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NEW NUMERICAL METHOD FOR THE ELECTRONIC PROPERTIES OF DISORDERED
SYSTEMS
A. Mackinnon, B. Kramer, W. Graudenz
To cite this version:
A. Mackinnon, B. Kramer, W. Graudenz. NEW NUMERICAL METHOD FOR THE ELECTRONIC
PROPERTIES OF DISORDERED SYSTEMS. Journal de Physique Colloques, 1981, 42 (C4), pp.C4-
63-C4-66. �10.1051/jphyscol:1981410�. �jpa-00220714�
JOURNAL DE PHYSIQUE
CoZZoque C4, suppZ6ment au nO1O, Tome
42,oetobre
2981page
c4-63NEW NUMERICAL METHOD FOR THE ELECTRONIC PROPERTIES OF DISORDERED SYSTEMS
A . MacKinnon, B . Kramer and W. ~ r a u d e n z *
Referat fur Theoretische Physik, PhysikaZisch Teehnische BundesanstaZt, 3300 Brazmschweig,
F.R. G.
* Universitat Dortmund, 4600 Dortmund,
F.R. G.A b s t r a c t . - We p r e s e n t r e s u l t s of c a l c u l a t i o n s of t h e l o c a l i s a t i o n l e n g t h and c o n d u c t i v i t y of 2 d i m e n s i o n a l s y s t e m s d e s c r i b e d by t h e Anderson H a m i l t o n i a n . A r e c e n t l y d e v e l o p e d method f o r c a l c u l a t i n g p r o p e r t i e s o f v e r y l o n g 1 D s y s t e m s i s g e n e r a l i s e d f o r l o n g s t r i p s o f v a r y i n g t h i c k n e s s . We o b s e r v e s c a l i n g b e h a v i o u r w i t h t h e t h i c k n e s s b u t n o e v i d e n c e o f an Anderson t r a n - s i t i o n i n 2D.
I n t y o d u c t i o n ,
-
There h a s been a c o n s i d e r a b l e u p s u r g e o f i n t e r e s t i n t h e problem of l o c a l i s a t i o n i n d i s o r d e r e d s y s t e m s . T h i s h a s been due t o new e x p e r i m e n t s on p s e u d o o n e , and two d i m e n s i o n a l s y s t e m s 11--51, and t o t h e development o f new methods of c a l c u l a t i o n , b o t h n u m e r i c a l C6-81 and a n a l y t i c a l [9-121.There i s some c o n f u s i o n i n t h e r e s u l t s o f t h e 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 . I n t h e a n a l y t i c a l work, where d=2 i s found t o be t h e lower c r i t i c a l dimension f o r t h e Anderson t r a n s i t i o n , s p e c i a l models o r a p p r o x i m a t i o n s a r e c h o s e n , whose consequences a r e n o t e a s i l y c o n t r o l l e d . I n t h e n u m e r i c a l work t h e s y s t e m s i z e s a r e p o s s i b l y t o o s m a l l t o g i v e c o n c l u s i v e r e s u l t s a b o u t t h e t r a n s i t i o n between l o c a l i s e d and e x t e n d e d s t a t e s C131. For i n s t a n c e , a l t h o u g h t h e r e was a n a l y t i c a l e v i d e n c e t h a t f o r d = l a l l s t a t e s a r e l o c a l i s e d f o r any n o n z e r o d i s o r d e r [14,151 and t h a t t h e d . c . c o n d u c t i v i t y i s z e r o C161, i t p r o v e d d i f f i c u l t t o d e m o n s t r a t e t h i s n u m e r i c a l l y C171. Q n l y t h e development o f a new n u m e r i c a l method whose s t o r a g e r e q u i r e m e n t i s i n d e p e n d e n t o f t h e s y s t e m s i z e made t h e t r e a t m e n t o f m a c r o s c o p i c a l l y l a r g e s y s t e m s ( l o 9 s i t e s ) pos- s i b l e and l i f t e d t h e c o n t r a d i c t i o n between a n a l y t i c a l t h e o r y and n u m e r i c a l c a l c u l a - t i o n C181.
I n t h i s p a p e r we s h a l l g e n e r a l i s e t h i s method t o h i g h e r d i m e n s i o n a l s y s t e m s , and p r e s e n t r e s u l t s f o r d-2 f o r b o t h t h e l o c a l i s a t i o n l e n g t h and t h e c o n d u c t i v i t y . Method. - The i d e a o f o u r method i s t o c a l c u l a t e a p h y s i c a l q u a n t i t y f o r a s y s t e m w i t h N a t o m s , s a y A(N), from t h e same q u a n t i t y f c r N - 1 atoms, i . e .
F o r example, t h e m a t r i x e l e m e n t o f t h e G r e e n ' s f u n c t i o n ( r e s o l v e n t ) , between t h e s t a t e s l o c a t e d a t t h e e n d s of a c h a i n > f N atoms d e s c r i b e d by t h e Anderson Ha- m i l t o n i a n w i t h d i s o r d e r e d d i a g o n a l e l e m e n t s ~ ( i ) , <1/ G(N)
\
N > , i s g i v e n bywhere Z i s t h e (complex) e n e r g y , V t h e ( n e a r e s t n e i g h b o u r ) o f f - d i a g o n a l e l e m e n t of t h e H a m i l t o n i a n , and <NIG(N)IN> t h e d i a g o n a l e l e m e n t o f t h e G r e e n ' s f u n c t i o n on t h e
Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981410
JOURNAL DE PHYSIQUE
~ t h s i t e . The l o c a l i s a t i o n l e n g t h X may t h e n be d e f i n e d a s
The same p r i n c i p l e can be a p p l i e d t o t h e c o n d u c t i v i t y a s given by t h e Kubo- Greenwood formula C181. The advantage of t h i s r e c u r s i v e method l i e s n o t only i n t h e n e g l i g i b l e computer s t o r a g e needed, b u t a l s o i n t h e f a c t t h a t t h e s t a t i s t i c a l
accu-
racy of t h e r e s u l t a f t e r N s t e p s may be c a l c u l a t e d and used a s a c r i t e r i o n f o r t e r -
-
minating t h e procedure [13,181.Thus, we have a method which c o n t a i n s no o t h e r r e s t r i c t i o n on t h e s i z e of t h e system t h a n t h e computer time a v a i l a b l e , and which allows u s t o go on i n c r e a s i n g t h e s i z e of t h e system u n t i l a
given
accuracy o f t h e r e s u l t i s reached. This allows f o r much more r e l i a b l e e x t r a p o l a t i o n s t h a n have been p o s s i b l e h i t h e r t o . A simple g e n e r a l i s a t i o n of t h e above method t o h i g h e r dimensions i n v o l v e s r e p l a c i n g t h e simple b a s i s f u n c t i o n s of t h e one dimensional system with c h a i n s (d-2) o r p l a i n s ( d = 3 ) of atoms g i v i n g two o r t h r e e dimensional systems, which a r e very long i n one d i r e c t i o n . The s c a l a r q u a n t i t i e s appearing i n t h e r e c u r s i v e r e l a t i o n s f o r d = l t h e n become m a t r i c e s and a l l p h y s i c a l p r o p e r t i e s depend on t h e t h i c k n e s s M of t h e system.For example, I < l I ~ ( i ) / i > l ~ i n eq. ( 4 ) becomes T r < l I G ( i ) l i > * < i ! G ( i ) l l > i n v o l v i n g a sum o v e r a l l combinations of two s i t e s a t o p p o s i t e ends o f t h e system.
R e s u l t s . - I n t h e r e s u l t s p r e s e n t e d i n t h e following we used t h e r e c t a n g u l a r d i s - t r i b u t i o n f o r t h e s i t e d i a g o n a l elements o f H , i . e .
The accuracy of our d a t a i s Ah = 0 . 0 1 X f o r t h e l o c a l i s a t i o n l e n g t h , and Ao = 0.10 f o r t h e c o n d u c t i v i t y .
Fig. 1. A ( M ) / M vs. A(-)/M f o r v a r i o u s W/V v a l u e s and M = 4 , 8 , 1 6 , 32.
I n s e r t : A(-) vs. W/V
The l o c a l i s a t i o n l e n g t h . - We have c a l c u l a t e d t h e l o c a l i s a t i o n l e n g t h A (M) of two dimensional systems with 4V i W _< 15V f o r M = 4 , 8 , 1 6 , 32. The d a t a were found t o obey a s c a l i n g law
A ( M ) / M = f { X ( - ) / M } ( 6 )
where A(-) i s t h e l o c a l i s a t i o n l e n g t h f o r M*. They do n o t f u l f i l t h e r e l a t i o n s h i p
f o u n d by P i c h a r d and Sarma E l 9 1 a l t h o u g h t h e i r method o f c a l c u l a t i o n i s s i m i l a r t o o u r s . We have s e e n no e v i d e n c e o f a "phase t r a n s i t i o n " . For W=5V and W=6V, X (M) s t a r t s l i n e a r i n M , a s o b s e r v e d i n r e f 1191 b u t becomes s u b l i n e a r f o r l a r g e r M. We c o n c l u d e t h a t t h e l i n e a r r a n g e is n o t i n d i c a t i v e o f t h e M* l i m i t .
For t h i s r e a s o n , and b e c a u s e we were u n a b l e t o f i t a c r i t i c a l form f o r h(m, W) unambiguously t o eq. ( 6 ) we a t t e m p t e d t o c a l c u l a t e X(m,W) w i t h o u t assuming c r i t i c a l b e h a v i o u r . S i n c e we e x p e c t f
{ X I
i n e q . ( 6 ) t o r i s e m o n o t o n i c a l l y and f{ X I =
x f o r s m a l l x {M>>X(m)} we i n v e r t e q . ( 6 ) and w r i t ewere g t o }
=
g ' { o } = o. Expanding g { x } i n some s e t o f f u n c t i o n s l e a d s t o a s y s t e m o f l i n e a r e q u a t i o n s f o r X(m, W) and f o r t h e e x p a n s i o n c o e f f i c i e n t s . The r e s u l t i n g s c a l i n g f u n c t i o n f {x} i s shown i n f i g . 1. X(m, W) i s shown i n t h e i n s e r t . Although A(m, W) grows r a p i d l y n e a r W=5V t h i s may n o t y e t b e c o n s i d e r e d e v i d e n c e f o r c r i t i c a l b e h a v i o u r . I t does s u g g e s t however t h a t , f o r W<5V, M must b e v e r y l a r g e b e f o r e X(M) c o n v e r g e s .F i g 2. ow2/v2 v s . yv/w2 f o r v a r i o u s y , M = 8 and W = 1, 2 , 4 , 5 , 6 , 8 , 1 0 , 12.
Continuous l i n e : S c a l i n g f u n c t i o n f o r d = l
The c o n d u c t i v i t y . - I n t h e c a s e o f t h e d c c o n d u c t i v i t y it i s n e c e s s a r y t o s t u d y t h e dependence on t h e t h r e e p a r a m e t e r s M , W , and t h e i m a g i n a r y p a r t o f t h e e n e r g y , y . The l a t t e r h a s p r o v e d t o b e e x t r a o r d i n a r i l y i m p o r t a n t f o r d-1, where a s c a l i n g law
w 2 0 = h { Y v/w2} ( 8 )
was found. F i g . 2 shows some d a t a f o r M=8, assuming t h e above s c a l i n g p r o p e r t y , t o g e t h e r w i t h t h e s c a l i n g f u n c t i o n h { x } o f d = l . The d a t a do n o t obey r e l a t i o n ( 8 ) . F o r l a r g e y
v/w2
t h e c o n d u c t i v i t y i s r e d u c e d compared w i t h M - 1 , whereas f o r s m a l l yv/w2 it i s i n c r e a s e d . T h i s i n c r e a s e i s c l e a r l y a consequence o f t h e l a r g e r l o c a l i - s a t i o n l e n g t h . The d e c r e a s e i n t h e "Drude" p a r t o f t h e c u r v e on t h e o t h e r hand i s due t o t h e f a c t t h a t t h e i m p o r t a n t l e n g t h i s now t h e p r o j e c t i o n o f t h e " i n e l a s t i c mean f r e e p a t h " ( l / y ) o n t o t h e a x i s .- 3
The c o n d u c t i v i t y i s o n l y weakly dependent on M f o r W<6V and y>10 V. T h e r e f o r e it i s d i f f i c u l t t o draw any r e l i a b l e c o n c l u s i o n a b o u t t h e M a l i m i t i n t h i s regime.
C o n c l u s i o n .
-
We have developed a method o f computing t h e l o c a l i s a t i o n l e n g t h and c o n d u c t i v i t y o f v e r y l o n g t h i n s y s t e m s and have a p p l i e d it t o t h e c a l c u l a t i o n o fJOURNAL DE PHYSIQUE
s t r i p s o f v a r y i n g t h i c k n e s s i n o r d e r t o s t u d y t h e t r a n s i t i o n t o 2 d i m e n s i o n s . We h a v e s e e n n o e v i d e n c e o f c r i t i c a l b e h a v i o u r i n two d i m e n s i o n a l s y s t e m s . The l o c a l i - s a t i o n l e n g t h s which we h a v e c a l c u l a t e d f o r W
=
5V a n d W = 6V a r e however compar- a b l e w i t h t h e t h i c k n e s s o f o u r s y s t e m s a n d w i t h t h e s i z e o f s y s t e m s c o n s i d e r e d i n most s i m u l a t i o n s up t o now. Thus, j u s t as i n o n e d i m e n s i o n , we a r e f o r c e d t o con- c l u d e t h a t t h e "Anderson t r a n s i t i o n s " s e e n i n p r e v i o u s n u m e r i c a l work a r e d u e t o t h e l o c a l i s a t i o n l e n g t h becoming c o m p a r a b l e w i t h t h e s y s t e m s i z e .Acknowledgements. - We acknowledge u s e f u l d i s c u s s i o n s w i t h G. C z y c h o l l , W . WGger, H. K i n g , J . S a k .
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