EFFECT OF DISORDER AND LONG RANGE STRAIN FIELD ON THE ELECTRON STATES

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EFFECT OF DISORDER AND LONG RANGE STRAIN FIELD ON THE ELECTRON STATES

A. Claesson

To cite this version:

A. Claesson. EFFECT OF DISORDER AND LONG RANGE STRAIN FIELD ON THE ELECTRON STATES. Journal de Physique Colloques, 1979, 40 (C6), pp.C6-39-C6-41. �10.1051/jphyscol:1979608�.

�jpa-00219024�

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JOURNAL DE PHYSIQUE Colloque C6, s u p p l h e n t au n06, tane 4 0 , juin 1979, page (26-39

E F F E C T O F D I S O R D E R AND LONG RANGE S T R A I N F I E L D ON THE ELECTRON S T A T E S

A. Claesson

Department of Theoretical Physics, Ume& University, S-901 87 ~me8, Sweden

Resume.- Les d i s l o c a t i o n s causent un champ de c o n t r a i n t e q u i a f f e c t e l e s 6 l e c t r o n s . Dans des semi- conducteurs c o v a l e n t s comme Ge e t S i des & t a t s f a i b l e m e n t l o c a l i s 6 s s o n t c r 6 6 s avec des n i v e a u x d 1 6 n e r g i e prPs des bords des bandes. On p e u t m o n t r e r c e l a th6oriquement en t r a i t a n t l e champ de con- t r a i n t e s6par6 du noyau de l a d i s l o c a t i o n en u t i l i s a n t une e q u a t i o n de masse e f f e c t i v e .

A b s t r a c t . - D i s l o c a t i o n s g i v e r i s e t o s t r a i n f i e l d s t h a t a f f e c t t h e e l e c t r o n s . I n c o v a l e n t semiconduc- t o r s l i k e Ge and S i weakly l o c a l i z e d s t a t e s a r e c r e a t e d w i t h energy l e v e l s n e a r t h e band edges. T h i s i s shown t h e o r e t i c a l l y by t r e a t i n g t h e s t r a i n f i e l d s e p a r a t e f r o m t h e d i s l o c a t i o n c o r e u s i n g an e f f e c t i v e mass e q u a t i o n .

L o o k i n g a t any t y p e o f d i s l o c a t i o n s - ^{edge as} p o t e n t i a l which i s a t t r a c t i v e on one s i d e and r e p u l - well as screw o r combinations o f b o t h - t h e t h e o r e t i -

c i a n w i l l soon r e a l i z e i t s complex a t o m i c s t r u c t u r e . There i s t h e b i g d i s o r d e r a t t h e core, where t h e p o s i t i o n s o f t h e atoms a r e n o t known f o r c e r t a i n , and around t h a t t h e l o n g range s t r a i n f i e l d .

I t i s a well-known and i n some way p u z z l i n g e x p e r i e n c e t h a t most o f t h e e l e c t r o n s seem t o be unaware o f d e v i a t i o n s f r o m a p e r f e c t s t r u c t u r e . B u t

s i v e on t h e o t h e r s i d e o f t h e d i s l o c a t i o n . Thus one c o u l d v e n t u r e on t h e guess t h a t t h e main e f f e c t o f t h e s t r a i n on e l e c t r o n s i s t o c o n f i n e them t o exten- ded areas on one s i d e o f t h e d i s l o c a t i o n . From t h i s would t h e n f o l l o w t h a t t h e " a n s a t z "

H = H o + H ( 2 )

i s r e a s o n a b l e f o r t h e s e e l e c t r o n s . I n o t h e r words we assume t h a t some e l e c t r o n s a r e o n l y a f f e c t e d b y t h e t h e r e a r e e l e c t r o n s which a r e a f f e c t e d and some o f s t r a i n w h i l e o n t h e o t h e r hand t h e r e a r e e l e c t r o n s them a p p a r e n t l y h e a v i l y . I f one t r i e s t o make t h e which w i l l o n l y n o t i c e t h e d i s l o c a t i o n c o r e . Appa- s i t u a t i o n s i m p l e and d i s r e g a r d s t h e e l e c t r o n - e l e c t r o n r e n t l y t h i s s e p a r a t i o n w i l l l e a d t o an i n t e r a c t i o n i n t e r a c t i o n one c o u l d t h i n k of an e f f e c t i v e H a m i l t o - between t h e c o r e and s t r a i n r e g i o n s which w i l l i n - n i a n o f t h e f o r m f l u e n c e some s t a t e s . However, s t a r t i n g w i t h t h e H = Ho + HE + Hc ( 1 ) assumption ( 2 ) t h e r e s u l t o f t h e c a l c u l a t i o n s m i g h t Here Ho r e p r e s e n t s t h e p e r f e c t c r y s t a l , HE t h e s t r a i n

c o n t r i b u t i o n and Hc t h e c o r e . I n any l a r g e , connected r e g i o n n o t c o n t a i n i n g t h e d i s l o c a t i o n t h e B l o c h e l e c t r o n w i l l j u s t n o t i c e t h e s t r a i n w h i l e i n a small domain around t h e c o r e t h e resemblance t o a l a t t i c e i s r a t h e r n o n - e x i s t e n t .

The s t r a i n - f i e l d i s d e f i n e d t h r o u g h t h e s t r a i n t e n s o r ^E ( r ) which i s s m a l l b u t f o r m a l l y o f i n f i n i -

FlV i

t e range (%?). L i n e a r e l a s t i c i t y t h e o r y , which should be v a l i d j u s t up t o t h e c o r e r e g i o n , g i v e s t h e we1 l- known s t r a i n - and s t r e s s f u n c t i o n s /I/. I t i s r a t h e r i l l u m i n a t i n g t o s t u d y s t r e s s c o n t o u r s g i v e n i n t h e

h o p e f u l l y j u s t i f y t h e guess.

We should s t r e s s t h a t t h e s e p a r a t i o n o f t h e s t r a i n c o n t r i b u t i o n f r o m t h e t o t a l d i s l o c a t i o n i s h e r e m a i n l y based on t h e o r e t i c a l m o t i v a t i o n s . A t t h e moment i t seems r a t h e r d i f f i c u l t t o unambigously i n - t e r p r e t t h e e x p e r i m e n t s a l o n g t h e s e l i n e s . :But we r e t u r n t o t h a t problem.

The way o f o b t a i n i n g a H a m i l t o n i a n o f t h e f o r m ( 2 ) f o r an e l a s t i c a l l y deformed c r y s t a l i s well-known and has been d e s c r i b e d i n t h e l i t e r a t u r e (see e.g.

/3/ o r / 4 / ) . F o r t h e p e r f e c t c r y s t a l we t a k e

H = - n v 2 + ? V ( 1 - ~ : ) 1 (3)

'

i t e r a t u r e Y e ' g . in as show the where ~7 are t h e p o s i t i o n s . In t h e strained

Strain effect as a slowly disturbance when crystal Ry ⁺ ⁺ . ^$^.F o r c o m p u t a t i o n a l reasons t h e c o r e r e g i o n i s excluded. One s h o u l d a l s o n o t i c e

u n s t r a i n e d e l e c t r o n c o o r d i n a t e s 11' a r e now i n t r o d u r t h e d i f f e r e n t c h a r a c t e r o f t h e s t r e s s c u r v e s on

ced a c c o r d i n g t o 1 = + ⁵. ^fand we f i n a l l y ob- o p p o s i t e s i d e s o f t h e edge d i s l o c a t i o n r e l a t e d t o

tain "" . . .

compression o r d i l a t a t i o n r e s p e c t i v e l y . T h i s i s i n 1

t i = - T V ~ + V(1-ji) +

I.lv ^{( 1 )} Puvv - ; ^s⁽³⁾

agreement w i t h t h e shape of Landauers /2/ ad hoc

a v ( g ) ,iq.r]

r v ^{( 4 )}

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979608

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C6-40 JOURNAL DE PHYSIQUE

Here we have accepted t h e slow v a r i a t i o n o f ~ ( 1 ) mentioned above; g are r e c i p r o c a l 1 a t t i c e vectors.

A c t u a l l y a l l coordinates i n ( 4 ) are u n s t r a i n e d and should c a r r y an index o. S(g) i s the s t r u c t u r e f a c - t o r , here assumed strain-independent f o r s i m p l i c i t y .

Having thus found a Hamiltonian o f t h e form ( 2 ) the way t o continue i s c l e a r although one even now has a choice. The slow v a r i a t i o n o f t h e s t r a i n t e n s o r suggests t h a t t h e p e r i o d i c f i e l d o f t h e un- d i s t u r b e d c r y s t a l should be replaced by a mass ten-

s o r which i s d e f i n e d by t h e unperturbed band s t r u c - F i q . 1 : Density o f s t a t e s a t the valence band from ture. How this can be done for semiconductors like th' s t r a i n f i e l h o f a 60" d i s l o c a t i o n i n Ge. E i s

measured from t h e band edge. N i s i n u n i t s o f

Ge and S i w i t h a p e r t u r b i n g p o t e n t i a l has been shown 2 LO2, where L i s t h e d i s l o c a t i o n length; E i s i n eV.

by L u t t i n g e r and Kohn /5/. They discuss t h e under- fi

l y i n g approximations thoroughly and show t h a t neglec- t i n g h i g h e r F o u r i e r components o f t h e p o t e n t i a l and

dropping c e r t a i n i n t e r b a n d terms i s e q u i v a l e n t t o an

!IL

^,

Jij

aporoximation where t h e e r r o r i s given by t h e squared r a t i o o f t h e l a t t i c e spacing t o t h e e x t e n t o f t h e im- - ⁹

p u r i t y s t a t e f o r a Coulomb-1 i k e p o t e n t i a l . F o l l o w i n g a Eg

t h e l i n e s i n /5/ one o b t a i n s from H, i n (4) a s t r a i n

f i e l d o o t e n t i a l , U@), for t h e electron in t h e effec- Fig. : states the strain at t h e valence and conduction band edges (schematic).

t i v e mass equation, U n i t s as i n f i g u r e 1.

r e l a t e d t o an extremum point!, o f a given band n.

I n /6/ we made a v e r y simple n e a r l y - f r e e e l e c t r o n c a l c u l a t i o n o f D (k ,n) f o r the 60" d i s l o c a t i o n a t

uv 4

t h e conduction band edge i n Ge i n order t o see i f our assumptions a t t h e beginning were reasonable. The p o t e n t i a l s obtained o b v i o u s l y have a form t h a t must r a t h e r e f f e c t i v e l y exclude t h e trapped e l e c t r o n s from t h e d i s l o c a t i o n core.

The s t r a i n p o t e n t i a l can a l s o be d e f i n e d phenomenological l y /7/

- ^"

Duv = 'd 'uv ' kou kov 'u

i s a u n i t v e c t o r along Jo and Eu, Ed are parame-

-0

t e r s given by e.g. p i e z o r e s i s t a n c e measurements / 8 / . R e a l i s t i c c a l c u l a t i o n s o f the eigenstates o f t h e e f f e c t i v e mass Hamiltonian corresponding t o t h e 60"

d i s l o c a t i o n along these l i n e s have been performed by S. Winter f o r Ge c l o s e t o t h e conduction band /9/

and t h e valence band E n t e r n a l Report, me: U n i v e r s i - t y , 19777. He f i n d s weakly l o c a l i z e d s t a t e s c l o s e t o b o t h band edges. The lowest s t a t e a t t h e conduction band has an energy 0.1 eV below t h e band. The s t a t e s a t t h e valence band a r e more s h a l l o w ( F i g . 1 and 2 ) . One might expect t o see these s t a t e s e.g. i n o p t i c a l a b s o r p t i o n experiments /lo/ although i n t e r m i n g l i n g o f d i s l o c a t i o n core s t a t e s seems t o complicate the p i c t u r e /11/.

However, as i s well-known, r e c e n t experiments apparently c o n f i r m t h a t t h e 60" d i s l o c a t i o n s i n Ge and S i a r e mostly d i s s o c i a t e d i n t o 90"- and 30" par- t i a l s . N a t u r a l l y one would l i k e t o study t h e e f f e c t o f t h e corresponding s t r a i n f i e l d . P r i n c i p a l l y t h e c a l c u l a t i o n s a r e t h e same as f o r t h e complete d i s l o - c a t i o n b u t t h e i n t e r a c t i o n s between the p a r t i a l s can cause some t r o u b l e c r e a t i n g e.g. a compliaated geo- metry. C a l c u l a t i o n s a t t h e conduction band i n Ge f o r each p a r t i a l s e p a r a t e l y have been performed and t h e combined e f f e c t when t h e separation i s 100 atomic u n i t s has a l s o been s t u d i e d /12/. The r e s u l t i s e s s e n t i a l l y t h e same as f o r the complete d i s l o c a t i o n w i t h weakly bound s t a t e s down t o 0.1 eV below t h e conduction band edge. The deepest s t a t e i s p o l a r i z e d along t h e Burgers v e c t o r corresponding t o t h e shape o f t h e p o t e n t i a l .

The conclusions we can draw from the c a l c u l a - t i o n s are thus t h a t 60" d i s l o c a t i o n s and t h e i r par- t i a l s w i l l c r e a t e shallow e l e c t r o n l e v e l s a t the band edges. The p o t e n t i a l s t h a t a c t as t r a p s f o r these e l e c t r o n s have complicated s t r u c t u r e s b u t t h e main f e a t u r e s are t h a t they exclude t h e e l e c t r o n s from one s i d e o f t h e d i s l o c a t i o n and b i n d them loose- l y t o the o t h e r . Thus t h e s t a t e s a r e e f f e c t i v e l y orthogonal t o the deep, t i g h t l y bound core s t a t e s .

There a r e approximations t h a t c o u l d be impor-

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A . Claesson

t a n t and i n t e r e s t i n g t o l o o k i n t o . The e f f e c t o f t h e References coupling the 'Ore region is not known' The size

/I/ H j r t h , J .pa and Lothe, J., Theory of D i s l o c a t i o n s , o f t h e t r a n s i t i o n domain i s however v e r y l i m i t e d . (McGraw-Hi 11 ) 1968.

F u r t h e r t h e e i g h t conduction band minima w i l l i n t e - /2/ Landauer, R., Phys. Rev. - 94 (1954) 1386.

r a c t and t h a t w i l l i n f l u e n c e t h e s t r a i n l e v e l s . /3/ Pikus, G.E. and B i r , G.L., Soc. Phys. S o l i d S t a t e There a r e a l s o exchange and c o r r e l a t i o n terms which - 1 (1960) 1502.

have been taken i n t o account i n /6/ i n an average /4/ Kleinman, L., Phys. Rev. 182 (1969) 686.

way and which a r e i m p l i c i t e i n t h e phenomenological /5/ L u t t i n g e r , J.M. and Kohn, W., Phys. Rev. 97

(1955) 869.

/6/ Claesson, A., Phys. S t a t u s S o l i d i ( b ) 61 (1974) 599.

/ 7 / H e r r i n g , C. and Vogt, E., Phys. Rev. 2 (1956) 944.

/8/ B a r a n s k i i , P.J. and Kolomoets, V.V., Phys. S t a t u s S o l i d i ( b ) - 45 (1971) K55.

/9/ Winter, S., Phys. S t a t u s S o l i d i ( b ) - 79 (1977) 637.

/ l o / Winter, S., Phys. S t a t u s S o l i d i ( b ) - 85 (1978)K95.

/11/ Marklund, S., Phys. S t a t u s Sol i d i ( b ) - 85 (1978) 673.

/12/ Winter, S., Phys. S t a t u s S o l i d i ( b ) 90 (1978) 289.