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DISORDER INDUCED DENSITY OF STATES WITHIN THE BAND GAP
H. Leschke, B. Kramer
To cite this version:
H. Leschke, B. Kramer. DISORDER INDUCED DENSITY OF STATES WITHIN THE BAND GAP. Journal de Physique Colloques, 1981, 42 (C4), pp.C4-59-C4-62. �10.1051/jphyscol:1981409�.
�jpa-00220713�
DISORDER INDUCED DENSITY OF STATES WITHIN THE BAND GAP
H. Leschke and B. ~ r a m e r *
U n i v e r s i t a t DiisseZdorf, 4000 DiisseZdorf, p. R. G.
* ~ h ~ s i k a Z i s c h l'eehnische BundesanstaZt, 3300 Braunschweig, F.R.G.
A b s t r a c t . - We d e r i v e e x a c t u p p e r and l o w e r bounds f o r t h e c o n f i g u r a t i o n a l l y a v e r a g e d p a r t i t i o n f u n c t i o n ( L a p l a c e t r a n s f o r m o f t h e d e n s i t y of s t a t e s ) o f a m u l t i - b a n d H a m i l t o n i a n c o n t a i n i n g a s t o c h a s t i c p o t e n t i a l . Simple examples a r e d i s c u s s e d and some c o n c l u s i o n s a r e drawn w i t h r e s p e c t t o t h e i n t e r p l a y of e n e r g e t i c a l and s t a t i s t i c a l c o r r e l a t i o n s between t h e bands i n d e t e r m i n i n g t h e d e n s i t y of s t a t e s i n t h e gap.
I n t r o d u c t i o n . - The d e n s i t y o f s t a t e s (DOS) i n t h e t a i l s o f t h e bands i n amorphous
--
s e m i c o n d u c t o r s can be r e l a t e d t o t h e f l u c t u a t i o n s o f a random p o t e n t i a l . Q u a n t i t a - t i v e l y t h i s was f i r s t e s t a b l i s h e d by t h e c l a s s i c a l argument o f I . M . L i f s h i t z 111 f o r t h e low e n e r g y t a i l o f a f r e e e l e c t r o n band s u b j e c t t o t h e p o t e n t i a l o f a r a n - dom a r r a y o f i m p u r i t i e s . A s i m i l a r argument f o r t h e DOS i n t h e gap o f amorphous s e m i c o n d u c t o r s i s s t i l l l a c k i n g , a l t h o u g h t h e r e a r e some q u a l i t a t i v e models u s e d i n t h e u n d e r s t a n d i n g of e x p e r i m e n t a l d a t a C21.Among t h e m a t h e m a t i c a l methods [ 3 1 f u n c t i o n a l i n t e g r a l s C41, and v a r i a t i o n a l a p p r o a c h e s C 5
1
u s i n g t h e " c o h e r e n t s t a t e " r e p r e s e n t a t i o n ( s e e e g . [ 6 1 ) p r o v i d e s y s t e m a t i c methods f o r t h e d e n s i t y o f t h e t a i l s t a t e s y i e l d i n g c o r r e c t l y t h e h i g h and low e n e r g y l i m i t s f o r n o n i n t e r a c t i n g f r e e e l e c t r o n s i n random i m p u r i t y poten- t i a l s C5,71, and r e a s o n a b l e a p p r o x i m a t i o n s f o r one d i m e n s i o n a l d i s o r d e r e d s y s t e m s C 7 , 9 1 .I n t h i s c o n t r i b u t i o n we want t o e x t e n d t h e c o h e r e n t s t a t e approach o f L u t t i n - g e r 151, which i s b a s e d on t h e p a r t i t i o n f u n c t i o n r e p r e s e n t a t i o n o f t h e DOS t o t h e model o f a d i s o r d e r e d s o l i d w i t h s e v e r a l bands.
The model.
-
We c o n s i d e r t h e H a m i l t o n i a nH = C E I k n > < k n j + C v I R n > < h ' / .
kn Rnn
'
(1)kn Rnn
'
/ k n > a r e t h e Bloch s t a t e s w i t h wave v e c t o r k and band number n c o r r e s p o n d i n g t o t h e band s t r u c t u r e ckn o f a r e g u l a r l a t t i c e d e s c r i b e d by t h e l a t t i c e v e c t o r R. [ h > a r e t h e c o r r e s p o n d i n g Wannier s t a t e s , t h u s
N i s t h e number o f l a t t i c e s i t e s .
v R n n l a r e s t o c h a s t i c p o t e n t i a l m a t r i x e l e m e n t s s u b j e c t some p r o b a b i l i t y d i s - t r i b u t i o n . We t a k e vRnn, a s s t a t i s t i c a l l y i n d e p e n d e n t f o r d i f f e r e n t l a t t i c e s i t e s
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981409
JOURNAL DE PHYSIQUE
w i t h a d i s t r i b u t i o n i n d e p e n d e n t o f R.
The method.
-
The DOS n ( E ) i s r e l a t e d t o t h e c o n f i g u r a t i o n a l l y a v e r a g e d p a r t i t i o n f u n c t i o n < Z ( 6 ) > v i an ( E ) dE = - N 1 < Z ( 6 ) >
,
( 3 )0
where t h e b r a c k e t s d e n o t e t h e a v e r a g e w i t h r e s p e c t t o t h e s t o c h a s t i c p o t e n t i a l . Z ( 6 ) i s , a s u s u a l , g i v e n by
Z ( 6 ) : = T r e-BH ( 4 )
The c a l c u l a t i o n o f Z(6) i s o n l y p o s s i b l e f o r s i m p l e models, and i n c e r t a i n l i m i t s a s , f o r i n s t a n c e , l a r g e and s m a l l 6. However, one may c o n s t r u c t u p p e r and lower bounds f o r <Z(B)>, < z ( @ ) > and < Z ( B ) > , r e s p e c t i v e l y , which a l l o w f o r t h e d e r i v a t i o n of c o n t r o l l a b l e a p p r o x i m a t i o n s i n t h e whole B r e g i o n , and from which n ( E ) may b e c a l c u l a t e d by i n v e r t i n g e q . ( 3 ) .
The u p p e r bound i s o b t a i n e d v i a t h e Golden-Thompson i n e q u a l i t y ( s e e e g . [ l o ] ) . Let
H = H + V ( 5 )
w i t h H t h e d i s o r d e r i n d e p e n d e n t p a r t o f U, and V t h e s t o c h a s t i c p o t e n t i a l . Then Tr .-BH
<
Tr e-@HO. .-B" - -- : Z ( 6 ) , ( 6 )
where t h e t r a c e h a s been t a k e n w i t h t h e Wannier s t a t e s , vR d e n o t e s t h e m a t r i x v Rnn
' '
and ~ ( " ' ( 6 ) is t h e p a r t i t i o n f u n c t i o n o f t h e nth band:
~ ( ~ ' ( 6 ) : = C e-'€kn ( 8 )
k
F o l l o w i n g C51 t h e lower bound i s c o n s t r u c t e d from J e n s e n ' s i n e q u a l i t y f o r con- v e x f u n c t i o n s , which i n t h e c a s e o f t h e p a r t i t i o n f u n c t i o n r e a d s
Z ( B )
= z <+ile-6Hl
+i>' 2 e-'<+iiH1'i>. ( 9 )I I
{Oil i s an a r b i t r a r y s e t o f n o r m a l i s e d , u n i t y r e s o l v i n g s t a t e s
[lo].
A p a r t i c u l a r u s e f u l c h o i c e i s t h e s e t o f s o - c a l l e d " c o h e r e n t s t a t e s " . I n o u r c a s e o f a d i s c r e t e l a t t i c e we may t a k e ( s e e eg. [ 6 , 8 1 )-iRCkf l k l n > < k ' n l e i k f i , R '
/
R1n><R'nlI $ n , k R > : = e k I$,>
.
( 1 0 )i s an a r b i t r a r y n o r m a l i s e d r e f e r e n c e s t a t e c o r r e s p o n d i n g t o t h e band number n.
I t i s e a s i l y v e r y f i e d t h a t
Using t h e s e s t a t e s i n e q . ( 9 ) we g e t
( $ n ) < e - B & l v R l n n l <$nl~-~'>/2>
<?(B)> = C Rn
-
N w i t hWe n o t e t h a t i n t h e l o w e r bound t h e bands a p p e a r decoupled i n c o n t r a s t t o t h e u p p e r bound. T h i s i s due t o t h e p a r t i c u l a r c h o i c e o f t h e c o h e r e n t s t a t e s .
Examples and d i s c u s s i o n .
-
A s y s t e m f o r which t h e i m p l i c a t i o n s o f t h e p r e s e n t ap- p r o a c h c a n b e s t u d i e d q u i t e t h o r o u g l y i s t h e H a m i l t o n i a n w i t h o n l y one band (n.1) and a Gaussian d i s t r i b u t e d p o t e n t i a l w i t h z e r o mean. Then t h e i n e q u a l i t i e s f o r<Z(B)> r e a d <v2 > B2
<v2>
z 0 ( $ ) e B 2 -ii L- < z ( B ) > I z0e ( 1 4 )
<$
[
R>1 '
i s t h e i n v e r s e p a r t i c i p a t i o n number o f t h e t r i a l s t a t e j $>, b e i n g 1 / N i fI$>
i s e x t e n d e d and 1 i f I$> i s l o c a l i s e d t o o n l y one s i t e . I t i s e a s y t o s e e t h a tf o r <v2> = o ( n o d i s o r d e r ) t h e u p p e r and lower bounds c o i n c i d e f o r t h e c h o i c e
<$lk> = 6k
.
Thus t h e d e n s i t y o f s t a t e s o f t h e o r d e r e d s y s t e m i s r e g a i n e d a s it .os h o u l d b e . On t h e o t h e r hand f o r = s o ( d i s p e r s i o n l e s s b a n d ) t h e u p p e r and lower bounds c o i n c i d e f o r <$IR> = 6 y i e l d i n g
R.0
which c o r r e s p o n d s t o a Gaussian b r o a d e n e d DOS.
F u r t h e r l i m i t i n g c a s e s can b e s t u d i e d by v a r y i n g I$> s y s t e m a t i c a l l y f o r l a r g e and s m a l l B i n o r d e r t o maximize < z ( B ) > . Thus one c a n d e r i v e , f o r i n s t a n c e t h e famous L i f s h i t z t a i l o f t h e DOS C5,81,
A model o f p a r t i c u l a r i n t e r e s t w i t h r e s p e c t t o amorphous s e m i c o n d u c t o r s i s t h a t o f a H a m i l t o n i a n d e s c r i b i n g two bands s e p a r a t e d by a gap u n d e r t h e i n f l u e n c e o f a s t o c h a s t i c p o t e n t i a l w i t h Gaussian p r o b a b i l i t y d i s t r i b u t i o n .
I f we t a k e t h e i n t e r b a n d m a t r i x e l e m e n t s o f t h e p o t e n t i a l vRnn, ( n n ' ) a s z e r o , t h e two bands r i g o r o u s l y d e c o u p l e . T h i s i s a l s o r e f l e c t e d i n t h e u p p e r bound f o r <Z(B)>. Thus, t h e DOS i s a s i m p l e a d d i t i v e s u p e r p o s i t i o n o f t h e DOS o f t h e s i n g l e bands i r r e s p e c t i v e o f w h e t h e r t h e r e is a s t a t i s t i c a l c o r r e l a t i o n between d i f f e r e n t i n t r a b a n d p o t e n t i a l m a t r i x e l e m e n t s o r n o t . I n t h i s c a s e t h e DOS i n t h e gap i s a s u p e r p o s i t i o n o f two L i f s h i t z t a i l s , e a c h o f which i s o b t a i n a b l e s e p a r a t e l y ( a n d a p p r o x i m a t e l y ) a s i n t h e one-band c a s e .
I f t h e i n t e r b a n d m a t r i x e l e m e n t s a r e n o n z e r o t h e s i t u a t i o n becomes e x t r e m e l y d i f f i c u l t a s t h e u p p e r bound f o r <Z(B)> depends on t h e s t a t i s t i c a l c o r r e l a t i o n s b e t - ween t h e p o t e n t i a l m a t r i x e l e m e n t s . The lower bound a s a sum o f i n d e p e n d e n t band c o n t r i b u t i o n s d o e s n o t . A c o u p l i n g between t h e band c o n t r i b u t i o n s i n t h e l o w e r bound can be i n t r o d u c e d by t a k i n g l i n e a r c o m b i n a t i o n s o f c o h e r e n t s t a t e s f o r s i n g l e b a n d s , f o r i n s t a n c e .
where X i s an a d d i t i o n a l v a r i a t i o n a l p a r a m e t e r . The r e s u l t i n g e x p r e s s i o n s a r e com- p l i c a t e d . Though t h e r e a r e n o t y e t any d e f i n i t e r e s u l t s d e r i v e d from t h i s a n s a t z , i t a p p e a r s t h a t t h e s h a p e o f t h e DOS depends f o r o u r model c r u c i a l l y on t h e s t a t i s t i - c a l c o r r e l a t i o n s between t h e v a r i o u s p o t e n t i a l m a t r i x e l e m e n t s .
JOURNAL DE PHYSIQUE
C o n c l u s i o n . - G e n e r a l i s i n g t h e " c o h e r e n t s t a t e " a p p r o a c h t o t h e b a n d t a i l s o f s i n g l e , n o n c o u p l e d b a n d s i n d i s o r d e r e d s o l i d s we h a v e d e r i v e d a s y s t e m a t i c v a r i a t i o n a l method f o r t h e d e n s i t y o f states i n t h e g a p o f amorphous s e m i c o n d u c t o r s . The a p p r o - x i m a t i o n o b t a i n e d i n t h e v a r i a t i o n a l p r o c e d u r e f o r t h e a v e r a g e d p a r t i t i o n f u n c t i o n i s c o n t r o l l e d by a n u p p e r l i m i t , w h i c h i s e x p l i c i t e l y d e r i v e d . We h a v e c o n s t r u c t e d t r i a l s t a t e s f o r t h e p r o b l e m o f two b a n d s c o u p l e d by a s t o c h a s t i c p o t e n t i a l as l i n e a r c o m b i n a t i o n s o f t h e c o h e r e n t s t a t e s c o r r e s p o n d i n g t o t h e n o n c o u p l e d b a n d s . From t h e a n a l y t i c f o r m o f t h e u p p e r a n d l o w e r b o u n d s f o r t h e p a r t i t i o n f u n c t i o n we c o n c l u d e t h a t , i f t h e b a n d s a r e c o u p l e d by a s t a t i s t i c a l i n t e r b a n d p o t e n t i a l t h e s h a p e o f t h e d e n s i t y o f g a p s t a t e s i s a l s o d e t e r m i n e d by t h e s t a t i s t i c a l c o r r e l a - t i o n s b e t w e e n t h e i n t r a b a n d p o t e n t i a l s c o r r e s p o n d i n g t o d i f f e r e n t b a n d s .
R e f e r e n c e s .
[11 LIFSHITZ, I . M . , Usp. F i z . Nauk
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,
a n d WEAIRE, D.,
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B e r l i n - H e i d e l b e r g - N e w York 1979 ) C41 FRIEDBERG, R . , a n d LUTTINGER, J . M . , Phys. Rev. B 12 ( 1 9 7 5 ) 4460LUTTINGER, J .M.
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( 1 9 8 0 ) 3345-1 7 1 GROSS, E . P . , J . S t a t . P h y s . 1 7 (1977') 265 C81 LU, Chih-Yuan, P h y s . Rev. B
