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EFFECT OF DISLOCATION DISSOCIATION ON THE LOCALIZED ELECTRON AND HOLE STATES
AT SCREW DISLOCATIONS IN GERMANIUM
H. Teichler
To cite this version:
H. Teichler. EFFECT OF DISLOCATION DISSOCIATION ON THE LOCALIZED ELECTRON
AND HOLE STATES AT SCREW DISLOCATIONS IN GERMANIUM. Journal de Physique Collo-
ques, 1979, 40 (C6), pp.C6-43-C6-45. �10.1051/jphyscol:1979609�. �jpa-00219025�
JOURNAL D E PHYSIQUE Colloque C 6 , s u p p l h e n t au n06, tune 4 0 , j u i n 1579, page C6-43
EFFECT
3 FDISLOCATION DISSOCIATION ON T H E LaCALIZED ELECTEON AilD HOLE S T A T E S AT SCREW DISLJCATIONS IN GERMANIUN
H. T e i c h l e r
*
Max-Planck-Tnstitut fiir Metallforschung, I n s t i t u t fiir Physik, 0-7000 S t u t t g a r t , G s r r n m y .
Resume.- On o r e s e n t e l e s r e s u l t a t s des c a l c u l s numeriques c o n c e r n a n t l e s e f f e t s de l a d i s s o c i a t i o n s u r l e s @ t a t s l o c a l i s e s d ' e l e c t r o n s e t de t r o u s a s s o c i e s ?I des d i s l o c a t i o n s - v i s dans l e germanium.
Les c a l c u l s s o n t bases s u r l ' a p p r o x i m a t i o n de l a masse e f f e c t i v e e t l ' u t i l i s a t i o n du o o t e n t i e l de d e f o r m a t i o n .
A b s t r a c t . - R e s u l t s o f numerical c a l c u l a t i o n s c o n c e r n i n g t h e e f f e c t o f d i s l o c a t i o n d i s s o c i a t i o n on t h e l o c a l i z e d e l e c t r o n and h o l e s t a t e s a t screw d i s l o c a t i o n s i n Ge a r e presented. The c a l c u l a t i o n s a r e based on t h e e f f e c t i v e mass a p p r o x i m a t i o n and use t h e d e f o r m a t i o n p o t e n t i a l approach.
E a r l i e r t h e o r e t i c a l s t u d i e s /1,2/ o f t h e l o c a - s t a t e s .
l i z e d e l e c t r o n and h o l e bands a t screw d i s l o c a t i o n s I n o u r c a l c u l a t i o n s s t r a i g h t d i s l o c a t i o n s a r e (SD's) have c o n s i d e r e d t h e bound s t a t e s a s s o c i a t e d c o n s i d e r e d w i t h t h e i r l i n e d i r e c t i o n o a r a l l e l t o t h e w i t h complete SD's. However, as shown b y r e c e n t weak [ 1 1 0 ] - d i r e c t i o n . The two pD's w i t h B u r g e r s v e c t o r s beam e l e c t r o n m i c r o s c o p y i n v e s t i g a t i o n s /3,4/ i n Ge bl= * (a/6)[211] and b 2 =
-
( a / 6 ) [ 1 2 i I (e.g., / 3 , 4 / ) and S i no st of t h e SD's a r e d i s s o c i a t e d i n t o two 30" a r e assumed t o f o r m i n t r i n s i c o r , a l t e r n a t i v e l y , ex- p a r t i a l d i s l o c a t i o n s ( p D ' s ) . N i t h r e g a r d t o t h i s wehave i n v e s t i g a t e d t h e e f f e c t o f t h e s p l i t t i n g o f t h e SD's on t h e l o c a l i z e d e l e c t r o n and h o l e s t a t e s i n d u - ced b y t h e l o n g range s t r a i n f i e l d s o f t h e d i s l o c a - t i o n s . I n o a r t i c u l a r we have c o n c e n t r a t e d on t h e q u e s t i o n up t o which d i s s o c i a t i o n w i d t h o f t h e SD t h e s t r a i n f i e l d s o f t h e two pD's a p p r o x i m a t e l y a c t on t h e e l e c t r o n s and h o l e s l i k e t h a t o f an u n s o l i t SO, and f r o m w h i c h d i s s o c i a t i o n w i d t h on one may t r e a t t h e bound s t a t e s a t t h e two pD's l i k e s t a t e s a t i n f i n i t e l y f a r s e p a r a t e d p a r t i a l s . Here we p r e s e n t f o r Ge r e s u l t s o f n u m e r i c a l c a l c u l a t i o n s .
Throughout i n o u r s t u d y a o n e - p a r t i c l e p i c t u r e has been used and Coulomb i n t e r a c t i o n s between t h e bound s t a t e s have been n e g l e c t e d . T h i s a p p r o x i m a t i o n i s v a l i d as l o n g as t h e system remains i n t h e n e u t r a l s t a t e , i . e . , t h e p e r t u r b a t i o n bands determined i n t h e p r e s e n t t r e a t m e n t d e s c r i b e t h e s i n g l e - p a r t i c l e e x c i -
t r i n s i c s t a c k i n g f a u l t s on t h e (111) p l a n e . The s t r a i n t e n s o r s ~ ( r ) o f t h e d i s l o c a t i o n s a r e c o n s t r u c -
*
t e d by s u p e r p o s i t i o n o f t h e s t r a i n f i e l d s o f t h e c o r - r e s p o n d i n g edge and screw d i s l o c a t i o n s i n an i s o t r o -
~ i continuum c / 5 / . ( P o i s s o n ' s r a t i o v = 0.3 has been used). The e a r l i e r i n v e s t i g a t i o n s /1,2/ have shown t h a t t h e s t r a i n - f i e l d induced s t a t e s a t S o ' s have r a t h e r s h a l l o w e n e r g y l e v e l s i n t h e band gap and a r e extended i n space. Consequently we have used t h e e f - f e c t i v e mass a p p r o x i m a t i o n t o d e t e r m i n e t h e energy l e v e l s and wave f u n c t i o n s by a v a r i a t i o n a l orocedure, where t h e e l e c t r o n - o r h o l e - d i s l o c a t i o n i n t e r a c t i o n
i s d e s c r i b e d b y a d e f o r m a t i o n o o t e n t i a l (see, e.g.
,
/6-a/). I n t h e framework o f t h e s e a p o r o x i m a t i o n s t h e d i s l o c a t i o n s e n t e r t h e H a m i l t o n i a n by t h e i r l o n g range s t r a i n f i e l d o n l y and hence g l i d e - and s h u f f l e - s e t c o n f i g u r a t i o n s need n o t be d i s t i n g u i s h e d .t a t i o n spectrum a t SD's i n t h e l i m i t o f v a n i s h i n g 1. L o c a l i z e d e l e c t r o n s t a t e s . - The l o c a l i z e d e l e c t r o n d e n s i t y o f e x c i t a t i o n s . S i n c e t h e e l e c t r o n s t a t e s s t a t e s a t t h e pD's a r e assumed t o be s p l i t o f f f r o m a s s o c i a t e d w i t h t h e c o r e s t r u c t u r e of t h e p D ' s w i l l t h e c o n d u c t i o n band minima. I n t h e d e f o r m a t i o n poten- be s t r o n g l y l o c a l i z e d a t t h e c o r e ( i f t h e y e x i s t a t t i a l approach t h e p o t e n t i a l energy o f t h e e l e c t r o n s a l l ) t h e y may, i n t h e n e u t r a l s t a t e o f t h e d i s l o c a - i n t h e s t r a i n f i e l d o f t h e d i s l o c a t i o n s depends upon t i o n , be o f m i n o r i m p o r t a n c e f o r t h e p r o p e r t i e s o f t h e v a l l e y where t h e e l e c t r o n s come f r o m /1,6/ and t h e s t r a i n - i n d u c e d extended s t a t e s . T h e r e f o r e we have on t h e B u r g e r s v e c t o r o f t h e pD. Most t i g h t l y bound c o m p l e t e l y d i s r e g a r d e d e f f e c t s o f p o s s i b l e c o r e a r e e l e c t r o n s f r o m v a l l e y s i n t h e (170) p l a n e o f x p r e s e n t address : I n s t i t u t f u r V e t a l l p h y s i k d e r * k-space. F o r t h e two 30' pD's t h e energy l e v e l s be- Uni v e r s i tlt, 0-3400 G t i t t i n g e n
,
Germany. l o n g i n g t o t h e s e f o u r v a l l e y s a r e i n t i m a t e l y r e l a t e d4
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979609
C6-44 JOURNAL DE PHYSIQUE
t h r o u g h t h e symmetry o f t h e system and t h e p o s i t i o n o f t h e s e l e v e l s may be deduced f r o m t h e l e v e l s o f t h e
[
111:-val l e y c o n s i d e r e d below. The H a m i l t o n i a n i s c o n s t r u c t e d as i n / I / . F o r t h e d e f o r m a t i o n p o t e n t i a l parameters t h e v a l u e s Ed = -6.9 eV, Z,, = 17 eV /6/a r e used. T r i a l f u n c t i o n s o f t h e f o r m
have been s e l e c t e d . (The x-y-plane i s p e r p e n d i c u l a r t o t h e d i s l o c a t i o n l i n e ; al, a2, and c a r e v a r i a t i o n parameters.) The p a r t i c u l a r f o r m of Y p e r m i t s t o eva- l u a t e i n c l o s e d form a l l i n t e g r a t i o n s o c c u r i n g i n t h e c a l c u l a t i o n o f t h e k i n e t i c and p o t e n t i a l energy o f t h e e l e c t r o n s i n t h e s t r a i n f i e l d s o f t h e pD's. Only t h e m i n i m i z a t i o n o f t h e t o t a l energy has t o be c a r - r i e d o u t n u m e r i c a l l y .
The main r e s u l t s o f o u r c a l c u l a t i o n s a r e : i ) I n t h e case o f an u n s p l i t SD t h e minimum o f t h e
one-dimensional band o f l o c a l i z e d e l e c t r o n s t a t e s i s about 55 meV below t h e c o n d u c t i o n band m i n i - mum Ec. T h i s v a l u e i s i n f a i r agreement w i t h t h e r e s u l t o f /1/ where b y t a k i n g i n t o account t h e a n i s o t r o p y o f t h e s t r a i n f i e l d t h e e s t i m a t e E
-
Ec =
-
45 meV has been o b t a i n e d .i i ) F o r i n f i n i t e l y f a r s e p a r a t e d p a r t i a l s l o c a l i z e d e l e c t r o n s t a t e s a r e a s s o c i a t e d w i t h b o t h pD1s w i t h l o w e s t energy l e v e l s f o r t h e b l - d i s l o c a t i o n
-
a t E
-
Ec =-
15 meV, f o r t h e b , - d i s l o c a t i o n-
a t E-
E_ =-
14 meV.R Cot. u.J
0 10 104 1000
I I I
I
F i g . 1 : Energy l e v e l s o f most t i g h t l y bound e l e c t r o n s t a t e s below t h e b o t t o m o f t h e c o n d u c t i o n band a t d i s s o c i a t e d screw d i s l o c a t i o n s i n Ge as f u n c t i o n o f d i s s o c i a t i o n w i d t h R f o r i n t r i n s i c ( s o l i d l i n e ) and e x t r i n s i c (dashed l i n e ) s t a c k i n g f a u l t geometry.
2. L o c a l i z e d h o l e s t a t e s . - I n t h e p r e s e n t p r e l i m i n a - r y c a l c u l a t i o n o f t h e l o c a l i z e d h o l e s t a t e s we have t a k e n i n t o account t h e screw component o f t h e 30' pD's o n l y . The h o l e - t y o e p e r t u r b a t i o n bands above t h e t o p o f t h e v a l e n c e band a r e e v a l u a t e d b y a p p l y i n g t h e L u t t i n g e r - K o h n e f f e c t i v e mass H a m i l t o n i a n ( e - g . , / 7 / ) t o t h e f o u r degenerated s p i n - o r b i t s t a t e s ( a b b r e v i a t e d by IJ,Jz> w i t h J = 3/2 and J Z = +3/2, +1/2) a t t h e v a l e n c e band maximum. F o r t h e r e c i p r o c a l - m a s s
L
i i i ) F o r f i n i t e v a l u e s of t h e d i s l o c a t i o n w i d t h R Parameters t h e v a l u e s r e p o r t e d by Lawaetz /7/ have t h e e n e r g i e s o f t h e bound s t a t e s a r e shown in been used and t h e d e f o r m a t i o n p o t e n t i a l c o n s t a n t s f i g u r e 1. ~ h ~ ~ ~ l e v e l s have been c a l c u l a t e d by have been t a k e n from P o l l a k and Cardona /8/. W i t h a s s o c i a t i n g t h e t r i a l f u n c t i o n (Eq. ( 1 ) ) w i t h one r e s p e c t t o t h e symmetry o f t h e system t r i a l f u n c t i o n s o f t h e two p a r t i a l s b u t t a k i n g i n t o a c c o u n t i n O f the
t h e p o t e n t i a l e n e r g y t h e s t r a i n f i e l d s o f b o t h y(x,y,z) -. fi e - ~ r ei (kxx+kzz)
-
p a r t i a l s . I n f i g u r e 1 t h e most t i g h t l y bound s t a t e s a r e d i s p l a y e d o n l y . The energy o f t h e l o c a l i z e d e l e c t r o n s t a t e s depends upon w h e t h e r an i n t r i n s i c o r an e x t r i n s i c s t a c k i n g f a u l t i s formed, t h e s t a t e s w i t h l o w e r energy o c c u r r i n g i n t h e case o f an i n t r i n s i c f a u l t .
i v ) N e g l e c t of t h e edge component i n t h e two pD's l e a d s t o bound s t a t e s a p p r o x i m a t e l y i n t h e mid- d l e between t h e l e v e l s g i v e n i n f i g u r e 1 f o r i n t r i n s i c and e x t r i n s i c s t a c k i n g - f a u l t geometry.
From t h i s we conclude t h a t t h e g r o s s - f e a t u r e s o f t h e e f f e c t o f d i 7 l o c a t i o n d i s s o c i a t i o n on t h e bound. e l e c t r o n s t a t e s may. be e s t i m a t e d b y d i s - r e g a r d i n g t h e edge components o f t h e pD1s a t a l l .
x 3 1 x 3 3
,
e 3 .IT, -
> )+
s i n X ( r e3.I-
2 ' 2 - > +3 1 .71 3 3
e1.1
5 , -
;? > +f
e ' ~ e ~ . l ~ ,-2 > ) I
( 2 )have been i n t r o d u c e d . Here z and k z denote t h e p r o - j e c t i o n s on t h e 1 1 1 0 7 d i r e c t i o n , x and kx t h o s e on t h e L O 0 1 1 d i r e c t i o n , and y i s p a r a l l e l t o
K i l O ] . The ei as w e l l as c , kx, and X a r e v a r i a t i o n - a l parameters. F o r d i f f e r e n t v a l u e s o f t h e d i s s o c i a - t i o n w i d t h R t h e c a l c u l a t e d one-dimensional bands o f l o c a l i z e d h o l e s above t h e v a l e n c e bands a r e d i s p l a y e d i n f i g u r e 2 as f u n c t i o n s o f kz. F i g u r e 3 shows t h e energy l e v e l s of t h e most t i g h t l y bound h o l e s t a t e s a t kZ = 0 as f u n c t i o n s o f R. A t f i n i t e
R
t h e h o l e - t y p e energy l e v e l s were c a l c u l a t e d b y u s i n g a s i m i -H. Teichler
l a r approximation as i n t h e case o f t h e bound e l e c - References
t r o n s t a t e s , i.e. t r i a l f u n c t i o n s o f t h e form (2) were associated w i t h each o f t h e pD's and i n t h e po- t e n t i a l t h e s t r a i n f i e l d s o f both p a r t i a l s were i n - c l uded.
Fig. 2 : One-dimensional bands o f l o c a l i z e d h o l e s t a t e s above t h e valence band maximum a t d i s s o c i a t e d screw d i s l o c a t i o n s i n Ge f o r various values o f t h e d i s s o c i a t i o n w i d t h R (as obtained by approximating t h e p a r t i a l d i s l o c a t i o n s by t h e i r screw components o n l y ) . k denotes t h e wave v e c t o r p a r a l l e l t o the d i s l o c a t f o n l i n e .
R [at. u]
Fig. 3 : energy l e v e l s o f most t i g h t l y bound h o l e s t a t e s above t h e t o p of t h e valence bands a t disso- c i a t e d screw d i s l o c a t i o n s i n Ge as f u n c t i o n s o f t h e d i s s o c i a t i o n w i d t h R.
3. Conclusion.- For t h e l o c a l i z e d e l e c t r o n and h o l e s t a t e s a t n e u t r a l SD's i n Ge our c a l c u l a t i o n s i n d i - c a t e t h a t t h e d i s s o c i a t i o n o f t h e SD i n t o 30' pD's i s o f marked i n f l u e n c e on t h e l e v e l p o s i t i o n s i f t h e s e p a r a t i o n o f t h e two p a r t i a l s i s l a r g e r than about
0
20 A, and t h a t f o r d i s s o c i a t i o n widths R l a r g e r than
0
500 A t h e two pD's may be approximately t r e a t e d l i k e independent d i s l o c a t i o n s , i .e., as being i n f i n i t e l y f a r separated.
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