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SOME PROBLEMS OF THE THEORY OF ELECTRON ENERGY SPECTRUM OF A
DISORDERED SEMICONDUCTOR
V. Bonch-Bruevich, A. Mironov
To cite this version:
V. Bonch-Bruevich, A. Mironov. SOME PROBLEMS OF THE THEORY OF ELECTRON ENERGY
SPECTRUM OF A DISORDERED SEMICONDUCTOR. Journal de Physique Colloques, 1981, 42
(C4), pp.C4-33-C4-36. �10.1051/jphyscol:1981404�. �jpa-00220708�
SOME PROBLEMS OF THE THEORY OF ELECTRON ENERGY SPECTRUM OF A DISORDERED SEMICONDUCTOR
V.L. Bonch-Bruevich and A.G. Mironov
Faculty of Physics. Moscow University, Moscow, 117234, U.S.S.B.
Abstract.
-
The paper c o n s i s t s of two p a r t s . I n t h e f i r s t one we c o n s i d e r t h e l o c a l i s a t i o n problem a t d=2. Assertion of a l l s t a t e s t o be l o c a l i s e d i s shown t o depend upon some dynamic p r o p e r t i e s of t h e system-
c o n t r a r y t o t h e d=l and &3 c a s e s where j u s t t h e geometrical c o n s i d e r a t i o n s a r e known t o be suf- f i c i e n t . The second p a r t c o n t a i n s an attempt t o understand what i s i n f a c t measured when one s t u d i e s l i g h t a b s o r t i o n byt h e s p a t i a l l y inhomogeneous f i l m s (of t h e a Si:H type?.
1. L o c a l i s a t i o n problem.
-
According t o [I- 41
a l l e l e c t r o n s t a t e s Ere expected t o be l o c a l i s e d i n a two-dimensional d i s o r d e r e d semi- conductor w i t h a random f i e l d p r e s e n t . However t h e c a l c u l a t i o n s pre- s e n t e d i n l 5 , 6J suggest that t h e r e s u l t might be i n f a c t model-de- pendent-
that i s ependent upon some p a r t i c u l a r dynamical p r o p e r t i - e s of t h e systemx7 .
The reason seems t o l i e i n t h e f a c t t h a t a t d=2 s m a l l E d e p e n d e n t c o r r e c t i o n s t o t h e dimensionless conductance g a r e important (L i s t h e Length of t h e sample). Hence i t seems worthwhile t o r e c o n s i d e r t h e c a s e d=2 somewhat t a k i n g account of some p o i n t s which were not p r e v i o u s l y considered.Accordin7 t o
4,
2 1 a t small v a l u e s of g ( v i r t u a l l y i n a dis- c r e t e spectrum t h e eadxng term a t t h e temperature I = 0 i s g == gaexp(-2ciL), where ga i s a c o n s t a n t and d'l i s t h e t y p i c a l loca- l i s a t i o n r a d i u s (we s l i h t l change t h e n o t a t i o n ) . This eq. seems almost obvious: due t o !he haevel r e p u l s i o n e f f e c t t h e i s o e n e r g e t ' c
Xx'i l e v e l s should belong t o t h e most d i s t a n t l o c a l i s a t i o n c e n t e r s
.
However t h e s o - c a l l e d d i s c r e t l e v e l s n e a r and above t h e ~ e r m i - l e v e l F have i n f a c t a n a t u r a l w i d t h
% .
Hence f i n i t e l e n g t h hops becoae p o s s i b l e w i t h no a c t i v a t i o n needed provided t h e hopping d i s t a n c e ex- ceeds t h e c o r r e l a t i o n l e n g t h R e n t e r i n g t h e two-level coffffft3ation f u n c t i o n . Using t h e well-known a o t t reasoning one o b t a i n s= e, CIFI/S).
(1X ) The same s t a t e m e n t i s t r u e f o r t h e problem of t h e minimum m e t a l l i c c o n d u c t i v i t y a t d=3
[
7).~ x ) ~ h u s we aba don he p a r t i c u l a r d e f i n i t i o n of g f o r f i n i t e samples adopted i n
21,
2 j i n f a v o u r of w h a t seems t o us a more d i r e c t one.However both d e f ~ n i t i o n s seem t o l e a d t o i d e n t i c a l r e s u l t s ( i n a non-correlated c a s e this i s v e r y f i a b l e immediately).
xxx)
Analogous r e s u l t s may be obtained u s i n g a mo.bwo-level c e s r e l a t i o n f u n c t i o n found i n r e f . The r. h. s. of eg.(?) should be r e p l a c e d then by 2,6 l n ( 4 , 3 /
s
) w i t h&
= q m r [%= m=i).
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981404
C4-34 JOURNAL DE PHYSIQUE
The p o s s i b i l i t y of " i s o e n e r g e t i c t l hopping o v e r t h e l e n g t h R
>
Ro i s p r o p o r t i o n a l t o t h e e m ( - 2 d R ) . T h i s r e d u c e s t o t h e p r e v i o u s l y used f o m , e x p ( - 2 d L), i f L-
Ro; t h e same h o l d s f o r g ( a t L < R such hops a r e o b v i o u s l y i m p o s s i b l e-
CY = g = 0 a t T=O under equil!?brium c o w d i t i o n s ) . On t h e o t h e r hand if L>>R t h e hops seem t o proceed most- l y o v e r t h e d i s t a n c e R which i s smal? compared t o L and i s determin- e d by t h e p e r c o l a t i v e arguments. The conductance t h e n becomes inde- pendent of L-
up t o t h e e x p o n e n t i a l l y s m a l l terms coming from t h e r a r e c a s e s of e x t r e m e l y l o n g hops. Thus a t L>>Ro9 = go +$#,M~(-Z~L)-
(2 To make a rough e s t i m a t e of Ro we put d = 1 0 ~ c m - ~ , T =k / g
== ( t h i s i s probably an o v e r e s t i m a t e ) ,
IF(
= 1,4 ev. ThenRo W 10-~cm, T h i s seems t o be b i g enough t o make s e n s i b l e t h e con- c e p t of a c o n t i n u o u s spectrum, i f any, t h e s e l f - a v e r a g i n g p r o p e r t y of g seems t o be p r e s e r v e d as w e l l s i n c e a l l t h e d i s c r e t e l e v e l s a r e e x p e c t e d t o o c c u r w i t h i n t
90
e sample. A t t h e same time, t h e l e n g t h Ro i s s m a l l enough t o f o r c e us t a k e account of y e t a n o t h e r l e n g t h pre- s e n t-
t h e c o r r e l a t i o n l e n g t hY ' ~
e n t e r i n g t h e b i n a r y c o r r e l a t i o n f u n c t i o n of random f i e l d f l u c t u a t i o n s . T h i s l e n g t h e n t e r s t h e cor- r e c t i o n s t o g when t h e l a t t e r i s b i g enough ( v i r t u a l l y i n a c n t i - nuous spectrum). I n t h e c a s e of a charged i m p u r i t y s c a t t e r i n g xg we o b t a i n (d=2, g a c o r r e s p o n d s t o L 3 00 ):Yet a n o t h e r c o r r e c t i o n , -a/g
,
where a>
0 i s a c o n s t a n t , i s t o be i n c l u d e d i n t h e r.h.s. of ( 3 )11,
27-
provided t h e c o r r e c t i o nl5,
6 , i s unimportant. (The l a t t e r c a s e i s f o r m a l l y o b t a i n e d by p u t t i n ga=O).
I n p r e s e n c e of new l e n g t h s , x4and R
,
t h e f u n c t i o n g(bL) ob- t a i n e d by forming a hypercube of b cubes ghould g e n e r a l l y be w r i t - t e n i n t h e form g(bL) = f r b , g(L),zL, LR-'I
and a c e r t a i n i n g r e - d i e n t of t h e r e n o r m a l l s a t l o n group p h ~ l o s o ~ h y i s g e n e r a l l y l o s t . Yet some analogue of t h e r e n o r m a l i s a t i o n group e q u a t i o n may s t i l l be ob- t a i n e d u s i n g t h e method of [I, 2 1 :Here L' ( 4 L) i s a v a r i a b l e parameter t o be i d e n t i f i e d w i " c L a f t e r t h e c a l c u l a t i o n i s over. Now this eq. i s g e n e r a l l y nonautdndmous, s i n c e may depend e x p l i c i t l y upon
8
L' and L'R -I. This however does n o t change t h e e s s e n c e of t h e argument based $on s t u d y i n g t h e s i g n of j3.
F o r o u r purposes i t i s s u f f i c i e n t t o c o n s i d e r . h o l i m i t - i n g c a s e s .a ) "Short" samples: L a R
.
Here t h e hops a r e o v e r almost a l l of t h e sample; t h e f u n c t i o n6 i 8
given byS i n c e t h e f i r s t term i n t h e r.h.s. of (5.b) i s small b u t f i n i t e t h e
X) Analogous r e s u l t i s o b t a i n e d when t h e s c a t t e r i n g i s due t o a smooth random f i e l d .
b ) "Long" samples: L>>R
.
The e x p o n e n t i a l s of t h e type ex*(-2xL) should now be n e g l e 8 t e d wherever p o s s i b l e and we have- . B a a d ~ & C f ( - ~ d ~ ) ~ - ~ ,
g e i ,
( a )1 'bF.
( b (6 )I n t h i s c a s e t h e r e s u l t s of [I, 23 seem t o be f o r m a l l y confirmed
(provided a 0 ).
2. 0 t i c a l a b s o r p t i o n .
-
We c o n s i d e r t h e f i l m s of t h e columnar- i e r u c t u r e c o n t a i n i n g f o r example, t h e " t r u e " a-Si:H ("col- dnE1') :nd t h e i n t e r m e d i a t e " t i s s u e " [+11J.
The essence of t h e problem may be seen by c o n s i d e r i n g t h e c y l i n d r i c a l columns of r a d i u s R p a s s i n g through t h e f i l m p e r p e n d i c u l a r t o t h e i l l u m i n a t e d s u r f a c e (which i s a plane z=O). Let t h e d i s t a n c e between t h e columns be h i g h enough; then t h e f i l m i s i n f a c t j u s t a system of independent "wave- guides" s e p a r a t e d by a t i s s u e . I n c o n t r a s t t o t h e conventional wave- guides t h e s e a r e f i l l e d by an absorbing m a t e r i a l and t h e " w a l l s "foxmed by a t i s s u e a r e anything but m e t a l l i c . A l l t h i s however does n o t change t h e mathematics t o o much. Let fi
, cL
andEr, s ,
be t h e r e a l p a r t s of t h e d i e l e c t r i c p e r m e a b i l i t y and h i g h frequency con- d u c t i v i t y of t h e colwnn m a t e r i a l and t i s s u e r e s p e c t i v e l y and l e t t h e a b s o r p t i o n i n t h e t i s s u e be much h i g h e r than i n t h e column m a t e r i a l . Then t h e e l e c t r i c f i e l d of t h e plane p o l a r i s e d wave i n t h e f i l m ,E o z p ( - i a * )
,
i s approximately given byThis eq. holds everywhere except t h e t h i n l a y e r n e a r t h e i l l u m i n a t e d s u r f ace. Here
9
andy
a r e t h e c y l i n d r i c a l c o o r d i n a t e s , t h e a x i s c o i n c i d i n g w i t h t h a t of a column,0
i s t h e u s u a l step-function, while t h e vector-functions Ei,n and E a r e expressed r e s p e c t i v e -e , n
l y i n terms of t h e c y l i n d r i c a l f u n c t i o n s J
(ae2p
) and A:(a : ' ? )
, n i s t h e number of t h e r o o t of a c e r t a i n ~ J a n s c e n d e n t a l equation f o rae F F ~
; a t (%wC$R/C<)*~ t h i s equation i s j u s t'&
(3e:' Ej..
ji/
(ae:) R ) = 0 .
The complex wave-numbercCw+ip,
is e a s i l y ob- t a i n e d i n terms of WL,
EL and X ~ ) R ; by d e f i n i t i o nP,>O .
Ther.h.s. of ( 7 ) i s a sum of terms w i t h v a r i o u s complex propagation c o n s t a n t s . If t h e f i l m is t h i c k enough, then j u s t one of t h e t e r n s s u r v i v e s . However even i n t h i s c a s e t h e a b s o r p t i o n c o e f f i c i e n t meas- ured i n a t r a n s m i s s i o n experiment d i f f e r s from that obtained f o r a homogeneous m a t e r i a l . The s u r v i v i n g term corresponds t o
where c i s t h e l i g h t v e l o c i t y i n a f r e e space,
2
= Z T C /W i s t h e l i g h t wave l e n g t h ; eq. (8) i s v a l i d provided A &TK L ?,
butt h i s is not n e c e s s a r i l y a s t r o n g i n e q u a l i t y . S i n c e jt mighe w e l l be of t h e o r d e r of
R,
t h e c o r r e c t i o n might i n f a c t be q u i t e n o t i c e a b l e . More important however i s t h e f a c t that i n t h e c a s e t y p i c a l f o r a t h i n f i l m many terms may s u r v i v e ( e s p e c i a l l y i n t h e o p t i c a l t a i l re-C4-36 JOURNAL DE PHYSIQUE
g i o n ) . I n t h i s , case t h e u s u a l concept of a s i n g l e a b s o r p t i o n c o e f f i - c i e n t might b e c o m e quite misleading. T h e a p p a r e n t a b s o r p t i o n c o e f f i - c i e n t o b t a i n e d f r o m a t r a n s m i s s i o n experiment becomes dependent upon t h e thickness of the f i l m .
R e f e r e n c e s .
1 . ABRAHAMS E . , ANDERSON P.W., LICCIARDELLO D.C., RAMAKRISHNAN T . V . , P h y s . R e v . L e t t . 4 2 ( 1 9 7 9 ) 6 9 3 .
2. ABRAHAMS E . , R A M A ~ I s H N A N T . V . , J . N o n - C r y s t . S o l . 35 ( 1 9 8 0 ) 1 5 3 . EFETOV K.B., LARKIN A . I . , KHMELNITZKII D . E . , J E T P , E s m a
30
( 1 9 7 9 )2 4 8 .
4 . GORKOV L . P . , LARKIN A . I . , KHMELNITZKII D . E . , J E T P
2
( 1 9 8 0 ) 1 1 2 0 . 5. LEE P . , P h y s . R e v . L e t t . 4 2 ( 1 9 7 9 ) 1 4 9 2 .6 . LEE P . , J . N o n - C r y s t . S o l F g ( 1 9 8 0 ) 2 1 .
7 . BONCH-BRUEVICH V . L . , J E T P , P i s m a
32
( 1 9 8 0 ) 2 2 2 . 8 . MIRONOV A.G.. T h e o r . Math. P h y s . ( i n p r e s s ) . 9 . KNIGHTS J . , j. c on-cryst. s o l : 35/!6 i i 9 8 o j 1 5 91 0 . BRODSKY M.N.. I n " F u n d a m e n t a l P h y s i c s o f A m o r p h o u s S e m i c o n d u c t o r s " , P r o c . o f K ~ O ~Summer O 1nstitute..Springer-veriag, 1 9 8 0 , p. 5 6 . 1 1 . PAUL W . , I b i d . p. 7 2 .
