TWO DIMENSIONAL ELECTRON LOCALIZATION

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TWO DIMENSIONAL ELECTRON LOCALIZATION

M. Pepper

To cite this version:

M. Pepper. TWO DIMENSIONAL ELECTRON LOCALIZATION. Journal de Physique Colloques,

1981, 42 (C4), pp.C4-17-C4-25. �10.1051/jphyscol:1981402�. �jpa-00220706�

TWO D I M E N S I O N A L ELECTRON L O C A L I Z A T I O N M. Pepper*

Cavendish Laboratory, Cambridge, U. K.

Abstract.- This paper gives a s h o r t review o f r e c e n t experimental work on weak l o c a l i z a t i o n i n s i l i c o n i n v e r s i o n l a y e r s . I t i s shown t h a t both t h i s e f f e c t and e f f e c t s a r i s i n g from t h e e l e c t r o n - e l e c t r o n i n t e r a c t i o n a r e present. These mechanisms can be d i s t i n g u i s h e d by t h e i r response t o a magnetic f i e l d , which enables a c o n f i r m a t i o n o f t h e p r i n c i p a l t h e o r e t i c a l p r e d i c t i o n s . The l o c a l i z a t i o n i s c u t o f f by a magnetic f i e l d which leads t o the r e t u r n o f m e t a l l i c conduction, and, i t i s suggested, a minimum m e t a l l i c conductance.

I n t r o d u c t i o n . - Two types o f two dimensional system have been o f use i n l o c a l i z a t i o n studies, these a r e t h i n metal f i l m s ( 1 ) and the i n v e r s i o n , o r accumulation, l a y e r a t a semiconductor surface ( 2 ) . The p r i n c i p a l d i f f e r e n c e between these systems i s t h a t t h e c a r r i e r c o n c e n t r a t i o n i n t h e metals (gated s t r u c t u r e s ) can be a l t e r e d by o n l y a small p r o p o r t i o n . On t h e o t h e r hand t h e semiconductor systems possess a lower c a r r i e r c o n c e n t r a t i o n ( a maximum o f about 2 . 1 0 ~ ~ e - c m - ~ ) b u t t h i s can be decreased c o n t i n u o u s l y t o a n e g l i g i b l y small value.

A considerable amount o f work has been performed on t h e n a t u r e o f l o c a l i z a t i o n i n s i l i c o n i n v e r s i o n l a y e r s (2, 3, 4). The main p o i n t s which have been e s t a b l i s h e d a r e as follo\vs.

1. A m o b i l i t y edge, Ec, separates extended s t a t e s from l o c a l i z e d s t a t e s i n t h e two dimensional band t a i 1. M o d i f i c a t i o n s t o t h e concept o f t h e 2D m o b i l i t y edge w i l l be discussed l a t e r .

2. The temperature dependence o f t h e conductance, a, i s i n agreement w i t h M o t t ' s p r e d i c t i o n t h a t , when s t a t e s o f t h e Fermi energy are l o c a l i z e d , conduction i s by e x c i t a t i o n t o , Ec, passing i n t o v a r i a b l e range hopping ( a = exp

-

( T , / T ) ~ / ~ as t h e temperature i s reduced.

3. When t h e Fermi energy, EF, i s a t Ec t h e conductance i s n o t l e s s than

2

0.1 e2/4i, ( 3 . 1 V 5 n-I), t h e minimum m e t a l l i c conductance, although values g r e a t e r than t h i s have been obtained. M e t a l l i c conduction a t values o f a below omi f o r a p a r t i c u l a r specimen are n o t found. T h i s increase i n t h e value o f omin has geen discussed i n terms o f an increase i n the l e n g t h o f t h e p o t e n t i a l f l u c t u a t i o n s ( 2 ) . 4. The l o c a t i o n o f Ec increases as t h e l o c a l i z e d s t a t e s a r e occupied, implying, t h a t , as i n the semiconductor i m p u r i t y band, b o t h d i s o r d e r and t h e e l e c t r o n - e l e c t r o n i n t e r a c t i o n a r e r e s p o n s i b l e f o r t h e l o c a l i z a t i o n .

R e c e n t l y , t h e o r e t i c a l arguments have been advanced suggesting t h a t a l l s t a t e s a r e l o c a l i z e d i n t v ~ o dimensions (Abrahams e t a l . (5),Gorkov e t a l . (6),

Houghton e t a l . ( 7 ) , Haydock (8), Kaveh and H o t t ( 9 ) . On these arguments t h e m o b i l i t y edge i s a l o c a l i z a t i o n edge s e p a r a t i n g two d i f f e r e n t types o f l o c a l i z e d s t a t e s . These suggestions, and the experiments which t h e y s t i m u l a t e d , w i l l now

"On leave from Plessey Research, Caswell, Towcester, Northants, U.K.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1981402

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be discussed.

Weak l o c a l i z a t i o n and i n t e r a c t i o n e f f e c t s i n 2D.

-

Abrahams and co-workers ( 5 ) f i r s t suggested t h a t a l l s t a t e s i n 2D a r e l o c a l i z e d w i t h t h e m o b i l i t y edge being a l o c a l i z a t i o n edae seoaratina s t r o n ~ l v

- "

and weaklv l o c a l i z e d s t a t e s . Thev f i n d t h a t the conductance-of a' 2D sysEem i s

( t h i s formula was subsequently obtained elsewhere (6, 3 ) ) . oQ i s the normal conduct- ance, ne2r/m, and t h e l o g a r i t h m i c term a r i s e s from t h e l o c a l i z a t i o n . A t zero Kelvin, L i s t h e specimen l e n g t h , b u t a t f i n i t e temperatures L i s t h e i n e l a s t i c d i f f u s i o n l e n g t h ( ~ ~ , $ ) 3 , LIN and L a r e t h e i n e l a s t i c and e l a s t i c mean f r e e paths r e s p e c t i v e l y . The p r e - l o g a r i t h m i c term i s s t r i c t l y v a l i d f o r small changes, a i s a constant pos- sessing values 1 o r

3

deprnding on t h e s t r e n g t h o f s p i n f l i p s c a t t e r i n g . As

" v a r i e s as T P / ~ where LTN v a r i e s as TP t h e temperature dependence o f t h e change o f u , 6 0 , i s

60 =

-

const. Ln T

Kaveh and M o t t have suggested t h a t a t zero K e l v i n t h e l o c a l i z a t i o n edge separates exponential band t a i l s t a t e s from s t a t e s which a r e power law l o c a l i z e d ,

$ % l / r e x p ( i k . r ) . A l t e r n a t i v e forms o f l o c a l i z a t i o n have been proposed by Pichard and SZriiia (10) and Kaydock (8). However, a t f i n i t e temperatures i t i s suggested t h a t t h e power law l o c a l i z a t i o n i s converted i n t o exponential l o c a l i z a - t i o n and here t h e decay l e n g t h i s t h e i n e l a s t i c d i f f u s i o n l e n g t h .

Recently Hodges (11) has explored t h e connection between these quantum treatments and t h e e a r l i e r , c l a s s i c a l , theorem o f Polya (12) t h a t t r u e two dimen- s i o n a l d i f f u s i o n does n o t occur.

Another t h e o r y which p r e d i c t s the same temperature dependence has been pro- posed by A l t s h u l e r , 'ronov and Lee (13) and i s based on t h e t h r e e dimensional work o f A l t s h u l e r and Aronov (14). Here, i n t e r f e r e n c e between e l a s t i c s c a t t e r i n g and t h e e l e c t r o n - e l e c t r o n i n t e r a c t i o n leads t o a s i n g u l a r i t y i n the d e n s i t y o f s t a t e s a t EF. This r e s u l t s i n a c o r r e c t i o n t o t h e conductance given by

Where the f a c t o r F a r i s e s from e l e c t r o n screening and tends t o zero o n l y a t t h e h i g h e s t values o f c a r r i e r concentration, n. F o r low values o f n i t i s approximately given by,

where K i s t h e 2D screening l e n g t h .

Kaveh and M o t t (18) have obtained t h e r e s u l t o f A l t s h u l e r e t a l . by a non- diagrammatic technique. They p o i n t o u t t h a t the two c o r r e c t i o n s should add, t h e l o c a l i z a t i o n a f f e c t i n g the d i f f u s i v i t y and t h e i n t e r a c t i o n a f f e c t i n g t h e d e n s i t y o f s t a t e s . From t h e temperature dependence o f conductance i t i s o n l y p o s s i b l e t o separate t h e l o c a l i z a t i o n and i n t e r a c t i o n mechanisms i f F and t h e i n e l a s t i c s c a t t e r - i n g mechanism a r e known p r e c i s e l y . However, t h e magneto-resistance and Hal 1 e f f e c t o f f e r a more p r e c i s e method o f d i s t i n g u i s h i n g between t h e two (16, 17).

Logarithmic c o r r e c t i o n s , magneto-resistance and H a l l e f f e c t . -

a) Magneto-resi stance

I n t h e presence o f a magnetic f i e l d , B, t h e c y c l o t r o n length, LC, becomes a l e n g t h s c a l e a f f e c t i n g the conductance. The a p p r o p r i a t e c o r r e c t i o n i s

e2 4 e B ~

6 0

C$(3

+ fi/4eB.rI,,D) + L n

(-)I.

( 3 )

where L D t h e new l e n g t h scale i s given by

This approximation agrees w e l l w i t h equation 3 except when LC >> L where i t f a i l s t o capture t h e B2 dependence o f t h e conductance c o r r e c t i o n .

Thus, as B increases LD becomes s v a l l e r and a negative magneto-resistance i s found. Even i f o t h e r mechanisms dominate t h e temperature dependence o f

resistance, t h e negative magneto-resistance w i l l be found i f t h e o t h e r s c a t t e r i n g processes a r e n o t s e n s i t i v e t o small values o f 6.

Recently i t has been suggested by Lee and Ramakrishnan ( P r i v a t e communication, t o be published) t h a t a magnetic f i e l d a f f e c t s t h e i n t e r a c t i o n mechanism when

ggB >> kT, t h e f a c t o r 2

-

2F being t u r n e d i n t o 2

-

^F. This increased t h e magnitude

o f t h e l o g a r i t h m i c c o r r e c t i o n p a r t i c u l a r l y a t low values o f n. As t h e l o c a l i z a t i o n i s a f f e c t e d through o r b i t a l motion, t h e n e g a t i v e magneto-resistance w i l l be

dependent on t h e component o f B normal t o t h e plane o f conduction (19). On t h e o t h e r hand, t h e enhancement o f t h e i n t e r a c t i o n c o r r e c t i o n i s a s p i n e f f e c t and so i s o n l y dependent on t h e magnitude o f B, n o t the d i r e c t i o n .

b) H a l l e f f e c t

I n a s i m i l a r manner t h e H a l l constant, R R = uxy/02x) o f f e r s a means o f d i s t i n g u i s h i n g between t h e two regimes. Futuybllka((~0) showea i n t h e l o c a l i z a t i o n case the c o r r e c t i o n i n ox i s t w i c e t h a t of u

,,

thus t h e r e i s no c o r r e c t i o n i n RH. However, A l t s h u l e r e$ a l . (13) suggest t t a t f o r t h e i n t e r a c t i o n regime t h e r e i s no change i n axy o n l y a,,, consequently the change i n RH i s r e l a t e d t o t h e change i n r e s i s t a n c e R by

O r , a1 t e r n a t i v e l y , f o r l o c a l i z a t i o n t h e Hal 1 m o b i l i t y , UH, decreases l o g a r i t h m i c a l l y w i t h decreasing temperature, whereas i n t h e i n t e r a c t i o n regime p~ increases w i t h the l o g a r i t h m o f decreasing temperature.

Experimental i n v e s t i g a t i o n s . - The f i r s t observation o f t h e l o g a r i t h m i c c o r r e c t i o n was by Dolan and Osheroff (1) who i n v e s t i g a t e d t r a n s p o r t i n t h i n metal f i l m s . Bishop, Tsui and Dynes (21) showed t h a t t h e c o r r e c t i o n was present i n i n v e r s i o n l a y e r s , t h i s being the e x p l a n a t i o n o f t h e weak temperature dependence i n t h e m e t a l l i c regime present i n e a r l i e r work b u t n o t i n v e s t i g a t e d , i.e. Adkins, P o l l i t t and Pepper (22) F i g u r e 3. As mentioned p r e v i o u s l y , i n t h e absence o f d e f i n i t e knowledge o f t h e i n e l a s t i c s c a t t e r i n g mechanism and t h e value o f F, conductance measurements cannot d i s t i n g u i s h between t h e l o g a l i z a t i o n and i n t e r a c t i o n regimes.

However, i t was shown by Uren, Davies and Pepper (23) t h a t a combination o f measurements as a f u n c t i o n o f e l e c t r i c and magnetic f i e l d s c o u l d d i s t i n g u i s h between t h e mechanisms, and t h a t b o t h were present. T h i s work and t h a t o f these authors w i t h Kaveh w i l l now be discussed.

Anderson e t a l . (24) i n i t i a l l y p o i n t e d o u t t h a t as t h e e l e c t r o n temperature Te v a r i e d w i t h e l e c t r i c f i e l d , F, as Fx, i n f o r m a t i o n on t h e electron-phonon c o u p l i n g can be obtained by a combination o f e l e c t r i c f i e l d and temperature. T h i s t o p i c w i l l n o t be discussed here b u t t h e measurement o f t h e l o g a r i t h m i c c o r r e c t i o n as a f u n c t i o n o f e l e c t r i c f i e l d i s . a v e r y convenient experimental technique. F i g u r e 1 shows t h e l o g a r i t h m i c g r a d i e n t found by Uren, Davies and Pepper as a f u n c t i o n o f m a g n e t i c f i e l d , B. Here t h e g r a d i e n t i s expressed as t h e change o f conductance p e r decade change o f e l e c t r i c f i e l d (due t o two l e n g t h scales being i m p o r t a n t i n t h e presence o f a

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Figure 1. The "Quasi-Loaarithmic" slope i s plotted against maonetic f i e l d ,

^{B ,}

(Tesla). The slope i s the r e s i s t a n c e chanoe per decade change i n e l e c t r i c f i e l d with the l a t t i c e a t - 85 mK, L was 19 nm. The c i r c u l a r and square points a r i s e from localization and interactions respectively, from Uren, Davies and P e ~ n e r (23).

magnetic f i e l d (equations

4

and

5 )

the dependence of

80

on e l e c t r i c f i e l d i s

"quasi-logarithmic"). I t i s seen t h a t the gradient i n i t i a l l y decreases

w i t h

increasing B and then increases, f i n a l l y staying f a i r l y constant. The magneto- resistance in the two regimes showed d i f f e r e n t behaviour, being negative when the gradient decreased with

B

and positive when i t increased with

B.

I t thus appeared t h a t the magnetic f i e l d was distinguishing between the two mechanisms. This i n t e r p r e t a t i o n was enhanced by the f a c t t h a t the second ( i n t e r a c t i o n ) mechanism was l e s s s t a b l e against increasing e l e c t r i c f i e l d , enabling extraction of the gradients f o r the two processes a t a p a r t i c u l a r value of magnetic f i e l d . From analysis of the dependence on electron temperature i t appeared as i f the condition f o r the magnetic f i e l d enhancement of the interaction process was gfiB

> kTe,

where g i s the g value and

6

i s the Bohr magneton. Uren, Davies and Pepper present r e s u l t s i l l u s t r a t i n g the decreased s t a b i l i t y of the interaction mechanism against an increase i n temperature.

Confirmation t h a t the magnetic f i e l d produced the interaction regime was pro- vided by measurements of the Hall e f f e c t . The r a t i o

BRH/RY =

2d9/? was found i n t h e interaction (second) regime, t h i s r e s u l t was obtained by Bishop, Tsui and Dynes, (25) although they did not separate the two mechanisms. The gradual increase in the r a t i o towards

2

as ~ F L increases i s shown i n Figure 2. These r e s u l t s on the r o l e of interactions were f o r values of c a r r i e r concentration such t h a t F was

s

0.85.

However, the magnitude of t h e pre-logarithmic f a c t o r was considerably greater than

(2-2F) and was consistent with 2-F, in agreement with the recent suggestion of Lee

and Ramakrishnan. Confirmation t h a t t h e magnetic enhancement of the interaction

regime i s a spin e f f e c t , and not related t o o r b i t a l motion, was provided by Davies,

Uren and Pepper

( 2 6 ) .

These authors found l i t t l e dependence in the magnitude of the

interaction correction on t h e direction of the magnetic f i e l d .

specimen r e s i s t a n c e i s shown a t t h e top. The c i r c u l a r p o i n t s were taken u s i n g a sample w i t h L = 360 nm, t h e closed c i r c l e s and open c i r c l e s are w i t h e l e c t r o n temperature determined by e l e c t r i c f i e l d and ambient temperature r e s p e c t i v e l y . The crosses were obtained w i t h t h e specimen used f o r F i g u r e 1, from Uren, Davies and Pepper (23).

The work o f Davies e t a l . a l s o showed t h a t i t was p o s s i b l e t o o b t a i n m e t a l l i c behaviour a t 50 mK by completely s e p a r a t i n g t h e i n t e r a c t i o n and l o c a l i z a t i o n c o r r e c - t i o n s . This was achieved by u s i n g specimens w i t h a l o n g e r e l a s t i c mean f r e e p a t h than Uren e t a1

.

The dependence o f r e s i s t a n c e on e l e c t r i c f i e l d f o r v a r i o u s values o f magnetic f i e l d i s i l l u s t r a t e d i n F i g u r e 3. I t i s seen t h a t t h e magnetic f i e l d i n i t i a l l y reduces and then e l i m i n a t e s t h e l o g a r i t h m i c c o r r e c t i o n , and m e t a l l i c behaviour i s found f o r values o f e l e c t r o n temperature down t o 50 mK. ( A t h i g h values o f e l e c t r o n teaoerature, a 1 K, t h e r e s i s t a n c e again increases w i t h i n c r e a s - i n g e l e c t r i c f i e l d . This i s due t o a change i n screening w i t h e l e c t r o n temperature and has no relevance t o t h i s work.) F u r t h e r increase i n B r e s u l t s i n a ~ o s i t i v e magneto-resistance and t h e r e t u r n o f t h e l o g a r i t h m i c c o r r e c t i o n , although here i t i s due t o i n t e r a c t i o n s r a t h e r than l o c a l i z a t i o n . The complete s e p a r a t i o n o f t h e

mechanisms a r i s e s from t h e increased e l a s t i c mean f r e e path L and a correspondingly enhanced d i f f u s i o n l e n g t h L.

A

s m a l l e r value o f 6 i s r e q u i r e d f o r t h e c y c l o t r o n l e n g t h LC t o become considerably s m a l l e r than L, so causing t h e disappearance o f t h e l o g a r i t h m i c dependence on e l e c t r o n temperature b e f o r e B i s s u f f i c i e n t l y h i g h t o induce the i n t e r a c t i o n mechanism. However, t h e n e g a t i v e r e s i s t a n c e i s s t i l l obtained, t h i s o n l y disappears when LC i s near 1. The g r e a t e r s t a b i l i t y o f t h e l o c a l i z a t i o n mechanism i s apparent as a n e q a t i v e magneto-resistance i s observed when t h e e l e c t r i c f i e l d i s increased s u f f i c i e n t l y t o erase t h e i n t e r a c t i o n c o r r e c t i o n .

Another m a n i f e s t a t i o n o f t h e decreased l o c a l i z a t i o n l e n q t h s c a l e i s t h a t , as t h e magnetic f i e l d increases, a h i q h e r v a l u e o f e l e c t r i c f i e l d i s r e q u i r e d t o induce t h e l o g a r i t h m i c behaviour. I n c r e a s i n g B, hence decreasing LC, n e c e s s i t a t e s a s m a l l e r v a l u e o f L, and g r e a t e r v a l u e o f e l e c t r i c f i e l d , f o r t h i s t o determine t h e l e n g t h scale.

Kaveh e t a l . (18) have shown t h a t t h e l o g a r i t h m i c c o r r e c t i o n i s determined by t h e s h o r t e s t l e n g t h a v a i l a b l e , regardless o f i t s o r i g i n . The l e n g t h can be t h e c y c l o t r o n l e n g t h o r the i n e l a s t i c length, depending on t h e values o f magnetic f i e l d and e l e c t r o n temperature as i s shown i n F i g u r e 4.

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F i g u r e 3. Sheet r e s i s t a n c e i s n l o t t e d a o a i n s t t h e l o q a r i t h m o f e l e c t r i c f i e l d L = 360 nm, l a t t i c e temperature = 50 mK, c a r r i e r c o n c e n t r a t i o n = 3.8 x 1011 cW2. The values o f magnetic f i e l d i n Tesla a r e A = 0, B = 0.008, C = 0.021,

D

= 0.047, E = 0.074, F = 0.1,

G

= 0.15, H = 0.26,

I

= 0.32, 3 = 0.52, from Davies, Uren and Pepper (26).

E x t r a c t i o n o f t h e i n e l a s t i c l e n q t h from t h e magneto-resistance has allole~ed in v e s t i - g a t i o n o f t h e temperature dependence o f t h e i n e l a s t i c s c a t t e r i n g r a t e (27, 28). It has been found t h a t t h e T2 law i s m o d i f i e d by t h e e l a s t i c s c a t t e r i n g and a compon- e n t v a r y i n g w i t h a lower power o f temperature i s a l s o present, F i g u r e 5. This i s i n q u a l i t a t i v e agreement w i t h t h e o r e t i c a l c o n s i d e r a t i o n s suqgesting a T component due t o t h e b l u r r i n g o f e l e c t r o n momentum when

~ F L

i s near one.

F i n a l l y , v e r y r e c e n t l y Poole, Pepper and Glew (29) have i n v e s t i g a t e d t h e s i t u a t i o n where t h e screening f a c t o r F i s near 0.5, and both i n t e r a c t i o n and l o c a l i - z a t i o n effects are present i n t h e absence o f a magnetic f i e l d . The l o c a l i z a t i o n i s suppressed by t h e a p n l i c a t i o n of a magnetic f i e l d , and w h i l s t t h e r e s i s t a n c e

decreases w i t h B t h e H a l l constant does n o t vary

-

i n accordance w i t h t h e suqgestion of Fukuyama (20). However, when t h e l o c a l i z a t i o n i s suppressed, a l o g a r i t h m i c c o r - r e c t i o n i s s t i l l found as a f u n c t i o n o f temperature. T h i s i s due t o t h e i n t e r a c t i o n mechanism and now t h e H a l l r a t i o o f 2 i s found. These r e s u l t s c o n f i r m t h e sugges- t i o n o f Kaveh and Flott (8, 15) t h a t t h e i n t e r a c t i o n and l o c a l i z a t i o n c o r r e c t i o n s a r e a d d i t i v e .

i n e l a s t i c l e n g t h determined by e l e c t r o n h e a t i n q and l a t t i c e h e a t i n g r e s p e c t i v e l y . The dashed l i n e i l l u s t r a t e s t h e r e s u l t s exnected i f t h e l e n g t h was determined by t h e energy change o f t h e e l e c t r o n acquired from t h e f i e l d d u r i n g a d i s t a n c e

1,

a s i t u a t i o n which was n o t achieved; from Kaveh, Uren, Davies and Pepper, (18).

Conclusion.- Good agreement now e x i s t s between experiment and t h e p r e d i c t i o n s o f t h e l o c a l i z a t i o n and i n t e r a c t i o n t h e o r i e s . By t h e a p p l i c a t i o n o f a magnetic f i e l d , o r a combination o f magnetic and e l e c t r i c f i e l d s , t h e two mechanisms can be

separated and c l e a r l y d i s t i n g u i s h e d by t h e behaviour o f t h e H a l l e f f e c t and

magneto-resistance. I t i s p o s s i b l e t o o b t a i n m e t a l l i c behaviour a t temoeratures as low as 50

mK,

althouqh i t i s n o t c l e a r i f t h i s s i t u a t i o n i s maintained down t o absolute zero. E v e n t u a l l y kT becomes s m a l l e r than q$R and t h e i n t e r a c t i o n correc- t i o n w i l l become anparent. Holvever, i f t h i s mechanism i s o n l y a ~ e r t u r b a t i o n m e t a l l i c behaviour w i l l be obtained a t 0 K and t h e minimum m e t a l l i c conductance w i l l be observed. The r o l e o f the magnetic f i e l d i n r e s t o r i n g m e t a l l i c c o n d i t i o n w i l l r e s u l t i n a f i e l d induced m e t a l - i n s u l a t o r t r a n s i t i o n .

I n t h e presence o f i n t e n s e maanetic f i e l d s these mechanisms w i l l n o t have s i g n i f i c a n c e f o r t h e quantized H a l l r e s i s t a n c e (30), here oxX i s n e g l i g i b l y small and ox i s u n a f f e c t e d by i n t e r a c t i o n s . However, as t h e Fermi energy i s passed throu& t h e Landau l e v e l s a whole s e r i e s o f metal i n s u l a t o r t r a n s i t i o n s w i l l be obtained. As described i n e a r l i e r work (31) t h e c y c l o t r o n o r b i t w i l l d e l o c a l i z e e l e c t r o n s when t h e l e n g t h scale o f t h e p o t e n t i a l f l u c t u a t i o n s i s long.

Acknowledgements.- The work reviewed here was performed i n c o l l a b o r a t i o n w i t h W.A. Davies, M. Kaveh, D.A. Poole and M.J. Uren and w i l l be f u l l y described

elsewhere. Throughout t h i s work we have enjoyed many discussions w i t h

Professor S i r Nevi11 Mott. T h i s work was supported by t h e Science Research Council and, i n p a r t , by t h e European Research O f f i c e o f t h e U.S. Army.

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