i s t h e l a t t i c e energy. For small n, Evib(TL, n)

E vib(TL, O), b u t t h i s e q u a l i t y d i s a p p e a r s when n i n c r e a s e s ( 7 ) . It i s worth n o t i n g t h a t e q u a t i o n (1) i s v a l i d when t h e one- e l e c t r o n band s t r u c t u r e parameters remain c o n s t a n t . I f t h e y v a r y , t h e n E (0) v a r i e s ,

g

f o r i n s t a n c e when t h e r e c o v e r i n g of atomic o r b i t a l s change d r a s t i c a l l y upon e l e c t r o n e x c i t a t i o n , s o t h a t bandwidths and energy s e p a r a t i o n s a r e modified.

Under non-equilibrium c o n d i t i o n s , it t u r n s o u t t h a t t h e c a r r i e r chemical p o t e n t i a l or thermal band gap i s given by (2) :

i n which H(T L ) and S(T L ) a r e t h e e n t h a l p y and entropy of t h e band gap r e s p e c t i v e l y . H(TL) - TLS(TL) i s g i v e n by e q u a t i o n (1) s i n c e t h e pV term i s n e g l i g i b l e .

The o r i g i n of S(T L ) comes from t h e f a c t t h a t v a l e n c e and conduction band s t a t e s a r e n o t n e c e s s a r i l y a s s o c i a t e d with t h e same k i n d s of o r b i t a l s . For i n s t a n c e , i n S i , v a l e n c e and conduction band edges s t a t e s a r e a s s o c i a t e d with bonding and anti-bon- ding o r b i t a l s r e s p e c t i v e l y . Then, an i n c r e a s e of n induces a change i n t h e v i b r a - t i o n a l p r o p e r t i e s of atoms, i . e . a change of t h e entropy of t h e system. A s i m i l a r s i t u a t i o n o c c u r s presumably i n Se, A s Se and GeSe2 -type m a t e r i a l s 2 3 (8), i n which v a l e n c e and conduction bands a r e a s s o c i a t e d with Se-lone p a i r s and anti-bonding

s t a t e s r e s p e c t i v e l y .

I n S i , a t low TL (G 50 K ) , EVib(TL, n) and S(TL) a r e v e r y s m a l l , s o t h a t E - - g , o p t

=

E (0) - bn1I3, provided n i s n o t t o o high, a s observed i n e l e c t r o n - h o l e

Eg,ch g

drops.

Up t o now, i t has been assumed t h a t t h e numbers of e l e c t r o n s and h o l e s a r e equal

i n t h e semiconductor. However it may appear t h a t , under c e r t a i n c o n d i t i o n s , e l e c -

t r o n s and h o l e s a r e d r i v e n t o d i f f e r e n t r e g i o n s i n t h e m a t e r i a l . Then t h e band gaps

may d i f f e r i n both r e g i o n s , s i n c e t h e c o n t r i b u t i o n s of e l e c t r o n s and h o l e s t o t h e electron-phonon coupling a r e n o t e q u a l : t h e h o l e c o n t r i b u t i o n i s 3 . 6 + 1.0 times t h e one of e l e c t r o n s ( 9 ) .

Even when t h e numbers of e x c i t e d e l e c t r o n s and h o l e s a r e e q u a l , i t i s n e c e s s a r y t o i n t r o d u c e more t h a n one " e l e c t r o n i c " temperature ( 5 ) . F i r s t , a s s a i d b e f o r e , one can c a l c u l a t e T v i a n and t h e law of mass a c t i o n . I n f i r s t approximation, assu-

e ,

= '

ming p a r a b o l i c bands, t h i s i s t h e temperature which e n t e r s e q u a t i o n ( 2 ) .

One may determine an o t h e r e l e c t r o n i c temperature, Te, v i a t h e p r o b a b i l i t y , D(E), of f i n d i n g a n e l e c t r o n a t a given energy l e v e l , E. This i s t h e temperature of t h e e l e c t r o n i c system which e n t e r s t h e Fermi-Dirac law i n t h e absence of TL. Simulta- neously, one h a s t o i n t r o d u c e a temperature f o r h o l e s , Th ( 1 ) . These a r e t h e tempe- r a t u r e s t o be included i n t r a n s p o r t t h e o r y ( 5 ) .

Let u s a l s o remember t h a t , under s t r o n g l y f a r from e q u i l i b r i u m c o n d i t i o n s , T may L be undefined, s i n c e t h e phonon d i s t r i b u t i o n i s no longer a c o n v o l u t i o n of t h e e q u i l i b r i u m (n-dependent) phonon d e n s i t y - o f - s t a t e s w i t h t h e Bose-Einstein d i s t r i - b u t i o n law a t a given TL. A s a f i r s t approach, t h i s p o i n t w i l l n o t be taken i n t o c o n s i d e r a t i o n h e r e .

3 . D i f f u s i o n of e l e c t r o n s and h o l e s . - The chemical p o t e n t i a l of e l e c t r o n s ( h o l e s ) ,

ve (uh), i s a f u n c t i o n of n and TL. As a r e s u l t , any g r a d i e n t of n ( o r T ) e , n and/or TL induces a c u r r e n t d e n s i t y p r o p o r t i o n a l t o a ( u /T )/ax, where x i s t h e

e e

s p a t i a l c o o r d i n a t e i n t h e d i r e c t i o n of t h e f l u x . The same i s t r u e f o r h o l e s . I n an i n t r i n s i c semiconductor, one o b t a i n s (3) :

where t h e term a n 2 I 3 accounts f o r t h e k i n e t i c energy of c a r r i e r s and E i s t h e g,o v a l u e of t h e band gap i n t h e n o n - i r r a d i a t e d r e g i o n .

When looking a t t h e d i f f u s i o n p r o p e r t i e s i n a n i n t r i n s i c semiconductor, t h e posi- t i o n of E r e l a t i v e t o t h e band edges i s not important. This i s obviously d i f f e - F r e n t i n e x t r i n s i c semiconductors o r n e a r metal-semiconductor i n t e r f a c e s . E F i s g i v e n from t h e v a l u e of T e , n ' When T e , n S TL > 300 K, E F i s always near midgap, a t Eo/2.

n

L e t u s examine t h e c a s e where TL remains c o n s t a n t i n an n-type semiconductor.

Within t h e i r r a d i a t e d r e g i o n , T i s h i g h , s o t h a t EF i s n e a r E 12 and c a r r i e r

e , n g

d i f f u s i o n i s d e s c r i b e d by t h e i n t r i n s i c c a s e ( 3 - 5 ) : up t o n , t h e e l e c t r o n - h o l e plasma i s d r i v e n from t h e low n t o t h e high n r e g i o n , due t o t h e (T S(TL)) term

e , n

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i n e q u a t i o n ( 3 ) , i . e . a self-confinement o c c u r s . When n exceeds nc, t h e e l e c t r o n - h o l e plasma tends t o expand up t o t h e s i t u a t i o n where n e q u a l s n . L e t u s n o t e t h a t n i n c r e a s e s w i t h i n c r e a s i n g T and T and with d e c r e a s i n g T ( 5 ) . Any o t h e r

e , n L

v a r i a t i o n of E (T T ) a f f e c t s n . I n ~ a r t i c u l a r , i t h a s been shown t h a t a g , c h L' e , n

m a l l T - g r a d i e n t i s a b l e t o i n c r e a s e d r a s t i c a l l y n a s compared with t h e iso-T

L L

c a s e ( 3 , 4 ) . I n t h e border r e g i o n , T approaches TL and band bending o c c u r s due e , n

t o t h e doping. As a r e s u l t , e l e c t r o n s and h o l e s a r e d r i v e n i n o p p o s i t e d i r e c t i o n s . But t h e y have o p p o s i t e e l e c t r i c c h a r g e s and i n t e r a c t e l e c t r o s t a t i c a l l y , so t h a t they a r e n o t f r e e t o move independently.

One t h e n a r r i v e s t o a s i t u a t i o n where a system w i t h high n (around n ) e x i s t s , sur- rounded by a space charge b a r r i e r s i m i l a r t o a p-n j u n c t i o n . The most important p o i n t i s t h a t n i n c r e a s e s with TL, whatever TL.

D i f f u s i o n i s n o t t h e o n l y p r o c e s s occuring a t high Te,n. Simultaneously, o t h e r phy- s i c a l mechanisms t a k e p l a c e , which i n f l u e n c e d r a s t i c a l l y t h e k i n e t i c s of t h e system.

The main ones a r e t h e v a r i o u s recombination p a t h s of c a r r i e r s and t h e c r e a t i o n of d e f e c t s w i t h i n and / o r around t h e i r r a d i a t e d a r e a .

The i n f l u e n c e of recombination and c a r r i e r d i f f u s i o n o n t h e k i n e t i c s of t r a n s f e r of energy t o t h e l a t t i c e h a s been r e c e n t l y examined by Yoffa (10, 11). A t h i g h e x c i t a - t i o n r a t e s , i t t u r n s o u t t h a t t h e l a s e r energy i s g i v e n t o t h e l a t t i c e w i t h i n a c h a r a c t e r i s t i c d e p t h determined p r i m a r i l y by c a r r i e r d i f f u s i o n . This i m p l i e s t h a t experimental parameters a r e s t r o n g l y dependent on t h e r a t i o of t h e " s t e a d y - s t a t e "

d e n s i t y of e x c i t e d c a r r i e r s , n , t o n . When n exceeds nc, t h e t r a n s f e r of energy t o t h e l a t t i c e t a k e s p l a c e over a l a r g e r e g i o n , s i n c e d i f f u s i o n of e l e c t r i c a l c a r - r i e r s i s extremely r a p i d (3, 4 ) . T h i s r e s u l t s i n a d e c r e a s e of t h e h e a t i n g r a t e of t h e s o l i d , i . e . T L remains r e l a t i v e l y s m a l l , a s observed e x p e r i m e n t a l l y ( 1 2 ) . When t h e same number of e l e c t r o n - h o l e p a i r s i s e x c i t e d over a longer time, n may be l e s s t h a n n . I n t h i s c a s e , h e a t i n g of t h e l a t t i c e o c c u r s w i t h i n t h e i r r a d i a t e d r e g i o n and TL i s determined p r i m a r i l y by t h e normal h e a t d i f f u s i o n , c o n t r a r y t o t h e n > n c a s e , where i t i s determined by e l e c t r i c a l c a r r i e r s d i f f u s i o n . The d i f - f e r e n c e between t h e two regimes i s f u r t h e r enhanced by t h e f a c t t h a t h e a t d i f f u s i o n d e c r e a s e s w i t h i n c r e a s i n g n ( 5 ) . One t h e n a r r i v e s t o t h e c o n c l u s i o n t h a t , f o r a given energy provided by t h e l a s e r beam : ( i ) it i s e a s i e r t o h e a t t h e s u r f a c e r e g i o n a t low than a t v e r y high power ; ( i i ) consequently, n i s much l a r g e r a t low t h a n a t v e r y h i g h power. It would be i n t e r s t i n g t o s t u d y t h i s p o i n t experimen- t a l l y . L e t u s remember t h a t t h i s i s v a l i d when S(TL) i s r e l a t i v e l y l a r g e , i . e . n o t a t v e r y low temperature.

Another important p o i n t i s t h e c r e a t i o n of d e f e c t s by h i g h power l a s e r i r r a d i a t i o n .

This i s w e l l known i n t h e c a s e of l a s e r annealing of amorphous S i (13). More specta- c u l a r i s t h e t r a n s i t i o n from c r y s t a l l i n e t o amorphous S i (14) which makes e v i d e n t t h e c r e a t i o n of d e f e c t s by l a s e r i r r a d i a t i o n . This m o d i f i e s d r a s t i c a l l y t h e physi- c a l p r o p e r t i e s of t h e system. Indeed, it seems n a t u r a l t o c o n s i d e r t h a t d e f e c t s a r e c r e a t e d near o r move throughout t h e s u r f a c e r e g i o n of t h e semiconductor. Due t o t h e i r l a r g e number ( t y p i c a l l y 1018 - 1019 cm-3 d a n g l i n g bonds i n amorphous S i ) , one of t h e r e s u l t s i s t h e displacement of Ep towards t h e conduction o r v a l e n c e band edge. From a comparison w i t h l a s e r a n n e a l i n g , i t i s more probable t h a t E F moves towards t h e conduction band i n S i (Wautelet and Laude, t o be p u b l i s h e d ) . Then, a s i t u a t i o n s i m i l a r t o a p-n j u n c t i o n appears near t h e s u r f a c e of t h e semiconductor.

I n t h i s c a s e , e l e c t r o n s and h o l e s a r e s p a t i a l l y s e p a r a t e d , i . e . t h e i r recombination i s d r a s t i c a l l y reduced. This may e x p l a i n s e m i - q u a n t i t a t i v e l y t h e long-lived (100 nsec) m e t a s t a b l e s t a t e observed under i n t e n s e l a s e r e x c i t a t i o n of S i (15). L e t u s n o t e t h a t t h i s recombination process i s simultaneous t o t h e m i g r a t i o n and a n n i h i l a - t i o n of d e f e c t s a t t h e s u r f a c e , so t h a t a proper e v a l u a t i o n of t h e l i f e t i m e of t h e m e t a s t a b l e s t a t e needs t o c o n s i d e r a l l t h e s e mechanisms.

Up t o now, t h e l a t e r a l d i s t r i b u t i o n of t h e energy provided by t h e l a s e r h a s n o t be taken i n t o account. Since >it i s n o t uniform, t h e r e e x i s t r e g i o n s where n may be l a r g e r t h a n n and o t h e r s where n i s smaller t h a n nc. This i m p l i e s t h a t d i f f e r e n t

c '

mechanisms of d i f f u s i o n of c a r r i e r s may c o e x i s t w i t h i n t h e same sample.

4. Conclusions.-. I n t h i s work, i t i s shown how a v e r y high d e n s i t y of e l e c t r o n - h o l e p a i r s m o d i f i e s t h e d i f f u s i o n p r o p e r t i e s of such p a i r s i n semiconductors a t high l a t t i c e temperature. I n p a r t i c u l a r , i t a r i s e s t h a t t h e t r a n s f e r of energy from t h e energy source t o t h e l a t t i c e depends on t h e experimental c o n d i t i o n s : s t a r - t i n g l a t t i c e temperature, l a s e r power. I t comes a l s o t h a t t h e c r e a t i o n of d e f e c t s d u r i n g l a s e r i r r a d i a t i o n may lead t o a long-lived m e t a s t a b l e s t a t e near t h e i r r a - d i a t e d s u r f a c e of t h e semiconductor.

The author acknowledge v e r y i n t e r s t i n g d i s c u s s i o n s w i t h P r o f . L. D. Laude and D r . R. Andrew. This work was supported by p r o j e c t IRIS of t h e Belgian t l i n i s t r y f o r Science P o l i c y .

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C7-136 JOURNAL DE PHYSIQUE

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