THEORETICAL DESCRIPTION OF THE INFLUENCE OF DISLOCATIONS ON THE LUMINESCENCE OF LIGHT-EMITTING DIODES

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THEORETICAL DESCRIPTION OF THE INFLUENCE OF DISLOCATIONS ON THE LUMINESCENCE OF LIGHT-EMITTING DIODES

L. Pasemann

To cite this version:

L. Pasemann. THEORETICAL DESCRIPTION OF THE INFLUENCE OF DISLOCATIONS ON THE LUMINESCENCE OF LIGHT-EMITTING DIODES. Journal de Physique Colloques, 1983, 44 (C4), pp.C4-423-C4-435. �10.1051/jphyscol:1983450�. �jpa-00223070�

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Colloque C4, suppldment au n09, Tome 44, septembre 1983 page C4-423

THEORETICAL DESCRIPTION OF THE INFLUENCE OF DISLOCATIONS ON THE LUMINESCENCE OF LIGHT-EMITTING DIODES

L. Pasemann

Sektion Physik der Karl-Marx-Universitut, Leipzig, D. H. G.

RCsumE - Dans 1 ' h v p o t h S s e q u ' u n e d i s l o c a t i o n i s o l 6 e dCcor6e p a r d e s i m p u r e t E s p e u t E t r e c o n s i d 6 r C e comme une i'nhomogdn6itE macros-.

c o p i q u e , on p r o p o s e un rnodsle s i m p l e a u i d C c r i t l ' i n f l u e n c e d e d i s l o c a t i o n s non c o r r 6 l e e s s u r l a l u m i n e s c e n c e d e d i o d e s C l e c t r o - l u m i n e s c e n t e s . L ' a c c o r d d e c e m o d s l e a v e c l e s donnCes e x p C r i - m e n t a l e s e s t r a i s o n n a b l e .

Abstract

-

of LED5 in reasonable agreement with experimental data.

1.

-

Measurements of the luminescent efficiency of light-emitting diodes (LEDs) have yielded a strong dependence of the efficiency on the dislocation density (e.g. /1,2/). However, up to now the theoretical fundamentals are not yet sufficient to provide a n unambiguous inter- pretation of such measurements.

A first promising model was proposed by Lax /3/. His model is based on following two main assumptions: (i) A single dislocation is considered as a macroscopic inhomogeneity of cylindrical shape with recombination only at the surface of the cylinder. (ii) The diode is considered as a semi-infinite semiconductor, i.e. bounded only by the p-n junction at which the injected minority carrier density is assumed to be constant. Lax described furthermore, the ratio of the total number of carriers with dislocations to that without dislocations by

= 1

-

Wo

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983450

C4-424 JOURNAL DE PHYSIQUE

where BN r e p r e s e n t s a c r o s s s e c t i o n f o r r e d u c t i o n of N, and ^fd i s t h e d i s l o c a t i o n density. The q u a n t i t y c was introduced more s o p h i s t i - c a t e d t o t a k e i n t o account t h e c o r r e l a t i o n of t h e d i s l o c a t i o n s .

c was assumed t o be

which l e a d s t o

However, equation ( 2 ) i s not c o n s i s t e n t with t h e assumption ( i ) of t h e model. Eq. ( 2 ) y i e l d s

but according t o assumption ( i ) t h e r a t i o n N

-

where Ad i s t h e a r e a of t h e c r o s s - s e c t i o n of t h e c y l i n d r i c a l d i s l o c a t i o n region. Besides, i n usual LEDs t h e d i s t a n c e between s u r f a c e and p-n j u n c t i o n i s about one d i f f u s i o n length. Thus, i t i s doubtful whether t h e e f f e c t of t h e s u r f a c e on t h e luminescence can be n e g l e c t e d a s assumed by Lax ( s e e above ( i i ) ) .

I n a d d i t i o n t o t h e model d e s c r i b e d above Lax considered a second model i n which t h e volume of t h e c y l i n d r i c a l d i s l o c a t i o n r e g i o n provides "bulkv recombination a t a h i g h e r r a t e t h a n t h a t of t h e surrounding medium. Unfortunately, Lax was mistaken i n h i s treatment what can be immediately seen i n h i s r e s u l t which does n o t y i e l d t h e s o l u t i o n without d i s l o c a t i o n s i f t h e recombination i n s i d e t h e cy- l i n d e r i s equal t o t h a t o u t s i d e t h e c y l i n d e r /3/.

Therefore, i n t h i s paper Lax's second model i s t r e a t e d once more, c o n s i d e r i n g a d d i t i o n a l l y , t h e i n f l u e n c e of t h e s u r f a c e of t h e sample.

This model has a l r e a d y been used s u c c e s s f u l l y f o r t h e i n t e r p r e t a t i o n of t h e EBIC ( e l e c t r o n beam induced c u r r e n t ) c o n t r a s t /4-7/.

2.

-

The assumption of t h e model p r e s e n t e d h e r e , i.e. t h e d i s l o c a t i o n i s considered a s a macroscopic inhomogeneity i s t h e most e s s e n t i a l

4.

one. This assumption means, t h a t t h e mean f r e e p a t h l e n g t h 4 of t h e c a r r i e r s h a s t o be small compared w i t h t h e r a d i u s rd of t h e d f s l o - c a t i o n c y l i n d e r . For a s i n g l e d i s l o c a t i o n which is not decorated by

impurities the main effect on the carriers is expected to be inside the dislocation core with a radial dimension of some 10 2. Thus, considering a "puren dislocation as a cylindrical region, where the recombination rate is different from that outside the cylinder, the assumption rd)l is not satisfied (see Tab. 1). In this case the

material

-I ^-

^{I Z}

dislocation is described more appropriately as a scattering center (e.g. /8/).

However, it was found that a single dislocation is only detectable by EBIC-measurements if it is decorated by impurities / 9 / . Such a decorated dislocation can be regarded more appropriately as a macroscopic inhomogeneity. Further, it appears to be reasonable to assume that only dislocations decorated by impurities as a result of various heat treatments occur i n LED-materials, so that the macroscopic model presented here should be sufficiently justified.

3. - INJECTED CARRIER DENSITY IN PRESENCE OF A SINGLE DISLOCATION Let us consider a semiconductor sample with a junction parallel to the surface in a distance d and with a single dislocation perpendicular to the surface (Fig. ^1). Inside a cylindrical dis- location region of space R d both the radiative ( and the

surf ace

radiative

-

r ecom b~nation region

Pig. 1

-

C4-426 JOURNAL DE PHYSIQUE

non-radiative l i f e t i m e ('CLmd ) a r e assumed t o be d i f f e r e n t from t h o s e ( Z W d , r,,rad ) o u t s i d e t h e d i s l o c a t i o n . A t t h e j u n c t i o n t h e i n j e c t e d m i n o r i t y - c a r r i e r d e n s i t y n ( 2 ) i s assumed t o be c o n s t a n t ( c f . / 3 / ) . Then, f o r t h e bulk n ( g ) obeys i n steady s t a t e t h e f o l - lowing two d i f f e r e n t i a l equations

f o r i n s i d e S&

,

D

where

A t t h e s u r f a c e , z=d

where

~ = m

i s t h e s u r f a c e recombination v e l o c i t y vS i n u n i t s of t h e d i f f u s i o n v e l o c i t y .

A t t h e junction, z=0

n = ii = const. (8)

I n t h e absence of t h e d i s l o c a t i o n t h e s o l u t i o n of t h e problem i s known / 1 o/ :

with

Mathematically, t h e expression ( 9 ) i s a l s o a s p e c i a l a o l u t i o n of t h e g e n e r a l problem (eqs. (4-8)) obeying t h e boundary c o n d i t i o n s and each of t h e two d i f f e r e n t i a l equations ( 4 ) s e p a r a t e l y , but i t does not d e s c r i b e t h e a d d i t i o n a l r a d i a l dependence of t h e c a r r i e r d e n s i t y r e s u l t i n g from t h e coupling of t h e two d i f f e r e n t i a l equations.

Now, i n t r o d u c i n g c y l i n d e r c o o r d i n a t e s ( 3 , ^{z )} and using f o r t h e s p e c i a l s o l u t i o n (9) t h e i n t e g r a l r e p r e s e n t a t i o n / 1 1 /

with

then f o r t h e i n t e r n a l s o l u t i o n ( 2 i n s i d e t h e c y l i n d e r ) one f i n d e

( L V = ~and f o r t h e e x t e r n a l s o l u t i o n )

(r

Here, I, and K O a r e t h e modified Besael f u n c t i o n s of t h e o r d e r z e r o of f i r s t and second kind, r e s p e c t i v e l y , and f denotes t h e r a d i a l d i s t a n c e from t h e d i s l o c a t i o n l i n e . The unknown c o e f f i c i e n t s h and g a r e obtained by t h e matching c o n d i t i o n s a t t h e s u r f a c e of t h e

c y l i n d e r . Let rd be t h e r a d i u s of t h e c y l i n d e r i t must hold

I

If-, ,

I n s e r t i n g ( 1 2 ) and (13) i n t o (14) one o b t a i n s f i n a l l y f o r t h e i n j e c t e d m i n o r i t y - c a r r i e r d e n s i t y :

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where

04 a

J -

.

4.

-

where i s t h e e f f i c i e n c y without d i s l o c a t i o n s , J / J o i s t h e r a t i o of t h e c u r r e n t of photons with d i s l o c a t i o n s t o t h a t without d i s l o c a - t i o n s , and Io/I i s t h e r a t i o of t h e d i f f u s i o n c u r r e n t a c r o s s t h e p-n j u n c t i o n s u r f a c e without d i s l o c a t i o n s t o t h a t with d i s l o c a t i o n s . Using ( 1 5 ) and c o n s i d e r i n g o n l y t h e l a y e r between s u r f a c e and junc- t i o n a s r a d i a t i v e recombination r e g i o n (Fig. 1 ) i n presence of a s i n g l e d i s l o c a t i o n one o b t a i n s f o r t h e i n t e g r a t e d q u a n t i t i e s :

and

where

and

1 dislocation p-n junction area From this it follows for the efficiency:

The quantity GN can be regarded as a cross section which describes the variation of J i n presence of one dislocation, and analogously,

8-- is a cross section for the variation of the junction current in presence of one dislocation.

Now, an arbitrary number of dislocations is considered, where the distance between two neighbour dislocations should be sufficiently large, eo that no correlation exists between them. I n this case the efficiency can be written as

C4-430 JOURNAL- DE PHYSIQUE

where

no. of d i s l o c a t i o n s J d p-n j u n c t i o n a r e a

Exactly, fd d e e c r i b e s t h e average d i s l o c a t i o n d e n s i t y of t h e sample.

A s a measure f o r t h e n e a r e s t d i s t a n c e between two u n c o r r e l a t e d d i s l o c a t i o n s t h e r a d i a l d i s t r i b u t i o n

of t h e i n j e c t e d m i n o r i t y c a r r i e r s around a s i n g l e d i s l o c a t i o n can be a n a l y s e d , where

d (4 -e*d/Lj ( 4 +tis- e

Q,

'

-

7

+

0

i s t h e c o n s t a n t r a d i a l d i s t r i b u t i o n of t h e c a r r i e r s i n a sample without d i s l o c a t i o n s . Two d i s l o c a t i o n s a r e r e g a r d e d t o have no i n - f l u e n c e upon a n o t h e r i f

Now, t h e t h e o r e t i c a l r e s u l t s may be a p p l i e d t o i n t e r p r e t e x p e r i - mental d a t a . Roedel e t a l . / 1 / have p u b l i s h e d a p l o t of e l e c t r o - luminescent e f f i c i e n c y v e r s u s d i s l o c a t i o n d e n s i t y f o r 45 i n d i v i d u a l GaAs-LEDs (Fig. 2). I n o r d e r t o g e t a r e a s o n a b l e agreement of t h e c a l c u l a t e d q - f d -behaviour w i t h t h e experimental one t h e both unknown q u a n t i t i e s ( r d / L ) and ( C /z'), which c h a r a c t e r i z e a s i n g l e d i s l o c a t i o n w i t h i n t h e model, were v a r i e d w i t h i n t h e i n t e r v a l 0.02 4 rd/L 6 1 end w i t h i n 2

<

r e e d - o, --

T'r, d

i.e. t h e r a d i a t i v e recombination r a t e i n s i d e t h e d i s l o c a t i o n r e g i o n was assumed t o be z e r o what a p p e a r s t o be r e a s o n a b l e i n GaAs-

m a t e r i a l .

Only one p a i r of v a l u e s which i s c o n s i s t e n t with t h e assumptions of t h e model was o b t a i n e d , namely

( r d / L ) = 0.55 and (Z/Z') = 100

t o f i t t h e e x p e r i m e n t a l p l o t ( s e e F i g . 2). F o r t h e c r o s s s e c t i o n s t h e v a l u e s

6 2 . 1 L 2 = 213 )am2 and

6, ^{8.97 L~ = 897}

pn2

were c a l c u l a t e d .

F i g . 2

-

The d o t t e d l i n e d e s c r i b e s t h e t h e o r e t i c a l b e h a v i o u r beyond t h e

maximum d e n s i t y of u n c o r r e l a t e d d i s l o c a t i o n s ( = 1.6 l ~ ~ c r n - ~ ) . P d r v c

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Fig. 3 - Experimental d a t a (from B r a n t l e y e t a l . 121; t h e c i r c l e s ) and t h e o r e t i c a l values o b t a i n e d u s i n g d = 6 p, S = d , ( ~ r ~ d / ~ & d ) = 0,

f r

and _F = 0.7, p ⁼ 35 f o r L = 6 p, _{r ra} = ₅ 3 0.7, 9 =

3

U U

f o r L = 5 p, ^{'d =} 7 ^{6m* 0.7,} p ⁼ 3 49 35 f o r L = 7 p. The d o t t e d l i n e s p o i n t out t h e range beyond t h e maximum d e n s i t y of u n c o r r e l a t e d d i s l o c a t i o n s

(&Ex

For Gap-LEDs

( r d / L ) = 0.7 and ( Z / Z ' ) = 35

were found i n t h e same manner a s d e s c r i b e d f o r GaAs i n o r d e r t o f i t t h e experimental d a t a measured by B r a n t l e y e t a l . / 2 / ( t h e c i r c l e s and t h e s o l i d l i n e i n Fig. 3). Here, t h e f o l l o w i n g i n p u t parameters were used: L = 6 p, d = 6 p, S = 0 0 , (cad /rLed ^{= 0.}

I n /2/ t h e d i f f u s i o n l e n g t h s a r e g i v e n f o r t h e 4 specimens con- s i d e r e d i n Fig. 3: L = 5

...

Table 2: Cross s e c t i o n s i n Gap To be a b l e t o g i v e t h e range of v a l i d i t y f o r t h e c a l c u l a t e d yipd)-curves shown i n Pigs, 2,:

t h e maximum d e n s i t y of un- c o r r e l a t e d d i s l o c a t i o n s must be determined. Por t h i s reason t h e normalized r a d i a l d i s t r i - b u t i o n (Q/Qo) of t h e i n j e c t e d m i n o r i t y c a r r i e r s around a s i n g l e d i s l o c a t i o n was c a l c u l a t e d b o t h

f o r GaAs a n d ' ~ a ~ - ( p i g . 4). A s i n p u t parameters t h e v a l u e s g i v e n above were used. Using t h e c r i t e r i o n ( 2 8 ) f o r t h e n e a r e s t d i s t a n c e ad between two u n c o r r e l a t e d d i s l o c a t i o n s t h e v a l u e s

f 27 p f o r GaAs,

were obtained. Now, t h e d i s l o c a t j o n s a r e

Fig. 4

-

The dashed l i n e s show t h e Q/Q, -behaviour a r i s i n g from a s i n g l e d i s l o c a t i o n , t h e s o l i d l i n e r e p r e s e n t s t h e s u p e r p o s i t i o n of t h e c o n t r i b u t i o n s from two neighbour d i s l o c a t i o n s .

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e q u i d i s t a n t l y d i s t r i b u t e d i n t h e sample. I n t h i s way one g e t 8 a hexagonal l a t t i c e ( F i g . 5) and, c o n s e q u e n t l y ,

mox

-

Sd,,' a ( 2 /

? T I

f o r t h e maximum d e n s i t y of u n c o r r e l a t e d d i s l o c a t i o n s . T h i s g i v e s 1 .6#1o5 f o r GaAs,

4.5r105 cm'2 f o r Gap ( c f . F i g s . 2 and 3 ) .

With t h e h e l p of t h e model p r e s e n t e d h e r e t h e a b s o l u t e l y maximum d i s l o c a t i o n d e n s i t y f o r which t h e l u m i n e s c e n c e s h o u l d e n t i r e l y e x t i n g u i s h c a n be g i v e n t o o ( s e e eq. ( 3 ) ) :

4 (1 . I * I o6 cme2 f o r GaAs,

& , n a r r k ~ d

1.8r106 cm'* f o r Gap.

I n comparison w i t h t h e e x p e r i m e n t a l p l o t s shown i n F i g s . 2 and 3 t h e s e &,or - v a l u e s a p p e a r t o be r e a s o n a b l e .

Fig. 5 - Arrangement o f u n c o r r e l a t e d d i s l o c a t i o n s a t maximum d e n s i t y , where a d d e n o t e s t h e n e a r e s t n e i g h b o u r d i s t a n c e . F o r Gap and GaAs t h e a d - v a l u e s a r e g i v e n by ( 2 9 ) .

Acknowledgement. - The a u t h o r wish t o t h a n k P r o f e s s o r K. Unger f o r many v a l u a b l e h i n t s and c r i t i c a l r e a d i n g o f t h e m a n u s c r i p t . I i e l p f u l d i s c u s s i o n s w i t h D r . B. R h e i n l h d e r a r e g r a t e f u l l y acknowledged t o o .

/I/ R.J. ROEDEL, A.R. VON NEIDA, R. CARUSO, and L.R. DAWSON, Extended Abstracts (Electrochemicsl Society, Princeton, N.J., 1977) Vol. 77-2, p. 857.

/2/ W.A. BRANTLEY, O.G. LORIMOR, P.D. DAPKUS, S.E. HASZKO, and R.H. SAUL, J.App1. Phys. 46 (1975) 2629.

/3/ M. LAX, J.App1. Phys. (1278) 2796.

/4/ C. DONOLATO, Optik (1978) 19; Appl. Phys. Letters 2

(1979) 80.

/5/ C. DONOLATO and H. KLANN, J.App1. Phys. 2 (1980) 1624.

/6/ L. PASEMANN, Ultramicroscopy 6 ^{(1 981 ) 237.}

/7/ L. PASEMANN, H. BLUMTRITT, and R. GLEICHMANN, phys.stat.

sol. (a) 70 (1982) 197.

/8/ W, SCKROTER and R. LABUSCH, phys.stat.so1. (b) (1969) 539.

/9/ J. HEYDENREICH, H. BLUMTRITT, R. GLEICHMANN, H. JOHANSEN in:

Scanning Electron Microscopy 1981 (~ditor P. Becker, 0. Johari), Chicago, SEM Inc. 1981/1, p. 351.

/lo/ 0. MADELUNG in: Handbuch der Physik (~ditor S. Fliigge)

Berlin-GSttingen-Heidelberg: Springer, 1957, Vol. XX, p. 118.

/11/ W. GROBNER, N. HOFREITER, Integraltafel Teil 1, Wien und Innsbruck, Springer, 1957, p. 128.