HAL Id: jpa-00220711
https://hal.archives-ouvertes.fr/jpa-00220711
Submitted on 1 Jan 1981
HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.
L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.
ANDERSON LOCALISATION IN TWO-BAND SYSTEMS
H. Aoki
To cite this version:
H. Aoki. ANDERSON LOCALISATION IN TWO-BAND SYSTEMS. Journal de Physique Colloques,
1981, 42 (C4), pp.C4-51-C4-54. �10.1051/jphyscol:1981407�. �jpa-00220711�
ANDERSON LOCALISATION
I N
TWO-BAND SYSTEMS H. AokiCavendish Laboratory, Madingley Road, Cambridge CB3 OHE, England
A b s t r a c t
-
The e l e c t r o n i c s t r u c t u r e of d i s o r d e r e d two-band systems i s i n v e s t i g a t e d s y s t e m a t i c a l l y f o r t h e f i r s t time t o s h e d l i g h t on t h e n a t u r e of l o c a l i s a t i o n i n r e a l i s t i c s y s t e m s . C h a r a c t e r i s t i c l o c a l i s a t i o n i n t h e models i n c l u d i n g i n t r i n s i c semiconductor and h y b r i d i s e d s-d systems i s s t u d i e d by b o t h t h e d i r e c t d i a g o n a l i - s a t i o n of t h e H a m i l t o n i a n and t h e d e c i m a t i o n method. We s t r e s s e d t h e importance of t h e i n t e r - b a n d n i x i n g , which produces i n t r i n s i c f e a t u r e s i n t h e e l e c t r o n i c s t r u c t u r e a r i s i n g from i n t e r p l a y of randomness and b a s i c e l e c t r o n i c s t r u c t u r e .INTRODUCTION
The l o c a l i s a t i o n of s t a t e s , which i s s p e c i f i c t o random s y s t e m s , i s e s s e n t i a l f o r u n d e r s t a n d i n g t h e e l e c t r o n i c p r o p e r t i e s of n o n - c r y s t a l l i n e m a t e r i a l s . Much of t h e work on t h i s problem have been done i n models i n which a s i n g l e band i s assumed.
I f , however, we t u r n t o non-simple bands such a s a semiconductor w i t h v a l e n c e and c o n d u c t i o n b a n d s , o r a h y b r i d i s e d s-d s y s t e m s , t h e l o c a l i s a t i o n due t o randomness g i v e s r i s e t o even more f a s c i n a t i n g e l e c t r o n i c s t r u c t u r e . I n t h e p r e s e n t p a p e r , we e l u c i d a t e t h e c h a r a c t e r i s t i c n a t u r e of l o c a l i s a t i o n i n t h e two-band s y s t e m s , and show t h a t t h e i n t e r e s t i n g f e a t u r e s a r i s e from an i n t e r p l a y of b a s i c band s t r u c t u r e and t h e e f f e c t of randomness. I n p a r t i c u l a r we emphasize t h e i m p o r t a n t e f f e c t of i n t e r - b a n d m i x i n g , which can d r a s t i c a l l y a f f e c t t h e c h a r a c t e r of e i g e n - s t a t e s . We i n v e s t i g a t e d t h e e i g e n s t a t e s by n u m e r i c a l l y d i a g o n a l i q i n g t h e model H a m i l t o n i a n f o r l a r g e systems. We a l s o u s e d t h e d e c i m a t i o n f o r r e a l - s p a c e r e - n o r m a l i $ a t i o n developed by t h e p r e s e n t a u t h o r [I] t o a n a l y s e t h e e l e c t r o n i c s t r u c t u r e d i a g r a m m a t i c a l l y .
FORMULATI ON
I n a two-band model, t h e H a m i l t o n i a n i s w r i t t e n i n t h e t i g h t - b i n d i n g form a s
where li> i s a Wannier s t a t e a t i - t h s i t e , ~ i l ' i s t h e e n e r g y of t h e u-th o r b i t a l a t i , and V i j u v i s t h e t r a n s f e r e n e r g y between u-th o r b i t a l a t i and V-th o r b i t a l a t j . The band mixing i s d e t e r m i n e d by ciAB and
vijAB.
To be s e c i f i c we adopt r e c t a n g u l a r d i s t r i b u t i o n s of random q u a n t i t i e s s u c h t h a t ciA, ciE and E i A B a r e d i s t r i b u t e d w i t h w i d t h s WA, WB and WAB c e n t r e d a t -Eo/2, E / 2 and EAB, r e s p e c t i v e l y . F o r s i m p l i c i t y we assume t h a t vijAA = VA, vijBB = VB,VijgB
= 0 f o r n e a r e s tneighbour ( i , j ) . The u n p e r t u r b e d band s t r u c t u r e i s d e t e r m i n e d by VA, Vg and Eo.
We s t u d y h e r e two t y p i c a l c a s e s : I n t h e f i r s t model we took -VA = VB = V = 1.0 and Eo 2ZV w i t h c o o r d i n a t i o n number Z , which r e p r e s e n t s a semiconductor w i t h a d i r e c t gap a t
r
p o i n t . I n t h e second model we have a narrow band embedded i n aArticle published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981407
JOURNAL DE PHYSIQUE
Fig. 1. ( a ) The A and B o r b i t a l s on a s q u a r e l a t t i c e a r e shown. The l i n e s between t h e o r b i t a l s s t a n d f o r t h e m a t r i x elements connecting them. The heavy l i n e and t h e wiggly l i n e show examples of t h e diagrams c o n t r i b u t i n g t o v i j A B ( l ) and V+ jBB ( I ) , r e s p e c t i v e l y . The f u l l (open) c i r c l e s r e p r e s e n t t h e r e t a i n e d (eliminated) s i t e s i n
fi
decimation. (b) The mixing of s t a t e s due t o inter-band e f f e c t s i s s c h e m a t i c a l l y shown. The shaded r e g i o n i n t h e d e n s i t y of s t a t e s , D , corresponds t o locali'sed s t a t e s i n t h e absence of i n t e r - b a n d mixing, H ~ ~ ,wide band with VA=-5.0, VB=-1.0 and Eo%O, which r e p r e s e n t s a h y b r i d i s e d s-d system as a n o t h e r p r o t o t y p e of two-band l o c a l i b a t i o n . The importance of t h e inter-band mixing i s r e a d i l y s e e n , s i n c e t h e amplitudes of t h e A and B band components, a i and b i r e s p e c t i v e l y , a r e governed by coupled e q u a t i o n s of motion d e r i v e d from t h e Hamiltonian (1). I n f i g u r e 1 we show t h e mixing of t h e two-band components due t o inter-band e f f e c t s by both diagrams and d e n s i t y of s t a t e s .
RESULTS
The Hamiltonian (1) i s expressed a s a 2Nx2N m a t r i x , where N i s t h e t o t a l number of s i t e s . Here we took 2N=512 corresponding t o a 16x16 s q u a r e l a t t i c e w i t h p e r i o d i c boundary c o n d i t i o n s i n both x and y d i r e c t i o n s . For each sample t h e random m a t r i x elements a r e g e n e r a t e d according t o t h e d i s t r i b u t i o n above. We o b t a i n e d t h e eigen- v a l u e s and e i g e n s t a t e s by d i a g o n a l y s i n g t h e m a t r i x . Each e i g e n s t a t e i s a l i n e a r combination of A and B components as
One of t h e most conspicuous r e s u l t s i s t h a t t h e r e i s a s i g n i f i c a n t c o r r e l a t i o n i n t h e A and B components f o r an e i g e n s t a t e . T h i s i s b e s t expressed a s a c o r r e l a t i o n f u n c t i o n ,
where
R ~ ~ - I R ~ - R ~ ! ,
angular b r a c k e t s s t a n d f o r an average over c o n s t a n t R;j and K i s a decay c o e f f x c i e n t which i s f i n i t e f o r l o c a l i ' s e d s t a t e s . The e x p o n e n t i a l form f o r f a i s c o n s i s t e n t w i t h an e x p o n e n t i a l envelope f u n c t i o n f o r l o c a l i s e d wave- f u n c t i o n s . I n f i g u r e 2 we show an example of t h e behaviour of t h i s c o e f f i c i e n t as a f u n c t i o n of energy. I t i s c l e a r l y seen t h a t K h a s a s h a r p peak a t t h e pseudo-gap where t h e valence and conduction bands merge. This r e s u l t shows t h a t t h e s t a t e s a r e l o c a l i k e d w i t h a s t r o n g c o r r e l a t i o n between t h e two-band components i n t h e pseudo-gap region. This c o r r e l a t i o n can be e x p l a i n e d i21 by a simple p e r t u r b a t i o n t h e o r y i f we r e g a r d t h e band mixing as a p e r t u r b a t i o n . We can look i t another way when t h e t a i l s of t h e valence and conduction bands o v e r l a p ( f i g . l ( b ) ) . The s p l i t t i n g caused by t h e band mixing produces, i n t h e z e r o t h approximation i n t h e degenerate p e r t u r b a t i o n , e i g e n s t a t e s w i t h bonding and anti-bonding c h a r a c t e r s composed of l o c a l i b e d s t a t e s i n A and B bands. These c o r r e l a t e d s t a t e s should have an e f f e c t on p h y s i c a l q u a n t i t i e s such a s p h o t o c o n d u c t i v i t y .energy i n t h e semiconductor model w i t h W~=Wg=2.9, E0=7.5, W~g=1.6, EAB=0.8 and -V~=Vg=1.0. The d e n s i t y of s t a t e s i s a l s o shown f o r comparison.
The e f f e c t of band mixing produces a n o t h e r i n t e r e s t i n g phenomenon i n t h e h y b r i d i s e d s-d model. I n t h i s c a s e , t h e presence of a wide s-band h a s a d r a s t i c e f f e c t on t h e narrow d-band. I n t h e case of a s i n g l e d - o r b i t a l i m p u r i t y i n s-band t h e impurity
l e v e l becomes a v i r t u a l bound s t a t e C31 with a width % ~ < k s l ~ s d l i d > ! $ D ( E F ) , where D i s t h e d e n s i t y of s e l e c t r o n s . I n t h e p r e s e n t c a s e , we a r e f a c e d w i t h a much more complicated s i t u a t i o n with a whole d-band which has a locali'sed regime due t o d i s o r d e r b e f o r e t h e h y b r i d i s a t i o n i s turned on. When t h e band mixing i s turned on, a c h a r a c t e r i s t i c h y b r i d i s a t i o n r e s u l t s . An example i s shown i n f i g u r e 3. I n t h i s f i g u r e we have p l o t t e d t h e t o t a l amplitude of t h e A-band component, A2 E
zil
a i l 2 , f o r each normali'sed e i g e n s t a t e ( 2 ) . For a r e l a t i v e l y small band mixing, we have aF i g . 3. The t o t a l magnitude of A- band component, A2, i s p l o t t e d v e r s u s energy i n t h e s-d model w i t h W~=4.0, Eo=O.O, VAZ-5.0, Vg=-l.O.
The below l e f t ( a ) i s a r e s u l t f o r WB=1.5, W~g=1.0, E A B = ~ . ~ , w h i l e t h e below r i g h t (b) i s f o r Wg=7.0, W ~ ~ = 7 . 0 , E ~ g = 3 . 5 . The d e n s i t y of
n
s t a t e s f o r t h e former case i s a l s oshown f o r comparison.
JOURNAL DE PHYSIQUE
l a r g e f l u c t u a t i o n i n A2 i n t h e r e g i o n of energy where s and d bands o v e r l a p . T h i s i s a g a i n e x p l a i n e d by a d e g e n e r a t e p e r t u r b a t i o n t h e o r y w i t h t h e band mixing a s a p e r t u r b a t i o n L21. For l a r g e band mixing, t h e z e r o t h o r d e r w a v e f u n c t i o n i t s e l f i s determined by t h e band mixing, and we have a smoother dependence of A2 on energy ( f i g u r e 3 ( b ) )
.
DISCUSSION
The e l e c t r o n i c s t r u c t u r e r e v e a l e d above comes b a s i c a l l y from t h e diagrammatic s t r u c t u r e shown i n f i g u r e 1. The d e c i m a t i o n method o f r e a l - s p a c e r e n o r m a l i s a t i o n developed by t h e p r e s e n t a u t h o r proved t o b e q u i t e powerful i n a n a l y s i n g a v a r i e t y of systems i n c l u d i n g non-simple systems L41 a s w e l l a s u s u a l two o r t h r e e dimen- s i o n a l systems [ l l . By t h e d e c i m a t i o n t r a n s f o r m a t i o n , t h e system can b e r e g a r d e d a s a n assembly of t h e s i t e s w i t h a l a r g e r l a t t i c e s p a c i n g ( f u l l c i r c l e s i n f i g u r e 1 ) i n t e r a c t i n g v i a e f f e c t i v e i n t e r a c t i o n s . The e f f e c t i v e i n t e r a c t i o n , V i j u v ( n ) , can b e e x p r e s s e d i n terms of t h e l o c a t o r e x p a n s i o n f o r t h e Green f u n c t i o n . A n e a t way of e x p r e s s i n g t h e decimated H a m i l t o n i a n can b e d e r i v e d i n which t h e o r i g i n a l e q u a t i o n f o r t h e Green f u n c t i o n , (E
-
H+
i 6 ) G = 1, i s reduced i n t ow i t h K E E-H+i6. The s u f f i c e s f o r K s t a n d f o r t h e p a r t of t h e m a t r i x r e p r e s e n t a - t i o n of K such t h a t (KAB);j = K i j ( i E A , ' j E B ) , e t c , where A (B) i s t h e r e t a i n e d ( e l i m i n a t e d ) s u b s e t o f b a s l s i n t h e d e c i m a t i o n . I f we t a k e A - o r b i t a l s a s t h e s u b s e t A i n t h e two-band model f o r t h e b a s i s above (v;jAB = O), we have an e f f e c t i v e i n t e r a c t i o n ,
where GB(E) i s t h e B-band Green f u n c t i o n . T h i s c o n c i s e l y e x p l a i n s why t h e band mixing g i v e s r i s e t o s u c h a d r a s t i c e f f e c t on t h e e l e c t r o n i c s t r u c t u r e , and a l s o shows t h a t t h e l o c a l i s a t i o n and d e l o c a l i s a t i o n of t h e h y b r i d i s e d s t a t e s i n t h e whole energy spectrum a r e determined s e l f - c o n s i s t e n t l y . I f , f o r i n s t a n c ~ , GB(E) h a s a s t r u c t u r e of extended s t a t e s , t h i s i s r e f l e c t e d i n t h e l o n g - r a n g e V ; . ~ ( E ) , t h u s a f f e c t s t h e A-band, and v i c e v e r s a . T h i s s t r u c t u r e of i n t e r a c t i o n s a l s o a f f e c t s t h e d i s t r i b u t i o n w i d t h of r e n o r m a l i s e d t r a n s f e r e n e r g i e s , which i s g a u s s i a n i n l o g a r i t h m i c s c a l e .
The c o n c e p t of two-band l o c a l i k a t i o n i n a s i m p l e model d e s c r i b e d h e r e can b e r e a d i l y a p p l i e d t o r e a l i s t i c systems such a s f o u r - f o l d o r t h r e e - f o l d c o o r d i n a t e d semi.conductors, i n which t h e i n t e r - b a n d mixing e l e m e n t s a r e between t h e b a s i s f o r s-p h y b r i d i s e d o r b i t a l s . As f o r t h e s-d s y s t e m , we can make t h e model more r e a l - i s t i c by i n c l u d i n g t h e f i v e - f o l d degeneracy of d - o r b i t a l s f o r t h e t r a n s i t i o n o r n o b l e m e t a l s .
Acknowledgements - Valuable d i s c u s s i o n s w i t h P r o f e s s o r S i r Nevi11 Mott a r e g r a t e f u l l y acknowledged. Work a t t h e Cavendish L a b o r a t o r y was s u p p o r t e d by t h e S c i e n c e Research C o u n c i l .
REFERENCES
1. AOKI H . , S o l i d S t a t e Cormnun. 3 1 (1979) 999; J. Phys. C13 (1980) 3369.
2. AOKI H . , J. Phys. C14 (1981), t o be p u b l i s h e d . 3. ANDERSON P.W., Phys. Rev. 124 (1961) 41.
4. AOKI H . , S o l i d S t a t e Commun. 37 (1981) 677.
