The diagonal and hall conductivity of a two-dimensional electron system in a strong magnetic field

~ o ~ Solid State Communications, Vol. 67, No. 5, pp. 499-503, 1988. ~8_~Printed in Great Britain.

0038-1098/88 $3.00 + .00 Pergamon Press plc

THE DIAGONAL AND HALL CONDUCTIVITY OF A TWO-DIMENSIONAL A STRONG MAGNETIC FIELD

ELECTRON

SYSTEM

G.Czycholl l n s t l t u t f u r T h e o r e t l s c h e Physlk C, RWTH Aachen, D-5100 Aachen, Federal Republic o f Germany I n s t l t u t ftlr Phystk,Hnlversit,'lt D o r t m u n d , D-4600 D o r t m u n d 50, Pederal Republic o f Germany (Received

IN

and

1 June 1988 by B.Miihlschlegel)

The t i g h t - b i n d i n g model o f e l e c t r o n s on a t w o - d i m e n s i o n a l s q u a r e lattice in a s t r o n g p e r p e n d i c u l a r m a g n e t i c field and a r a n d o m p o t e n t i a l is investigated. For a finite s y s t e m of 16.16 a t o m s w i t h periodic b o u n d a r y c o n d i t i o n s all site m a t r i x e l e m e n t s o f t h e o n e - p a r t i c l e Green f u n c t i o n are c a l c u l a t e d numerically by m a t r i x inversion. From t h e s e m a t r i x e l e m e n t s t h e density of s t a t e s , t h e diagonal conductivity Oxx and t h e Hall conductivity Oxy can be calculated. In spite of t h e relatively small s y s t e m size t h e conductivities Oxx and Oxy ( c a l culated as a f u n c t i o n of t h e band filling) show already t h e behaviour, which is c h a r a c t e r i s t i c f o r t h e i n t e g e r q u a n t i z e d Hall effect, namely regions o f vanishing Oxx and t h e Hall p l a t e a u s in Oxy a t i n t e g e r m u l t i p l e s o f e Z / h .

It is c o m m o n l y accepted t h a t t h e model of a t w o - d i m e n s i o n a l (2d) e l e c t r o n s y s t e m in a r a n d o m p o t e n t i a l and a p e r p e n d i c u l a r s t r o n g m a g n e t i c field s h o u l d exhibit t h e normal (integer) q u a n t i z e d Hall effect(QHE) 1'2. According t o t h i s picture t h e disc r e t e highly d e g e n e r a t e Landau levels, which f o r m t h e s p e c t r u m o f free 2d e l e c t r o n s in a s t r o n g m a g netic field, are b r o a d e n e d into Landau b a n d s due t o t h e disorder, and w i t h i n each Landau band one e x p e c t s delocalized s t a t e s a r o u n d t h e b a n d c e n t e r and localized s t a t e s at t h e b a n d edges. The diagonal conductivity Oxx vanishes and t h e Hall conductivity Oxy is q u a n t i z e d in i n t e g e r m u l t i p l e s o f eZ/h, if t h e Fermi energy falls into t h e regime of localized s t a t e s , so t h a t - as a f u n c t i o n o f t h e b a n d filling o r t h e inverse m a g n e t i c field - Oxy exhibits t h e c h a r a c t e r i s t i c p l a t e a u s a t t h e values l e Z / h (1~ ~1) and Oxx v a n i s h e s s i m u l t a n e o u s l y . Up t o now a q u a n t i t a t i v e c o n f i r m a t i o n of t h i s picture does n o t yet exist. There are many a r g u m e n t s a n d r e s u l t s which c o n f i r m t h e e x i s t e n c e o f localized s t a t e s and o f a t l e a s t one delocalized s t a t e within each Landau band, b u t a s i m u l t a n e o u s c a l c u l a t i o n o f b o t h t r a n s p o r t coefficients, Oxx and 6xy , s t a r t i n g f r o m t h e Kubo f o r m u l a and r e p r o ducing t h e above m e n t i o n e d b e h a v i o u r is still missing. Therefore, it is n o t y e t clear, if t h e d i s o r d e r model can q u a n t i t a t i v e l y explain t h e QHE e x p e r i m e n t s , concerning, f o r Instance, t h e w i d t h of t h e QHE plateaus . The QHE should, o f course, also exist f o r t h e model o f t w o - d i m e n s i o n a l lattice e l e c t r o n s within a r a n d o m p o t e n t i a l and a s t r o n g m a g n e t i c field. For lattice e l e c t r o n s w i t h o u t d i s o r d e r it is a p p r o p r i a t e

t o i n t r o d u c e a m a g n e t i c field via t h e Pelerls s u b s t i t u t i o n 3'4 ;the H a m i l t o n i a n o f t h e s y s t e m w i t h m a g n e t i c field is t h e n given by E(~- ~ , i f £ ( ~ is t h e b a n d dispersion w i t h o u t m a g n e t i c field, i~ is t h e m o m e n t u m o p e r a t o r and ~ d e n o t e s t h e v e c t o r potential. For t h e 2d tight-b_inding model in a p e r pendicular m a g n e t i c f i e l d (TBMF-model) on a s q u a r e l a t t i c e t h e Pelerls s u b s t i t u t i o n leads t o t h e Hamiltonian s HO=

E t ( Inx+l ny> + I n x - I ny> + nxny (1) + eZwi~nxl nxny+l > + e-Z~i~nxl n x n y - l > ) < n x n y I

Here eB a2 - - 2~ c

(2)

is a d i m e n s i o n l e s s m e a s u r e o f t h e s t r e n g t h o f t h e m a g n e t i c field B, which lies in z-direction, and t h e Landau gauge h a s b e e n used, i.e. ~=(O,Bx,O); a d e n o t e s t h e l a t t i c e c o n s t a n t , e t h e e l e c t r o n charge, and c t h e velocity o f light. The p a r a m e t e r ~ can also b e I n t e r p r e t e d as t h e n u m b e r o f m a g n e t i c flux q u a n t a p e r unit cell (1%= 1 t h r o u g h o u t t h i s paper). Energies will b e m e a s u r e d in u n i t s o f t h e hopping m a t r i x e l e m e n t t(=l), and Inxny> d e n o t e s t h e W a n nier s t a t e localized a t t h e l a t t i c e site ..i~"=(nxa'nya)" The s p e c t r a l p r o p e r t i e s o f t h i s model are very i n t e r e s t i n g , showing, f o r instance, s e l f - s i m i l a r i t y , etc., as d i s c u s s e d by H o f s t a d t e r s. Therefore, t h e t r a n s p o r t p r o p e r t i e s o f t h i s model would b e I n t e r esting, even if it w o u l d have n o t h i n g to do w i t h t h e QHE.

499

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TWO-DIMENSIONAL ELECTRON SYSTEM IN A STRONG MAGNETIC FIELD

For t h e c a l c u l a t i o n o f t r a n s p o r t q u a n t i t i e s t h e model has t o b e e x t e n d e d by some s c a t t e r i n g m e c h chanlsm, and as t h e t r a d i t i o n a l d i s o r d e r (localization) e x p l a n a t i o n of t h e QHE s h a l l be checked, t h e model (1) is e x t e n d e d by a r a n d o m site diagonal potential V = ~ an Inxny>
Here E is t h e Fermi energy and j~ d e n o t e s t h e u- c o m p o n e n t of the current operator, G(zl=(Z-Ho-V1-1 is the o n e - p a r t i c l e Green function, G ' ( z ) = - i z - n o - V ) - z its derivative, and f~=Na 2 d e n o t e s t h e v o l u m e of t h e s y s t e m iN n u m b e r o f lattice sites). In W a n n l e r r e p r e s e n t a t i o n t h e c o m p o n e n t s o f the c u r r e n t are explicitely given by

(3) jx = i e a E Inxny>(
The ~n are a s s u m e d t o be u n i f o r m l y d i s t r i b u t e d over a n energy interval o f w i d t h W, which is t h e m e a s u r e o f t h e d i s o r d e r s t r e n g t h ; so for vanishing m a g n e t i c field (B=0) t h e model (1,3) is j u s t t h e 2d A n d e r s o n model 6 of a disordered system, which is t h e s t a n d a r d model f o r t h e investigation o f A n d e r s o n localization 7. So it s e e m s t o be quite n a t u r a l to use t h e model (1-3) f o r t h e investigation of 2d localization in a m a g n e t i c field and t h e QHE. But m o s t work on t h e QHE s t a r t s f r o m t h e Landau model, i.e. free e l e c t r o n s in a magnetic field, ext e n d e d by a random p o t e n t i a l 2 There exists only few work on t h e t r a n s p o r t properties o f the 2d disordered TBMF model. I know only o f two previous i n v e s t i g a t i o n s by Aoki a and by Schwettzer, MacKlnnon and Kramer 9. Aoki a exactly diagonalized t h e H a m i l t o n i a n (1,3) for a finite 2d s y s t e m o f 16.16 lattice s i t e s and c a l c u l a t e d t h e Hall conductivity Oxy as a f u n c t i o n of t h e b a n d energy E (i.e. t h e p o s i t i o n of t h e Fermi energy) s t a r t i n g from t h e Kubo formula. He did n o t c a l c u late Oxy as a f u n c t i o n of t h e band filling E = J d E ' o(E')

Vol. 67, No. 5

(4)

(6)

jy = i e a ~,1 nxny >( e-Z~t~nx
the the

e2 4~ N

~ ' Anm(E) Amn(E) nm with (7) Anm(E)= (< nx+l nyl - e 2

°xY (E) = - 2~ N f E dE' B(E') with B(E) = ~ [ Cnm (E) Dmn(E) - Enm(E) Amn(E) ] nm (8) C n a_i l( E l = ( e - 2 n i ~ n x < n .z~. n 3 .+11-eZ~i~nx Dmn(E) = (< mx+l my I -

Enm(E)= (e -zwi~nx< nxny+l ~- e2~i~n×< n x ny-I I) G2(E+i~) Imxmy>

(9(E) density of s t a t e s ) , and t h e diagonal c o n d u c t i v ity Oxx was also n o t calculated. Schweitzer et al. 9 c a l c u l a t e d t h e Hall c o n d u c t i v i t y Oxy according t o t h e Kubo f o r m u l a f o r t h e o r d e r e d TBMF model w i t h in t h e s p e c t r a l gaps and f o r t h e disordered TBMF model as a f u n c t i o n of t h e b a n d filling v f o r a special s t r i p g e o m e t r y (essentially Infinite l e n g t h in x-direction, finite w i d t h up t o 64. s i t e s in y - d i r e c tion). They obtained t h e plateau behaviour of Oxy(~) with a r a t h e r n a r r o w w i d t h of t h e plateau, however. But no meaningful results for t h e diagonal conductivity Oxx could b e o b t a i n e d j°. Thus a calculation o f b o t h t r a n s p o r t coefficients, Oxx and Oxy, as a f u n c t i o n of t h e b a n d filling v does n o t yet e x i s t for t h e disordered 2d TBMF model. For t h e c a l c u l a t i o n of t h e c o m p o n e n t s octB o f t h e conductivity t e n s o r I s t a r t f r o m t h e Kubo f o r mula, w h i c h f o r zero frequency and zero t e m p e r a t u r e may be w r i t t e n as 11'12 %~(E)=-

1

; ~ E ' Tr[(G(E+ial-G(E'-ia))jaG (E+i~)jl5 (S? -(G(E'+iS)-G(E'-i~)) j!sG' (E'-i~)j~ ]

For t h e density of s t a t e s o n e has, o f course, p(E) = - - ~ T r ImG(E+i~) = (9) = - l ~ n 2 ~ i A c a l c u l a t i o n o f t h e conductivities Oxx and Oxy according t o t h e s e equations (7) and (8) requires t h e knowledge o f all site m a t r i x e l e m e n t s of t h e o n e particle Green f u n c t i o n G(z). I c a l c u l a t e d t h e s e matrix elements numerically by a standard (Gaussian) m a t r i x inversion a l g o r i t h m . So far I i n v e s t i g a t e d only relatively small s y s t e m s up to 16,16 lattice sites. To avoid b o u n d a r y e f f e c t s I used periodic boundary conditions in b o t h directions. Of course, periodic boundary conditions in x - d i r e c t i o n are only p o s s i b l e for certain d i s c r e t e values of ~,namely only if the magnetic period due to the phase f a c t o r of t h e hopping m a t r i x e l e m e n t s in (1) Is c o m m e n s u r a b l e with t h e s y s t e m size. Therefore, for a l e n g t h o f 16 sites in x - d i r e c t i o n I choose the values ¢x=1/8, 2/8, and 3 / 8 f o r the d i m e n s i o n l e s s m a g n e t i c field s t r e n g t h . For a finite s y s t e m o f

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TWO-DIMENSIONAL ELECTRON SYSTEM IN A STRONG MAGNETIC FIELD

N=2$6 sites the Green function has N discrete poles along t h e real energy axis. A finite imaginary p a r t 5 has t o be c h o s e n t o e n s u r e t h a t the matrix (E+iS-Ho-V) can be inverted. To avoid unrealistic s t r u c t u r e s due t o t h e p o l e s 5 m u s t be c h o s e n a t least o f t h e same magnitude as the distance b e t w e e n adjacent poles. But as in a magnetic field t h e s p e c t r u m o f t h e TBMF model is confined t o r a t h e r narrow energy intervals due t o t h e f o r m a t i o n o f (quasi) Landau b a n d s s, relatively small values o f 5 can be c h o s e n d e s p i t e t h e small s y s t e m size. It is well known f r o m t h e l i t e r a t u r e 1334 t h a t the i n t r o d u c t i o n o f a small energy imaginary part is d a n g e r o u s as it may hide localization due t o t h e mixing o f several locall zed s t a t e s . Nevertheless, some tendency towards localization m i g h t be observable by such a m e t h o d , in particular as t h e localization l e n g t h m u s t be very s h o r t within a magnetic field, as the experimental static conductivity really vanishes. So localization should probably more easily be detected for the 2d T B M F model than for 2d systems without a magnetic field. In the following I present and discuss the numerical results, which are obtained for a disorder strength of W=0.2 and an energy imaginary part 5=0.03. As the spectrum is symmetric around zero, the calculations are performed for energies from the interval -g.
501

2.

p(E}

....L .......L

-4.

-3.

-2.

-i.

I~.

E

Fig.h

Energy d e p e n d e n c e o f t h e d e n s i t y o f s t a t e s o(E) o f the 2d TBMF model f o r a d i s o r d e r W=0.2, a d i m e n s i o n l e s s magnetic field ~=1/8, and a Lorentzian broadening 5=0.03

5.

4.

tD

tD

2.

-4.

-3.

-2.

-I.

~. E

Fig.2 : Diagonal conductivity Oxx (dashed line) and Hall conductivity Oxy(full line) as a f u n c t i o n o f t h e position o f t h e Fermi energy E f o r W=0.2, ~=1/8 and 5=0,03

4.

I

3.

%1= 2.

tD ×

-4.

-B.

-2.

-h

Fig.3: The s a m e as in Fig.2 f o r c~=1/4

502

T W O - D I M E N S I O N A L E L E C T R O N SYSTEM IN A STRONG M A G N E T I C F I E L D

~=1/4, f o r which only t w o s u b b a n d s occur in t h e lower h a l f o f the s p e c t r u m . Only t h e l o w e s t s u b h a n d around E=-2.7 has a "'Landau band" c h a r a c t e r and we observe a non-vanishing Oxx and an increase o f Oxy from 0 t o 1, as expected. Within t h e central subband Oxy d r o p s t o zero, and both, Oxx and Oxy,exhibit relatively large fluctuations, which are probably due t o t h e f a c t t h a t t h e w i d t h o f t h e c e n t r a l subhand is already larger t h a n 116, i.e. t h e d i s t a n c e b e t w e e n adjacent eigenenergies is relatively large so t h a t ~=0.03 is probably t o o small within t h i s energy region and e f f e c t s resulting f r o m t h e d i s c r e t e pole s t r u c t u r e appear. Fig.4 s h o w s t h e OxxlE) and Oxy(E) curves f o r ct=3/8. For this p a r a m e t e r t h e s p e c t r u m is s p l i t t e d again into eight s u b b a n d s , which w i t h o u t d i s o r d e r are c e n t e r e d around +2.SS,+-2.35,+-1.95, and +_0.1516. So in t h e l o w e r half o f the s p e c t r u m t h r e e s u b b a n d s are relatively close t o g e t h e r within the interval -2.6~E~-1.9. Due t o the disorder and the additional Lorentzlan broadening the gaps between these adjacent s u b b a n d s are s m e a r e d out, because o f which Oxx is n o n - v a n i s h i n g within the whole interval and s h o w s only minima within the f o r m e r gap regions (for i n s t a n c e near E=-2.1). Oxy increases rapidly within t h e f i r s t s u b b a n d up t o a value larger t h a n 6, t h e n it d r o p s even more rapidly within the second s u b b a n d b e c o m i n g negative down t o a value b e l o w -2 and it Increases again within t h e third s n b b a n d reaching exactly t h e value 1 (e2/h), if all t h r e e o u t e r m o s t subbands are filled. This behavionr is typical f o r rational values o f ct=p/q with p>117,9. Oxy is n o t exactly at an i n t e g e r multiple o f e 2 / h in t h e gaps b e t w e e n t h e t h r e e adjacent "Landau hands", as it should be (at least f o r t h e o r d e r e d infinite system) 9.17 according to Laughlin's a r g u m e n t s la, probably because we do n o t have t r u e gaps between these subbands due to the broadening.The Oxy(E) curves presented in Figs.2-¢ e s s e n t i a l l y agree with Aoki's c o r r e s p o n d i n g r e s u l t s obtained by m a t r i x diagonalization 8. Even details (as the dip near E:-I,05 f o r u=l/8) were also observed by Aoki. For ~=1/a. Aoki IRef.8b) obtained large f l u c t u a t i o n s also within t h e subband around E=-2.7, which I also o b s e r v e d w h e n choosing a s m a l l e r M:O.OI). In f a c t t h e value ~=0.03 was chosen as large to avoid unrealistically s t r o n g fluctuations for all p a r a m e t e r s investigated; f o r ~:1/8 and ~=3/8 n o n - f l u c t u a t i n g r e s u l t s could be obtained also for ~=0.01, at least within the o u t e r m o s t t h r e e s u b b a n d s . For t h i s 5 the Oxy-CUrve reaches a b s o l u t e l y slightly larger values in t h e t w o "gaps" b e t w e e n the t h r e e adjacent s u b b a n d s f o r a=3/8, which are t h e n also quantitatively in a g r e e m e n t with Aokl's r e s u l t for ~=3/8. But Aokl could not obtain the c o r r e s p o n d i n g Oxx-result and he did n o t determine 0xy as a f u n c t i o n o f the band filling v, which r e s u l t s are here p r e s e n t e d In Figs. 5-7 for the same p a r a m e t e r s as before

Vol. 67, No. 5

I~.

. . . . . . . . -4.

J -~.

.........

J

. . . . . . . . .

-2.

.........

z -1.

i 61~

E

Fig,4; The same as in Fig.2 f o r a=3/8

5.

4.

%1= b 2.

I~.l

l~.'2

l~.'3

l~.'4

~.'5 -o

Fig.S: Diagonal conductivity Oxx (dashed line) and Hall conductivity Oxy as a f u n c t i o n o f the band filling v f o r W=0.2,~=1/8 and ~:0.03

/'\"

3.

:

i

',,J',

i:

%J=

×

/

Fig.6: The same as in Fig.5 for c~=I/4

Vol. 67, No. 5

TWO-DIMENSIONAL ELECTRON SYSTEM IN A STRONG MAGNETIC FIELD

5.

%1=

\/ ......... [~.

, ....... ~.I

~.L..~....J {~.2

......... ~.3

J ......... 13.4

~.S

FIR.7: The s a m e as in Fig.5 f o r ct=3/8

(~=l/8,1/¢,and 3/8,~=0.03). Clearly we have t h e regions o f vanishing Oxx and s i m u l t a n e o u s p l a t e a u s of Oxv a t 1 e 2 / h (for all t h r e e p a r a m e t e r s ) and a t 2 e 2 / h ~(for ct=l/8). Por c~=1/8 (Pig.S) Oxy(V) is l o w e r t h a n t h e classical s t r a i g h t line ( t h r o u g h t h e p o i n t s (0.,0.),(0.12S,1.)) f o r s m a l l v and it b e c o m e s l a r g e r t h a n t h e classical value a t a b o u t v~O.07 a n d reaches t h e q u a n t i z e d p l a t e a u value (with an accuracy o f 99.8%) already f o r v<0.125, i.e. b e f o r e t h e l o w e s t s u b b a n d Is c o m p l e t e l y filled. The same holds t r u e f o r t h e s e c o n d s u b b a n d in t h e case ~=1/8 a n d f o r t h e f i r s t s u b b a n d f o r ~=1/4. A similar r e s u l t f o r Oxy(V) f o r ~=1/8 and only t h e l o w e s t s u b b a n d was o b t a i n e d by Schweitzer e t al. 9, using t h e s t r i p g e o m e t r y and fixed b o u n d a r y conditions. With this method, however, no s i m u l t a n e o u s r e s u l t s f o r Oxx could be o b t a i n e d 1°. Obviously, it Is n o t n e c e s s a r y t o have a relatively

503

large s y s t e m ("infinite" in one direction) t o r e p r o d u c e t h i s plateau behavlour; f u r t h e r m o r e , my c a l c u l a t i o n s s h o w t h a t t h i s b e h a v l o u r does n o t depend o n t h e b o u n d a r y c o n d i t i o n s and t h a t it is n o t n e c e s s a r i l y t h e b o u n d a r y s t a t e s o r edge c u r r e n t s , which carry t h e Hall c u r r e n t 9, as f o r periodic b o u n d a r y c o n d i t i o n s edge states and c u r r e n t s do n o t exist. But the Oxy(V)-plateaus and regions of vanishing Oxx(V) o b t a i n e d here are still relatively narrow. W h e r e a s in good QHE s a m p l e s t h e plateau w i d t h may be o f a b o u t 90% o f a Landau b a n d width, t h e w i d t h o f t h e c a l c u l a t e d p l a t e a u s is only o f t h e m a g n i t u d e o f 10g o f a s u b b a n d , which is also t r u e f o r t h e r e s u l t o f Ref.9. According t o t h e d i s o r d e r (Anderson localization) model of t h e QHE t h e w i d t h o f t h e Hall p l a t e a u s s h o u l d increase w i t h increasing disorder. I could not observe a c o r r e s p o n d i n g tendency, b u t t h e plateau w i d t h o b t a i n e d w a s e s s e n t i a l l y t h e same also f o r t h e d i s o r d e r s t r e n g t h s W=O.S and W=I. Within t h e r a n g e i n v e s t i g a t e d it did also n o t depend on t h e e n e r g y imaginary p a r t S. So t h e only possibility l e f t is a n increase o f t h e w i d t h o f t h e Hall p l a t e a u s w i t h increasing s y s t e m size; o t h e r w i s e a n o n - i n t e r a c t i n g 2d s y s t e m w o u l d n o t be able t o a c c o u n t for t h e e x p e r i m e n t a l l y observed plateau widths, and o t h e r (e.g. i n t e r a c t i o n , s c r e e n i n g ) e f f e c t s w o u l d have to be included even for the integer QHE. Such c a l c u l a t i o n s f o r l a r g e r s y s t e m s are planned, b u t they will p r o b a b l y require t h e use o f a v e c t o r computer (at l e a s t if it is n o t p o s s i b l e t o develop a more e f f i c i e n t numerical a l g o r i t h m ) . The p r e s e n t c a l c u l a t i o n s have b e e n p e r f o r m e d o n the SIEMENS-7890-F computer of the Hochschulrechenzentrum Dortmund.

References: 1. K.v.Klitzing, G. Dorda, M.Pepper, Phys.Rev.Letters 4 5 , 494 (1980) 2. For an overview see t h e contributions in: The Q u a n t i z e d Hall Effect, R.E.Prange and S.M.Girvin(eds.), G r a d u a t e T e x t s in C o n t e m p o r a r y Physics, S p r i n g e r - V e r l a g Berlin Heidelberg N e w - Y o r k 1987. P o t a review on localization in a m a g n e t i c field and the QHE see: T.Ando, Prog.Theor.Phys. Suppl. 8 ~ , 69 (1985) 3. R.E.Pelerls, Z. Phys. 8 0 , 763 (1933) 4. J.M.Luttinger, Phys.Rev. 8 4 , 814 (19S1) S. D.R.Hofstadter, Phys.Rev. B I 4 , 2239 (1976) 6. P.W.Anderson, Phys.Rev. 109, 1492 (19S8) 7. For a review on A n d e r s o n localization see: P.A.Lee, T.V.Ramakrishnan, Rev.Mod.Phys. 5 7 , 287 (1985) 8. H.Aoki, Phys.Rev.Letters SS, 1136 (198S); Proc.LITPIM Suppl. (PTB-PG-1), p.302,

L.Schweltzer and B.Kramer (eds.), PTB B r a u n s c h w e i g 1984 9. L.Schweitzer, B.Kramer, A.MacKinnon, Z. Physik B_5~9 379 (1985) 10. B.Kramer, private commounication 11. A.Basttn, C.Lewiner, O . B e t b e r - M a t i r e t , P.Nozieres, J.Phys.Chem.Solids 3 2 , 1811 (1971) 12. L.Smrcka, P.Streda, J.Phys. CIO, 21S3 (1977) 13. G.Czycholl, B.Kramer, Z. Physik B39 , 379 (198S) G.Czycholl, B.Kramer, A.MacKinnon, Z. Physik B43, S (1981) 14. D.J.Thouless, S.Klrkpatrlck, J.Phys. C14 , 23S (1981) 1S. B.Kramer, L.Schweitzer, A.MacKinnon, Z. Physik I356, 297 (1984) 16. G.Czycholl, W.Ponischowskl, t o be p u b l i s h e d 17. D J . T h o u l e s s , M.Kohmoto, M.P.Nightingale, M.den Nijs, Phys.Rev.Letters 4 9 , 405 (1982) 18. R.B.Laughltn, Phys.Rev. B23, $632 (1981)