DEEP ELECTRON LEVELS AND FURTHER EFFECTS OF TOPOLOGICAL DISORDER BY DISLOCATIONS

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DEEP ELECTRON LEVELS AND FURTHER EFFECTS OF TOPOLOGICAL DISORDER BY

DISLOCATIONS

H. Teichler, H. Veth

To cite this version:

H. Teichler, H. Veth. DEEP ELECTRON LEVELS AND FURTHER EFFECTS OF TOPOLOGICAL DISORDER BY DISLOCATIONS. Journal de Physique Colloques, 1983, 44 (C4), pp.C4-93-C4-101.

�10.1051/jphyscol:1983411�. �jpa-00222850�

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Colloque C4, supplément au n°9, Tome 44, septembre 1983 page C4-93

D E E P ELECTRON LEVELS A N D FURTHER EFFECTS OF T O P O L O G I C A L D I S O R D E R B Y D I S L O C A T I O N S

H. Teichler and H. Veth

Inatilut ftlr Metallphysik dev Universitai GOtt-ingen und Sondorfcpsahungs- beveieh 126, GOttinqen, F.R.G.

Résumé - On présente une revue brève des calculs des niveaux profonds associés aux dislocations. On décrit en particulier des résultats obtenus par une technique de récursion qui permet de conserver la symétrie de translation. Enfin, certains effets liés au "potentiel de jauge" des dislocations sont mentionnés.

Abstract - Recent calculations of deep levels at dislocations are briefly reviewed. In particular results of a recursion approach maintaining the translation symmetrie are presented. In addition, effects of the "gauge potential" of dislocations are sketched.

I. INTRODUCTION

The topological disorder introduced in a lattice by dislocations manifests itself in a non-vanishing closing vector of the Burgers circuit /l/ (and thus identifies the dislocation as a topological defect in the lattice 12/) where the closing vector corresponds with the Burgers vector /l/ of the dislocation. The interrelationship between topological disorder by dislocations and the electronic properties of semiconductors in the first instance follows from the fact that the topological disorder for a variety of dislocations leads to dangling bonds in the dislocation cores.

These dangling bonds - as proposed by Shockley /3/ long ago - give rise to deep levels in the band gap and hence influence the electrical properties of the material.

(For a review of the experiments in this field see, e.g. /4/.) - However, it should be stated that there may exist also other sources of deep levels in the dislocation cores, e.g. extremely stretched bonds /5,6/.

At present the theoretical investigations of the deep electron levels at the dangling, bonds in the dislocation cores inevitably are entangled with the topological disorder in the system: the change in topology brought in by a dislocation prevents one from characterizing (isolated) dislocations in a simple manner by a local pertubation potential / 7 / and hence demands elaborated approaches to determine the corresponding deep electron levels.

With regards to this special theoretical methods have been put forward /8,9/. The major part of our contribution shall be devoted to a critical summary and discussion of the present status of these methods and of the results derived therefrom.

In addition to these "short-range" effects of topological disorder in the dislocations cores the global modified topology of a crystal with dislocations - as characterized by the Burgers circuit - also influences the electronic excitations in the system on a "long-range" scale. As realized, e.g. by Kawamura /10/ in his numerical investigations of the scattering of electrons at screw dislocations (in a simple cubic tight-binding model) those parts of an incident plan wave which circumvent the dislocation line on the left hand side and those which pass it on the right hand side exhibit a phase-mismatch when meeting again behind the dislocation which yields a rather strong scattering of the electrons. This behaviour has been put into analogy /10/ with the Bohm-Aharonov-effect /ll/. Recently it has been shown 111

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1983411

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C4-94 JOURNAL DE PHYSIQUE

t h a t t h i s p a r t i c u l a r f e a t u r e o f d i s l o c a t i o n s may be d e s c r i b e d i n a continuum approach by a n o n - v a n i s h i n g " g a u g e - p o t e n t i a l " which a c t s on t h e e l e c t r o n s i n a d d i t i o n t o t h e commonly used d e f o r m a t i o n - p o t e n t i a l s /12,13/.

-

E f f e c t s o f t h i s g a u g e - p o t e n t i a l s h a l l be o u t l i n e d i n a f i n a l s e c t i o n o f t h e p r e s e n t c o n t r i b u t i o n .

11. DEEP LEVEL CALCULATIONS FOR DISLOCATIONS

S i n c e weak-beam e l e c t r o n microscopy i n v e s t i g a t i o n s (e.g., /14,15/) r e v e a l e d t h a t d i s l o c a t i o n s i n t h e diamond and s p h a l e r i t e s t r u c t u r e a r e d i s s o c i a t e d a n t o p 8 r t i a l s we shad1 r e f e r i n t h e f o l l o w i n g i n most cases t o t h e p r e d o m i n a t i n g 30

-,

60

-,

and 90 - p a r t i a l d i s l o c a t i o n s /16,17/ i n g l i d e s e t c o n f i g u r a t i o n . E x c e l l e n t r e v i e w s o f t h e e a r l i e r t h e o r e t i c a l i n v e s t i g a t i o n s o f t h e deep e l e c t r o n l e v e l s a s s o c i a t e d w i t h t h e s e d i s l o c a t i o n s have been g i v e n by Jones /8/ and Marklund /18/. W i t h r e g a r d s t o t h a t h e r e o n l y some g e n e r a l statements about t h e so f a r p u b l s i h e d papers i n t h i s f i e l d s h a l l be g i v e n which m i g h t be o f importance f o r c h a r a c t e r i z i n s r e c e n t and, e v e n t u a l l y , f u t u r e developments.

A l t h o u g h t h e p u b l i s h e d c a l c u l a t i o n s of t h e deep e l e c t r o n l e v e l s a t d i s l o c a t i o n s d i f f e r w i t h many r e s p e c t s t h e y share some g e n e r a l f e a t u r e s i n common:

a ) The l e v e l s i n t r o d u c e d by t h e d i s l o c a t i o n s a r e e x t r a c t e d by comparing t h e c a l c u - l a t e d energy s p e c t r a o f c r y s t a l s c o n t a i n i n g d i s l o c a t i o n s w i t h t h o s e o f i d e a l c r y s t a l s .

b) The a t o m i c p o s i t i o n s i n t h e c r y s t a l s w i t h d i s l o c a t i o n s a r e taken f r o m p e r i p h e r a l s t u d i e s (such as g e o m e t r i c c o n s t r u c t i o n s , r e s u l t s o f l i n e a r e l a s t i c i t y theory, l a t t i c e - m o d e l c a l c u l a t i o n s ) .

c ) One-electron models a r e used.

The approach ( a ) seems t o be i n d i s p e n s a b l e s i n c e , as mentioned i n t h e i n t r o d u c t i o n , t h e t o p o l o g i c a l d i s o r d e r b r o u g h t i n by a d i s l o c a t i o n p r o v e n t s one from t r e a t i n g d i s l o c a t i o n s i n a p e r t u r b a t i v e way / 7 / . Thus s u f f i c i e n t a c c u r a t e c a l c u l a t i o n s o f t h e deep l e v e l s t r u c t u r e w i l l need l a r g e n u m e r i c a l e f f o r t s .

-

Concerning ( b ) one has t o t a k e i n t o account t h a t assumptions a b o u t t h e a t o m i c arrangement may be o f marked i n f l u e n c e on t h e r e s u l t s o f l e v e l s t r u c t u r e c a l c u l a t i o n s . T h i s has been r e v e a l e d , i n p a r t i c u l a r , by Y a r k l u n d ' s s t u d i e s /18,19/ a b o u t t h e e f f e c t s o f d i s l o c a t i o n s - c o r e geometry and back-bond a n g l e s on t h e p o s i t i o n s o f dangling-bond bands.

-

When s t u d y i n g t h e deep l e v e l s of uncharged d i s l o c a t i o n s i n elemental semiconductors t h e use o f a o n e - e l e c t r o n approach, i t e m ( c ) , seems t o be no r e s t r i c t i o n . A t charged d i s l o c a t i o n s , however, t h e deep l e v e l s may be s h i f t e d r e l a t i v e t o t h e l o c a l band edges (H. Veth and M. Lannoo, t o be p u b l i s h e d ) w h i c h t h e n demands more e l a b o r a t e t r e a t m e n t s . The u s e o f a o n - e l e c t r o n a p ~ r o a c h may be a poor a p p r o x i m a t i o n , i n p a r t i c u l a r , f o r d i s l o c a t i o n s i n compounds s i n c e t h e r e , as r e a l i z e d i n s u r f a c e model c a l c u l a t i o n s /20/, c h a r g e - t r a n s f e r may o c c u r i n t h e p a r t i a l l y i o n i c bonds which s t r o n g l y may i n f l u e n c e t h e b o u n d - s t a t e energy l e v e l s .

D e s p i t e t h e above mentioned s i m i l a r i t i e s t h e c a l c u l a t i o n s p r e s e n t e d so f a r d i f f e r a p p r e c i a b l y w i t h some r e s p e c t s :

i ) Some o f them /15,8/ make u s e o f t h e t i g h t - b i n d i n g i n t e r p o l a t i o n scheme ( w i t h e x p o n e n t i a l s c a l i n g under d e f o r m a t i o n s ) , model t h e c r y s t a l w i t h d i s l o c a t i o n by a s p h e r i c a l c l u s t e r o f a l i m i t e d number o f atoms ( t y p i c a l l y 700 t o 1500) around t h e d i s l o c a t i o n c o r e , and d e t e r m i n e t h e deep l e v e l s from a c o n t i n u e d - f r a c t i o n d e n s i t y - o f - s t a t e s f o r m u l a t i o n /21/.

i i ) Some o f them /9,18/ make u s e o f t h e t i g h t - b i n d i n g scheme as s t a t e d above b u t a p p l y a s u p e r - c e l l a p p r o x i m a t i o n by c o n s i d e r i n g p e r i o d i c arrangements o f p a i r s o f a l t e r n a t e d i s l o c a t i o n s and c a l c u l a t e t h e complete e i g e n - v a l u e s p e c t r a o f these arrangements ( h e r e a l s o a r e c e n t s t u d y / 2 2 / o f d i s l o c a t i o n s i n diamond has t o be mentioned which employs t h e s u p e r - c e l l approach b u t i s based on d i s t a n c e dependent t i g h t - b i n d i n g parameters f r o m m u f f i n - t i n c a l c u l a t i o n s ) .

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a t o m i c m o d e l - p o t e n t i a l s and a p p l y t h e s u p e r - c e l l method.

i v ) F o r completeness h e r e a l s o two i n v e s t i g a t i o n s /25,26/ s h o u l d b e mentioned which c o n s i d e r t h e deep l e v e l s a t d a n g l i n g bonds a l o n g rows o f vacancies and o f h i ~ h e r vacancy c l u s t e r s i n S i . The method used i n these papers i s based on d e s c r i b i n g t h e d e f e c t s by l o c a l p e r t u r b a t i o n p o t e n t i a l s , an approach h a r d l y t o a p p l y t o d i s l o c a t i o n s because o f t h e a s s o c i a t e d changes i n t o p o l o g y .

The s u p e r - c e l l method a s w e l l as t h e s p h e r i c a l c l u s t e r approach have p a r t i c u l a r m e r i t s b u t on t h e o t h e r hand a r e c h a r a c t e r i z e d by s p e c i f i c l i m i t a t i o n s . Due t o c o m p u t a t i o n a l reasons t h e s u p e r - c e l l method i s a p p l i c a b l e t o systems w i t h s m a l l numbers o f n o n - e q u i v a l e n t atoms i n a c e l l o n l y ( t y p i c a l l y a b o u t 56 t o 88 atoms), w h i c h demands n a r r o w p a i r s o f d i s l o c a t i o n s (hence i n t e r a c t i o n s between l o c a l i z e d s t a t e s a t d i f f e r e n t d i s l o c a t i o n s c a n n o t b e e x c l u d e d ) and r a t h e r s i m p l e d i s l o c a t i o n - c o r e s t r u c t u r e s . The s p h e r i c a l c l u s t e r approach f a i l s i n r e s o l v i n g t h e d i s p e r s i o n o f t h e d e e p - l e v e l bands a l o n g t h e d i s l o c a t i o n s s i n c e i t s s p h e r i c a l geometry

a r t i f i c i a l l y breaks t h e t r a n s l a t i o n symmetry o f t h e d i s l o c a t i o n s . (Moreover, i n i t s p r e s e n t a p p l i c a t i o n s i t has a r a t h e r l i m i t e d energy r e s o l u t i o n o n l y . ) W i t h r e g a r d s t o t h i s we r e c e n t l y have developed a method combining t h e m e r i t s o f b o t h approaches, i.e., a method a p p l i c a b l e t o i s o l a t e d d i s l o c a t i o n s w i t h c o m p l i c a t e d c o r e s t r u c t u r e s b u t m a i n t a i n i n t h e t r a n s l a t i o n i n v a r i a n c e o f t h e system. T h i s method and r e s u l t s

!

o b t a i n e d r e c e n 1 ~ / 6 / t h e r e f r o m s h a l l b e b r i e f l y o u t l i n e d i n t h e n e x t s e c t i o n . I n o r d e r t o complete t h e p r e s e n t s e c t i o n we s h a l l have a s h o r t g l a n c e on deep l e v e l c a l c u l a t i o n s f o r d i s l o c a t i o n s i n compounds. F o r GaAs /27/ and CdTe /28/ such c a l c u l a t i o n s have been c a r r i e d on i n t h e framework o f a o n e - e l e c t r o n

bight-hinging

p a r a m e t r i z a t i o n o f t h e H a m i l t o n i a n i n a s u p e r - c e l l approach f o r t h e 30

-

^{and 90}

-

p a r t i a l s . The r e s u l t s depend a p p r e c i a b l y on t h e a t o m i c arrangements and t i g h t - b i n d i n g p a r a n e t e r s e t s b u t , n ~ v e r t h e l e

8

s, t h e y show some g e n e r a l t r e n d s : ( i ) t h e d a n g l i n g - b o n d bands o f t h e 30

-

and 90 - p a r t i a l s o f As(g) and Te(g) t y p e a r e

p r e d i c t e d t o l i e i n t h e bandgaps below t h e c o r r e s p o n d i n g bands o f Ga(g) r e s p . Cd(g) t y g e d i s l o c a t i o n s , and ( i i ) i n t h e case o f s c r e w - d i s l o c a t i o n d i s s o c i a t i o n i n t o 30 - p a r t i a l s a s u b s t a n t i a l charge t r a n s f e g s h o u l d o c c u r Between t h e two p a r t i a l s y i e l d i n g , e.g., f o r CdTe /28/ t h a t t h e 30 T e ( g ) - and 30 C d ( g ) - p a r t i a l s i n v o l v e d may have n e g a t i v e l y and p o s i t i v e l y charged c o r e s w i t h 1/2 e l e c t r o n ( h o l e ) i n excess

p e r dangl i ng bonds.

W i t h r e g a r d s t o t h e l a t t e r f i n d i n g t h e o n e - e l e c t r o n approach seems t o be a poor a p p r o x i m a t i o n . E s t i m a t e s o f t h e deep e l e c t r o n l e v e l s o f d i s l o c a t i o n s beyond t h e one- e l e c t r o n p i c t u r e s have been d e r i v e d by Farvacque and F e r r e /29/ w i t h i n t h e m o l e c u l a r - o r b i t a l method /20/ where i n t r a - and i n t e r - a t o m i c Coulomb e f f e c t s and changes i n t h e charge t r a n s f e r w i t h i n t h e p a r t i a l l y i o n i c bonds i s t a k e n i n t o account. The method has been a p p l i e d t o t h e 60°-shuffle-set-dislocation i n InSb, GaAs, ZnSe and CdTe /29/

y i e l d i n g i n each case t h e e x i s t e n c e o f d a n g l i n g bond s t a t e s i n t h e band gaps.

U n f o r t u n a t e l y , r e l i a b l e v a l u e s f o r t h e l e v e l p o s i t i o n s i n t h e band gaps c a n n o t b e deduced f r o m /29/ s i n c e i n t h i s approach t h e b u l k bands a r e b a d l y r e p r e s e n t e d . C a l c u l a t i o n s combining t h e m o l e c u l a r - o r b i t a l approach w i t h a t i g h t - b i n d i n g band p a r a m e t r i z a t i o n now a r e under way f o r CdTe (M. G r o h l i c h , p e r s o n a l communication).

111. TIGHT-BINDING

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RECURSION APPROACH MAIPITAINING THE TRANSLATION INVARIANCE I n i t s p r e s e n t form / 6 / t h e approach i s based on t h e t i g h t - b i n d i n g i n t e r p o l a t i o n scheme /30/ w i t h e x p o n e n t i a l s c a l i n g o f t h e t i g h t - b i n d i n g parameters under d e f o r m a t i o n as used i n /5,8,9,18/. W i t h i n t h i s approach t h e " p a r t i a l d e n s i t y of s t a t e s " N ( E , j ) o f s u i t a b l e s t a r t o r b i t a l s I j> i s e v a l u a t e d by a Lanczos r e c u r s i o n t r e a t m e n t /21/ i n terms o f c o n t i n u e d f r a c t i o n s , where

( la>: s t a t i o n a r y s t a t e s o f t h e system). The t r a n s l a t i o n i n v a r i a n c e o f c r y s t a l s k!i t h d i s l o c a t i o n s a l o n g t h e d i s l o c a t i o n l i n e i m p l i e s t h a t t h e e i g e n s t a t e s o f t h e system can be l a b e l e d b y B l o c h wave-vectors

k

p a r a l l e l t o t h e d i s l o c a t i o n l i n e 131. I n t h e

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r e c u r s i o n t r e a t y e n t s t a r t - o r b i t a l s a r e used w h i c h a r e Bloch-wave l i k e l i n e a r con- b i n a t i o n s o f sp - n y b r i d e s w i t h wave number k a l o n g a n i n f i n i t e row o f atoms p a r a l l e l t o t h e d i s l o c a t i o n . T a k i n g advantage o f t h e f a c t t h a t t h e k v a l u e i s m a i n t a i n e d d u r i n g t h e r e c u r s i o n procedure t h u s g i v e t h e p a r t i a l d e n s i t y o f s t a t e s f o r t h i s row o f h y b r i d e s f o r f i x e d v a l u e o f k. Comparison o f t h i s d e n s i t y o f s t a t e s w i t h t h a t of a s i m i l a r l i n e a r c o m b i n a t i o n o f h y b r i d e s i n t h e p e r f e c t l a t t i c e t h e n y i e l d s t h e deep e l e c t r o n l e v e l s o f t h e d i s l o c a t i o n s as f u n c t i o n s o f k.

The d e s c r i b e d approach corresponds t o embedding t h e d i s l o c a t i o n c o r e i n a c y l i n d r i c a l c l u s t e r o f atoms where t h e d i a m e t e r o f t h e c l u s t e r i n c r e a s e s w i t h i n c r e a s i n g number o f r e c u r s i o n steps. Up t o 110 r e c u r s i o n s t e p s a r e e v a l u a t e d i n some cases /6/

( c o r r e s p o n d i n g t o c y l i n d r i c a l c l u s t e r s o f a b o u t 22.000 rows o f atoms) g i v i n g a r a t h e r h i g h energy r e s o l u t i o n as demonstrated b y F i g . 1. ( F i g . l d i s p l a y s a n averaged d e n s i t y o f s t a t e s N(E;k) o b t a i n e d b y a v e r a g i n g o v e r a l l t h e N(E;k)-curves w i t h 90 t o 110 s i g n i f i c a n t r e c u r s i o n s t e p s and i n f i n i t e r e p e t i t i o n o f t h e l a s t c o e f f i c i e n t s i n each case. Note t h a t t h e v a l u e o f t h e bandgap i n m(E;k) c o i n c i d e s w i t h i t s v a l u e i n t h e u n d e r l y i n g t i g h t - b i n d i n g b a n d s t r u c t u r e w i t h o u t any a d j u s t m e n t . )

By means o f t h i s approach t h e deep e l e c t r o n l e v e l s f o r 30'-, 60°, and 9 0 O - ~ a r t i a l d i s l o c a t i o n s i n Ge and S i have been e v a l u a t e d / 6 / f o r c o r e c o n f i g u r a t i o n s w i t h d a n g l i n g bonds and w i t h r e c o n s t r u c t e d bonds /9,18/ where i n a l l cases t h e a t o m i c arrangements i n t h e d i s l o c a t i o n c o r e 8 have begn t a k e n f r o m l a t t i c e - m o d e l s i m u l a t i o n s as i n /32/. F o r t h e r e c o n s t r u c t e d 30

-

and 90 - p a r t i a l d i s l o c a t i o n s / 9 / i n Ge and S i t h e c a l c u l a t i o n y i e l d s resonances i n t h e v a l e n c e and c o n d u c t i o n bands o n l y b u t no deep l e v e l s i n &he gaps. The one-dimensional d e e p - l e v e l bands o b t a i n e d f o r t h e u n r e c o n s t r u c t e d 90 - p a r t i a l s a r e d e p i c t e d i n F i g . 2. As f a r as comparison w i t h e a r l i e r c a l c u l a t i o n s a r e p o s s i b l e (e.g. / 9 / ) t h e r e s u l t s a r e i n c l o s e agreement.

The u n r e c o n s t r u c t e d 3 0 O - ~ a r t i a l s s h a l l n o t b e c o n s i d e r e d h e r e s i n c e t h e y seem t o be u n s t a b l e compared w i t h t t ! e i r r e c o n s t r u c t e d v e r s i o n s /9/.

[%

1 -Averaged p a r t i a l d e n s i t y o f s t a t e s f o r 410,-rows o f sp - h y b r i d e s o w i t h 3 o i n S i ( a ) i n t h e p e r f e c t l a t t i c e and ( b ) f o r d a n g l i n g bonds i n a 90

-

p a r t i a l d i s l o c a t i o n /6/

F i g . 2 - Deep-level bands o f u n r e c o n s t r u c t e d 9 0 ' - ~ a r t i a l d i s l o c a t i o n s i n S i and Ge

(6)

Si

Ge

P a r t i c u l a r p r o g r e s s has been made f o r t h e 6 0 O - ~ a r t i a l s where t h e k-dependence o f t h e l e v e l s t r u c t u r e c o u l d n o t be r e s o l v e d i n t h e e a r l i e t ; c a l c u l a t i o n s . F i g . 3 d i s p l a y s t h e bands o f t h e i n c o m p l e t e l y r e c o n t r u c t e d 60 - p a r t i a l s i n S i and Ge c o n t a i n i n g one row o f w e l l separated d a n g l i n g bonds w i t h d i s t a n c e m a

( c o r r e s p o n d i n g t o c o r e s t r u c t u r e F i g . 3 o f Ref. /5/). The bands o r i g i n a t i n g from t h e d a n g l i n g bonds l i e i n t h e lower p a r t o f t h e bandgap near t h e v a l e n c e band edge and a r e p a r t l y f i l l e d . F o r S i an a d d i t i o n a l empty band i s found below t h e c o n d u c t i o n band edge w h i c h i s due t o a s t r o n g l y d i s t o r t e d c o v a l e n t bond ( a s a l r e a d y deduced from t h e s p h e r i c 8 1 c l u s t e r c a l c u l a t i o n s / 5 / ) .

-

A l s o a m o d i f i e d i n c o m p l e t e l y r e c o n s t r u c t e d 60 - p a r t i a l c o r e s t r u c t u r e has been i n v e s t i g a t e d w h i c h c o n t a i n s rows o f p a i r s o f weakly i n t e r a c t i n g dangl i n g bonds w i t h p e r i o d i c i t y a ( c o r r e s p o n d i n ? t o F i g . 5 o f Ref. / 5 / ) . I n t h i s case t h e i n t e r a c t i o n between t h e d a n g l i n g bonds w i t h i n each p a i r y i e l d s a s p l i t t i n g of t h e d a n g l i n g bond l e v e l s . As shown i n F i g . 4 f o r S i t h e c a l c u l a t i o n l e a d s t o an empty band i n t h e upper p a r t o f t h e gap ( w h i c h c o n t a i n s t h e a n t i - b o n d i n g combinations o f t h e d a n g l i n g bond s t a t e s w i t h i n each p a i r ) and a resonance i n t h e v a l e n c e band ( b i n d i n g c o m b i n a t i o n of t h e d a n g l i n g bonds i n each p a i r ) . Because of t h e l a r g e s e p a r a t i o n o f 6 a between t h e d i f f e r e n t d a n g l i n g bond p a i r s n e i t h e r t h e bound s t a t e s n o r t h e resonances e x h i b i t a n o t i c e a b l e d i s p e r s i o n .

0.5

slse, ^-

Deep l e v e l s t r u c t u r e o f t h e ^I

Incomp e t e l y r e c o n s t r u c t e d 60'-

I

& k 7

p a r t i a l ( w i t h p a i r s o f d a n g l i n g bonds) i n S i

I n a d d i t i o n t o t h e s t r a i g h t p a r t i a l - d i s l o c a t i o n s d i s c u s s e d so f a r a l s o l o c a l geome- t r i c d e f e c t s i n t h e d i s l o c a t i o n c o r e s ( s u c h as k i n k s o r r e c o n s t r u c t i o n d e f e c t s /42/) have been i n v e s t i g a t e d b y t h e c y l i n d r i c a l c l u s t e r approach / 6 / . I n t h e s e cases p e r i o d i c arrangements o f d e f e c t s have been s t u d i e d where t h e d i s t a n c e s between t h e d e f e c t s a r e chosen so l a r g e t h a t no n o t i c e a b l e i n t e r a c t i o n between t h e d e f e c t s

o c c u r s (e-g., no v i s i b l e d i s p e r s i o n i n t h e d e f e c t "bands"). As an example F i g . 5, p r e s e n t s t h e a t o m i c arrangement f o r an a r r a y o f r e c o n s t r u c t i o n d e f e c t s i n t h e 90

-

p a r t i a l d i s l o c a t i o n ( o b t a i n e d from l a t t i c e - m o d e l s i m u l a t i o n s f o r S i ) . By t h e r e c u r s i o n method t h e c o r r e s p o n d i n g energy l e v e l i n S i i s found a t E

+

0.02 eV ( f o r Ge t h e c a l c u l a t i o n g i v e s a resonance j u s t below t h e t o p o f t h e v a l e x c e band).

S i m i l a r c a l c u l a t i o n s y i e l d f o r t h e d a n g l i n g bond s t a t e o f a r e c o n s t r u c t i o n d e f e c t i n i n t h e ( r e c o n s t r u c t e d ) 3 0 O - ~ a r t i a l a bound s t a t e l e v e l a t E + 0.17 eV i n S i (Ev + 0.2 eV i n Ge) and f o r t h e d a n g l i n g bond i n a k i n k o f t # e u n r e c o n s t r u c t e d

(7)

9 0 O - ~ a r t i a l s a l e v e l a t E + 0.48 eV i n S i ( E

+

0.35 eV i n Ge).

-

I n a l l t h e s e cases

-

and from p e r i p h e r x l s t u d i e s

-

one f i n & t h a t t h e dangling-bond l e v e l p o s i t i o n p r i m a r i l y i s determined by t h e "back-bond" geometry, i .e., by t h e

arrangement o f t h e r e m a i n i n g t h r e e neighbours o f t h e dangling-bond atom, where t h e l e v e l r a i s e s w i t h i n c r e a s i n q " i n w a r d

r e l a x a t i o n " ( d e c r e a s i n g h e i g h t o f t h e dangling-bond atom above t h e basal p l a n e t h r o u g h i t s t h r e e n e i g h b o u r s ) . S i m i l a r e f f e c t s o f back-bond geometry have been d i s c u s s e d by M a r k l u n d 118,191 i n t h e con- t e x t of " i n w a r d r e l a x a t i o n " i n t h e c o r e s o f s t r a i g h t p a r t i a l s .

From t h e s e c a l c u l a t i o n s t h e f o l l o w i n g g e n e r a l c o n c l u s i o n s can be drawn:

( i ) i n t h e t i g h t - b i n d i n g approach t h e F i g . 5 - Arrangegent o f atoms i n t h e d a n g l i n g bonds g i v e r i s e t o p a r t i a l l y r e c o n s t r u c t e d 90 - p a r t i a l w i t h p e r i o d i c f i l l e d bands i n t h e l o w e r p a r t o f t h e gap. array O f reconstruction defects

( i i ) t h e d a n g l i n g bond s t a t e s a r e s t r o n g l y l o c a l i z e d near t h e dangling-bonds and t h e i r l e v e l p o s i t i o n s p r i m a r i l y a r e determined by t h e l o c a l geometry.

( i i i ) r e c o n s t r u c t i o n o f bonds l e a d t o a s e p a r a t i o n i n t o empty and f i l l e d bands ( n o t e t h a t a s i m i l a r s e p a r a t i o n may a r i s e f r o m a Hubbard s p l i t t i n g o f t h e bands as i n v e s t i g a t e d by Usadel and S c h r o t e r 1431).

( i v ) s t r o n g l y deformed bonds may g i v e empty s t a t e s i n t h e upper p a r t o f t h e

bandgap and resonances i n t h e v a l e n c e

s . . .

6

-

D a n g l i n g bond l e v e l i n t h e

band. p z r t i a l i n SI f o r v a r y i n g h e i g h t

d o f t h e d a n q l i n q bond atom above i t s - - basal plane.

There e x i s t f u r t h e r e f f e c t s which may i n f l u e n c e t h e dangling-bond l e v e l p o s i t i o n , e.g., changes i n t h e chemical s h i f t o f t h e l o c a l l e v e l s due t o changes o f t h e number and p o s i t i o n o f n e i g h b o u r i n g atoms. These e f f e c t s a r e n o t t a k e n i n t o a c c o u n t by t h e p r e s e n t t i g h t - b i n d i n g c a l c u l a t i o n s . The o r d e r o f magnitude o f t h i s e f f e c t may be e s t i m a t e d b y making use o f t h e G a u s s i a n - o r b i t a l r e p r e s e n t a t i o n f o r t h e l o c a l o r b i t a l s combined w i t h t h e Gaussian m o d e l - p o t e n t i a l s f o r atoms as i n /23,24/.

W i t h i n t h i s model one f i n d s f o r S i t h a t t a k i n g ayay one a t o m i c p o t e n t i a l l e a d s t o an upward s h i f t o f t h e n e i g h b o u r i n g d a n g l i n g sp - h y b r i d o f a b o u t 0.34 eV. From t h i s e s t i m a t e one may i n f e r t h a t t h e change i n chemical s h i f t s may r a i s e t h e d a n g l i n g bond band8 by about 0.34 eV and t h u s b r i n g t h e d e e p - l e v e l s t r u c t u r e o f t h e u n r e c o n s t r u c t e d 90 - p a r t i a l i n r a t h e r c l o s e agreement w i t h t h e e x p e r i m e n t a l f i n d i n g s o f H a l l - e f f e c t /33/ and p h o t o c o n d u c t i v i t y 1341 measurements.

I n a d d i t i o n t o t h e s h o r t - r a n g e d c o r e e f f e c t s t h e d i s l o c a t i o n s i n f l u e n c e t h e e l e c t r o n i c e x c i t a t i o n s i n semiconductors on a l o n g - r a n g e s c a l e b y t h e i r f a r - r e a c h i n g d e f o r m a t i o n s and t h e g l o b a l change i n t o p o l o g y . As can be e a s i l y seen i n a t i g h t - b i n d i n g approach, t h e deformations on t h e one hand and t h e m o d i f i e d t o p o l o g y on t h e o t h e r hand l e a d t o

.

d i f f e r e n t mechanisms b y which d i s l o c a t i o n s a c t on t h e e l e c t r o n s . The l o c a l d e f o r m a t i o n s a l t e r t h e t i g h t - b i n d i n g m a t r i x elements ( m o d e l l e d by an e x p o n e n t i a l s c a l i n g i n t h e p r e v i o u s s e c t i o n s ) . The change i n t o p o l o g y makes t h a t e l e c t r o n t r a n s f e r processes a l o n g c l o s e d p a t h s i n t h e r e a l c r y s t a l c i r c u m v e n t i n g t h e d i s l o c a t i o n l i n e when p r o j e c t e d s t e p - b y - s t e p o u t o an i d e a l r e f e r e n c e l a t t i c e l e a d t o open e l e c t r o n p a t h s i n t h e r e f e r e n c e system ( l i k e t h e Burgers c i r c u i t / I / ) . T h i s l a t t e r phenomenon i s a n o t h e r r e p r e s e n t a t i o n o f t h e phase-mismatching e f f e c t 1101 d e s c r i b e d i n t h e i n t r o d u c t i o n .

(8)

t h e d i s l o c a t i o n s a r e c h a r a c t e r i z e d by t h e i r d i s t o r s i o n ( t e n s o r - ) f i e l d Q ( r ) /35/.

(The c a r t e s i a n components o f

B

lnay be i n t r o d u c e d v i a B ( r ) i j = 2 . u . as tfie s i n g l e - v a l u e d p a r t i a l d e r i v a t i v e s o f - t h e m u l t i p l e - v a l u e d d i s p l a c e m e n t f i d l d u ( r ) o f t h e d i s l o c a t i o n . ) The symmetric p a r t o f @ p r e s e n t s t h e l o c a l deformations-g; t h e skew- symmetric p a r t ( " r o t a t i o n " ) d e s c r i b e s t h e t o p o l o g i c a l d i s t u r b a n c y and 7 s t h e source o f t h e n o n - v a n i s h i n g c l o s i n g v e c t o r o f t h e B u r g e r s c i r c u i t .

Sor f a r t h e d e f o r m a t i o n p a r t o f t h e d i s t o r s i o n f i e l d has been accounted f o r by 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.g. /12,13,36/). B o u n d - e l e c t r o n l e v e l c a l c u l a t i o n s by t h i s approach y i e l d f o r u n s p l i t d i s l o c a t i o n s i n Ge and S i s h a l l o w l e v e l s ( u p t o 0.1 eV) below t h e c o n d u c t i o n bands /12,13,37/ and (up t o 0.02 eV) above t h e v a l e n c e bands /12,13/. D i s s o c i a t i o n i n t o p a r t i a l s reduces t h e d e p t h o f t h e l e v e l s /13,38/.

( S i n c e i n t h i s approach t h e d i s l o c a t i o n s e n t e r by t h e i r s t r a i n f i e l d s o n l y , 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 b e d i s t i n g u i s h e d . )

-

I n a d d i t i o n t o t h a t t h e d e f o r m a t i o n - p o t e n t i a l has been used t o a n a l y s e q u a l i t a t i v e l y t h e s c a t t e r i n g of e l e c t r o n s and h o l e s a t n e u t r a l and weakly charged d i s l o c a t i o n s as d e p i c t e d b y t h e H a l l m o b i l i t y ( s e e e.g. / 4 / and / 3 9 - 4 1 / ) .

A c c o r d i n g t o / 7 / t h e skew-symmetric p a r t o f

fi

g i v e s r i s e t o a n o n - v a n i s h i n g "gaugeH- p o t e n t i a l 8 w h i c h m o d i f i e s t h e energy-momentum r e l a t i o n f o r t h e e l e c t r o n s i n t h e presence 07 d i s l o c a t i o n s i n a s i m i l a r way a s t h e v e c t o r p o t e n t i a l i n t h e presence of a magnetic f i e l d does: The canonic momentum o p e r a t o r i s changed i n t o t h e "gauge- i n v a r i a n t " momentum o p e r a t o r pt! and t h e band s t r u c t u r e energy o p e r a t o r i s changed from E($) i n t o E(fi+!) w i t h -

8 = E'(L).

p

-.

(gT means t h e transposed of

e ) .

Up t o now consequences of t h e g a u g e - p o t e n t i a l have been e v a l u a t e d f o r t h e p o s i t i o n o f t h e s h a l l o w l e v e l s and f o r t h e e l e c t r o n s c a t t e r i n g a t d i s l o c a t i o n s . I n t h e c a l c u l a t i o n o f t h e s h a l l o w e l e c t r o n s t a t e s w i t h i n t h e d e f o r m a t i o n p o t e n t i a l o f screw d i s l o c a t i o n s t h e g a u g e - p o t e n t i a l i n c r e a s e s t h e k i n e t i c energy necessary f o r l o c a l i z i n g t h e e l e c t r o n s n e a r t h e d i s l o c a t i o n and t h e r e b y reduces t h e d e p t h o f t h e l e v e l s by about 10 Z (J. G a t n i e j e w s k i , p e r s o n a l communication). Furthermore, t h e g a u g e - p o t e n t i a l l e a d s t o an a d d i t i o n a l s c a t t e r i n g o f e l e c t r o n s a t t h e screw d i s l o c a t i o n s i n Ge and S i which i s a t l e a s t as i m p o r t a n t as t h e d e f o r m a t i o n - p o t e n t i a l s c a t t e r i n g . Both e f f e c t s seem t o be due t o r a t h e r general f e a t u r e s o f t h e g a u g e - p o t e n t i a l and n o t be caused by p a r t i c u l a r p r o p e r t i e s o f t h e c o n s i d e r e d d i s l o c a t i o n s . W i t h r e g a r d s t o t h i s i t may be t h a t i n c l u s i o n o f t h e g a u g e - p o t e n t i a l f o r a r b i t r a r y d i s l o c a t i o n s w i l l reduce t h e s p l i t t i n g o f t h e s h a l l o w bound-state l e v e l s b u t i t may i n c r e a s e t h e e s t i m a t e s f o r e l e c t r o n s c a t t e r i n g a t n e u t r a l d i s l o c a t i o n s and t h u s may h e l p t o g e t a more q u a n t i t a t i v e d e s c r i p t i o n i n t h e l a t t e r f i e l d .

/ 1/ J. P . H i r t h and J. L o t h e : Theory o f D i s l o c a t i o n s (McGraw H i l l , New York 1968) / 2/ M. K h a n , L. M i c h e l and G. Toulouse, J. de Physique 38 (1977)

-

L

-

195

/ 3/ W. Shockley, Phys. Rev.

-

91 (1953) 228

/ 4/ R. Labusch and Ll. S c h r o t e r , i n : D i s l o c a t i o n s i n S o l i d s (ed. F. R. N. Nabarro, North-Holland,Amsterdam 1980) V o l . 5, p . 128

/ 5/ R. Jones and S. Marklund, phys. s t a t . s o l . ( b ) 101 (1980) 585

-

/ 6/ H. Veth, s u b m i t t e d as Thesis, Univ. G o t t i n g e n

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C4-100 JOURNAL DE PHYSIQUE

/ 7/ H. T e i c h l e r , Phys. L e t t e r s - 87 A (1981) 113

/

8/ R. Jones, J. de Physique

40

(1979) C6

-

33 / 9/ S. Marklund, phys. s t a t . s o l . ( b )

2

(1979) 83

/ l o /

K. Kawamura, Z. P h y s i k B - 29 (1978) 101

/11/ Y.Aharonovand D. Bohm, Phys. Rev.

-

115 (1959) 485 /12/ A. Claesson, J . de Physique - 40 (1979) C6

-

39 /13/ H. T e i c h l e r , J . de Physique

40

(1979) C6

-

43

/14/ A. Gomez, D. J. H. Cockayne, P. 0 . H i r s c h and k . V i t e k , Philos.Mag.31 - ( 1 9 7 5 ) 1 0 5 /15/ K. Wessel and H. Alexander, P h i l o s . Mag.

2

(1977) 1523

/16/ H. Alexander, J. de Physique - 40 (1979) C6

-

1 /17/ P. B. H i r s c h , J. de Physique

-

40 (1979) C6

-

27

/18/ S. Marklund, 3 r d Gordon Conference on " P o i n t and L i n e D e f e c t s i n Semiconductori', Plymouth, USA (1981)

/19/ S. Marklund, t h i s c o n f e r e n c e

/20/ P. M a s r i and M. Lannoo, S u r f . S c i .

52

(1975) 377

/21/ R, Haydock. V. Heine and M. J. K e l l y , J . Phys. C

-

8 (1975) 2591 /22/ A. Persson, phys. s t a t . s o l . ( b )

-

113 (1982) 253

/23/ J . E. N o r t h r u p , M. L. Cohen, J. R. Chelikowsky, J. Spence and A. Olsen, Phys. Rev.

4

(1981) 4623

/24/ J. R. Chelikowsky, Phys. Rev. L e t t e r s

-

49 (1982) 1569

/25/ M. J. K i r t o n , S. Brand and M. Jaros, P h i l o s . Mag. B - 42 (1980) 577 /26/ M. J. K i r t o n and M. Jaros, J. Phys. C

-

14 (1981) 2099

/27/ R. Jones, S. Oberg and S. Marklund, P h i l o s . Mag. B - 43 (1981) 829 /28/ S. Uberg, phys. s t a t . s o l . ( b ) - 108 (1981) 357

/29/ J. L. Farvaque and D. F e r r 6 , Revue Phys. Appi.

15

(1980) 33 /30/ K. C. Pandey and J. C. P h i l l i p s , Phys. Rev. B e (1976) 750

/31/ H. T e i c h l e r , i n : L a t t i c e D e f e c t s i n Semiconductors ( e d . F. A. H u n t l e y , I n s t . Phys. Conf. Ser., London and B r i s t o l 1975) V o l .

-

23 p. 374 /32/ S. Marklund, phys. s t a t . s o l . ( b ) - 100 (1980) 77

/33/ W. S c h r t i t e r , E. Scheibe and H. Schoen, J . Microscopy,

-

118 (1980) 23 /34/ D. Mergel, Thesis, Techn. U n i v . C l a u s t h a l 1982

/35/ E. Kroner: K o n t i n u u m s t h e o r i e d e r Versetzungen ( S p r i n g e r , B e r l i n 1958) /36/ R. W. Keyes, i n : S o l i d S t a t e P h y s i c s (eds. F. S e i t z and 0. T u r n b u l l ,

Academic Press, New Yorl: 196C) V o l . 11, p. 149

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/38/ S. Winter, phys. s t a t . s o l . ( b )

-

90 (1978) 289 /39/ F. Duster, Diploma Thesis, U n i v . G c t t i n g e n 1972

/40/ F. D u s t e r and R. Labusch, phys. s t a t . s o l . ( b ) - 60 (1973) 161 /41/ F. Duster, Thesis, Univ, G o t t i n g e n 1976

/42/ P. B. H i r s c h , J. M i c r o s c o p y - 118 (1980) 3