8/14/2019 Many-electron magnetoconductivity in 2D electrons on liquid helium

1/4

E L S E V I E R Surface Science 361/362 (1996) 835-8 38

s u r f a c e s c i e n c e

M a n y - e le c t r o n m a g n e t o c o n d u c t iv i t y i n 2 D e le c tr o n son l iq u id h e l iu m

P . J. R i c h a r d s o n a, A . B l a c k b u r n b, K . D j e r f i " , M . I . D y k m a n c, C . F a n g - Y e n d ,P . F o z o o n i ~, A . K r i s t e n s e n " , M . J . L e a " '* , A . S a n t r i c h - B a d a l ~ Physics Department, Roya l HoUoway, University of London, Egham, Surrey T W2 0 OE X, U Kb Department o f Electronics and Computer Science, Southampton University, Southampton 80 17 IBJ , UK* Department o f Physics and Astronomy, Michigan State University, M I 48824, US A

d Department o f Physics, Stanford Un i~rsfly, Stanford, CA 94 305, US ARece ived 21 June 1995 ; accep ted fo r pub l ica t ion 8 Augus t 1995

The mag ne toco nduc t iv i ty o f ~ (B) o f nonde genera te two->1, there i s s hor t -r a nge o r de r i n t he e l e c t r on sys t e m a nd , f o r F > 127,a 2D e l e c t r on c r ys t a l f o r ms . A f unda me nta l que s -t i on i s t he e x t e n t t o w h ic h t he se i n t e ra c t i ons i n f lu -e n c e t h e c o n d u c t iv i t y . I t i s n o w k n o w n t h a t

* Cor respon d ing au tho r . Fax : +4 4 1784 472794 ;e - m a il : m . l e a ~ h b n c ~ c . u k .

f luc tua t ing inte rna l e lec t r ic f ie lds , magni tude E l ,c on t r o l t he d i f f us ion o f c yc lo t r on o r b i t s a n d h e nc ethe ma gne to r e s i s t i v i t y p (B ) [ 1 ] a n d m a g n e t o c o n -d u c t iv i t y ~(B ) [ 2 ] . W e p r e s e n t n e w m e a s u r e m e n t sof (B) be low 1 K in th e e lec t ron f luid (see Ref. E3]f o r som e p r e v ious w or k) , a n d ob t a in va lue s O f E l ,i n g o o d a g r e e m e n t w i t h M o n t e C a r l o s i m u l at io n s .T he t r a ns i t i on t o t he 2 D so l id i s a l so obse r ved .

2 . E xpe r ime n ta lT h e m a g n e t o c o n d u c t i v i t y w a s m e a s u r e d u s in g a4 r a m d i a m e t e r C o r b i n o d i s k ( se e F ig . 1 ) m a d e1L~r~g opt ica l l i tho grap hy an d the prec is ion devicef a b r ic a t io n t e c h n i q u e s o f th e S o u t h a m p t o n

0039-6028/96 /$15 .00 Cop yr igh t O 1996 E lsev ie r Sc ience B.V. Al l r igh ts re se rvedP H S 0 0 3 9 - 6 0 2 8 ( 9 6 ) 0 0 5 4 5 - 6

8/14/2019 Many-electron magnetoconductivity in 2D electrons on liquid helium

2/4

8 3 6 P.J. Richardaon et aL Surface S cience 361/362 (1996) 835-838

: ~ o . 1

x

0.01 , ~ . . . . . . . . . . . . . .0 . 1 1 10B ( T )

F i g . 1 . h e / f u r v e r s u s B f o r n - - 0 . 5 1 ( b ) , 1 . 2 1 ( e) , 1 .6 0 ( d ) a n d2 . 0 7 ( e) x 1 0 a m - 2 a t 4 k I -I z . T h e s o f i d l i n e s s h o w t h e m a n y -e l e c t r o n t h e o r y .

U n i v e r s i t y Mi c ro e l ec t ro n i c s Cen t r e [2 ] . A cen t r a le l ec t ro d e A w as s u r ro u n d ed b y a r i n g E w h i chs ep a ra t ed t h e an n u l a r r ece i v i n g e l ec t ro d e B i n t ot h r ee s eg men t s B1 , B2 an d B3 , a l l en c l o s ed b y ag u a r d e l e c t r o d e G . E l e c t r o n s w e r e p r o d u c e d b yg l o w d i s ch a rg e an d h e l d o n t h e h e l i u m s u r f ace( t y p i ca l l y 5 0 / ~m ab o v e t h e e l ec t ro d es ) b y D Cp o t en t i a l s . T h e g ap s i n t h e e l ec t ro d e p a t t e rn w ereo n l y 1 0 # m s o t h e e l ec t ro n s h ee t w as v e ry h o m o -g en eo u s . A n A C v o l t ag e Vo ( t y p ica l l y 1 -1 0 m V ) a ta f r eq u en cy f (= co/ 2n ) b e t w ee n 2 an d 7 0 k H z w asap p l i ed t o e l ec t ro d e A an d a(B) w as o b t a i n ed f ro mt h e p h as e o f t h e A C cu r r en t t o e l ec t ro d e B . T h ee l e c tr o n d e n s it y w a s f o u n d fr o m t h e - r e D C b i a sv o l t ag e o n e l ec t ro d e E n eed ed t o " cu t -o f f " t h i scurren t .

3 . Resu l t s and di scuss ion

3.1 . The Drude reg ionIn low f i elds , B < 0 .5 T , the d a ta fo r a (B ) fo l lowthe s imple Drude- l ike resu l t , even fo r c lass ica l ly

s t rong f i e lds wi th /~B < 500,~ ( 0 ) n ea(B) - I + (lzB--~ a n d ~ ~-. 2 f o r / z B > > I

( I )w h e r e ~ ( 0 ) = n ep ,. T h e m o b i l i ty a w a s o b t a i n edf r o m t h e B 2 d e p e n d e n c e o f o- - ] (B ) [ 4 ] i n g o o da g r e e m e n t w i t h m e a s u r e m e n t s a t B = O b yM eh ro t r a e t al. [ 5 ] . T h e m o b i l i t y i s d en s i t y d ep en -

den t be cause the ver t i ca l e l ec t ri c p ress ing f ie ld , andthe e lec t ron-r ipp lon coupl ing , increases wi th den-s it y . F i g . 1 s h o w s m eas u rem en t s o f ne/ laa(B) v e r s u sB for Severa l dens i t i es , us ing the exper imenta l/~ va lues . In the Dr ud e reg ion , the d a ta a l l l ie onthe l ine , n e / va (B ) = B 2 (l ine a) .Ini t ial ly , these resul ts seem to confirm a s ingle-p a r t i c l e ap p ro ach t o mag n e t o co n d u c t i v i t y . I n f ac tt h ey co me ab o u t t h ro u g h e l ec t ro n - - e l ec t ro n i n t e r -ac t i o n s . F i r s t , t h e man y -e l ec t ro n co r r e l a t i o n t i mei s l es s than the re laxat ion t ime z f rom scat t er ingb y 4 H e v a p o u r a t o m s a n d r i p pl o n s [ 6 ] . T h i sch an g es t h e av e rag i n g o v e r t h e Bo l t zman n d i s -t r i b u t i o n an d , f o r t h e en e rg y -d ep en d en t r i p p l o ni n t e r ac t i o n , d ec reas e s t h e mo b i l i t y , co mp ared t oi n d ep en d en t e l ec t ro n s , i n s a t i s f ac t o ry ag reemen tw i t h ex p e r i men t . T h e ag reemen t o f t h e ex p e r imen -tal /~ valu es in zer o a nd classical ly s t ro ng (/~n >> 1)f i e lds impl ies tha t the re laxat ion ra te i s indep end en tof magnet i c f i e ld , z - l ( O ) = z - l ( B ) e v e n w h e r eL an d au l ev e l q u an t i s a t i o n ( en e rg y s p ac i n g fu o ~ ,o ~ = e B / m i s t h e cy c l o t ro n f r eq u en cy ) mi g h t b eex p ec t ed t o ch an g e t h e d en s i t y o f s t a te s an d t h escat t er ing ra te fo r quas i -e las t i c sca t t er ing . But thef luc tuat ing in ternal e l ec t r i c f i e ld Ef p roduces anen e rg y u n ce r t a i n ty , e E f : ~ T o v e r an e lec t ron ic ther -ma l w av e l en g t h ; t , r an d , f o r kT>>eE-gr>hoJo ,s mea r s o u t t h e L an d au l ev e l s . T h i s a l s o h o l d s fo re E f R o > h o Jo ( R e = q--~-k-T/eB is the c lass ica l cyclo -t ro n o rb i t r ad i u s) , w h i ch co r r e s p o n d s t o B < Bo ,an o n s e t fi e ld fo r d ev i a ti o n s f ro m t h e D ru d e m o d e l( t y p i ca ll y 0 .2 t o 1 T ) . T h e D ru d e m o d e l i s fo l l o w edbec ause o f the in ternal e l ec t r i c f ie lds , and the m any -e lec t ron theo ry g ives ne / /m = B 2 fo r / tB >> 1 , inde-pen den t o f n , in th i s reg ion .3.2. M any -elec tron magnetoconduct ivi ty

A dis t inc t ive fea tu re in Fig . I i s tha t the co nduc -t i v it y s a t u r a t e s ab o v e t h e D ru d e r eg i o n . Mo reo v e r ,t h e p l o t s o f ne/lur(B) are t h en d en s i t y an d t em-p e ra t u r e d ep en d en t . T h e en e rg y u n ce r t a i n t yeEtR~

8/14/2019 Many-electron magnetoconductivity in 2D electrons on liquid helium

3/4

P.J. Richardaon et , ,I /Surface Sciau~ 361/362 (1996) 835-838 83 7where K is a dimensionless factor, depending onthe scattering mechanism. For short range scatter-ing, such as 4He atoms, K m K s ( h o 4 / k T ) re fl ec tsthe change in the cyclotron orbit radius, and hencethe diffusion length, from R~ for t ic%/kT> 1 , K s = 4 ( t w ~ J k T ) - ' / 2 / ~ . Fo r dpplonsK - Kr depends on hcoo /kT in the same way butthe interaction strength also changes with field asthe domina nt scattering ripplon wavelength is pro-portional to the cyclotron orbit radius. This extrafactor is numerically close to unity in these experi-ments. Thus, for hCOo/kT= 1.34B/T>> 1, ne/p.a(B)decreases with field as seen in Fig. 1 above 2 T. Animportant conclusion is that, above the Druderegion, a - l ( B ) i s proportional to the internalelectric field El.3.3. Th e internal electric f ieid

For a classical electron fluid with thermal fluc-tuations, the mean-square internal electric field hasa nearly Gaussian distribution and can be writtenas:

n3/2kT F 1-')Ef = - ~ ( = E ~ F ( F ) , (3)where Eo sets the scale for the internal fields t r e ng t h . T h e d i m e n s i o n l e s s f t m c t i o n F ( / " ) i s p l o t -t e d in Fig. 2, from Monte Carlo computer simula-tions [8], for 10 k T , using theinternal fields from Fig. 2, as shown in Fig. 1 forseveral densities at 0.6 K. As B increases, the datacross over from the many-elect ron low field theory(equivalent to the Drude model, line a) to a densitydependent behaviour, Eq. (2) (lines b, c d and e).Conversely, the measured ~-I(B), at B = 2 T ,was used to obtain experimental values of theinternal electric field Ef as plotted in Fig. 3 versusEo from Eq.(3). The points come from over40 combinations of density and temperaturebetween 0.6 and 0.9 IC Within the errors t hemeasured field E f = v E o w i t h v=3.11 +0.10 is inclose agreement with v = ~/F = 3.07 + 0.03 from theMonte Carlo ca lculations for 20 < F < 70.3.4 . Th e 2D electron sol id

At lower temperatures, the 2D electron systemforms a classical solid below Tm=0.225x10-6n 1/2 K. The zero-field mobil ity decreases atthe transition [5] but little is known about themagnetotransport. Hysteresis in the highly non-linear magnetoconductivity has been interpre tedas shear-induced melting [9] or the decoupling of

2 0 0 0I I I I 1

o .O K o ~ v . 8 O ~ v / / /5 0 0

o X -] 0 .7o o . s o ~2"looo -

5 0 0

0 I I I I I0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 8 0 0

g 0 (V/m)F i g 3 . T h c i n t e r n a l f i el d E t , m c a m ~ c d f o r a r a n g e o f e l e c t r o ndc ~t i ~ a t 0 .6 , 0 .7, 0 .8 and 0 .9 K , vcrsm the re fc~ ncc f ie ld Eo .

8/14/2019 Many-electron magnetoconductivity in 2D electrons on liquid helium

4/4

83 8 P . J . R i c h a r d so n e t a S u r f c e S c ie n c e 3 6 1 /3 6 2 ( 1 9 9 6 ) 8 3 . $ - 8 3810lO

l o 7

07"0.1B(T)

'(b)' " '!

o . i ' o!4r (K)F i g . 4 . ( a) 1 / a ve r s u s B a t 0 .2 5 a n d 0 .I K f o r n = 0 . 8 x 1 0 ~ m - 2 ,(b) 1/ a versus T at 0=2 an d 2 T for n = 0.9 x 10u m -2 .t h e c r y s t a l f r o m co m m en su r a t e " d im p les" in t h eh e l iu m su r f ace F10 ] . F ig . 4 a sh o ws a - l ( B ) in t h ef lu id a t 0 .25 K (good agree me nt wi th Eqs. (1 ) an d( 2 )) an d in th e so l id a t 0 .1 K f o r n =0 .8 0 x1012 m -2 (Tin= 0 .1 9 K) , a s su m in g th a t t h e p h asesh i fts a re d ue to ma gne toco ndu ct iv i ty (i_e . neg lec t -in g Lo r en tz f o r ce co u p l in g to sh ea r m o d es in t h ec r y s t al ) . Th e m an y - e l ec t r o n a (B ) i s n o t ex p ec tedto ch an g e su b s t an t i a l ly a t t h e t r an s i t i o n f r o m ah ig h ly co r r e l a t ed f lu id to a c r y s t a l f o r co m p ar a -t ive ly h igh f ie lds where 2 n t / 2 h n F / m ~ o o < < 1 [7 ,11] .This i s seen in Fig . 413 wh ere the cha nge in a-I(B,T)i n l o w f i e ld s co r r e sp o n d s to a c h an g e in m o b i l i t y[ 5 ] , t h o u g h th e B 2 Dr u d e r eg io n is n o lo n g e r c l ea rexper im enta l ly a t 0 .1 K. H owe ver , a t h igh er f ie ldsthe t ran si t ion (a t the sa me Tin) i s mu ch less p ro-

n o u n c e d . T h e m a n y - e l e c t ro n t h e o r y m u s t n o w b eex ten d ed to in c lu d e th e r eg io n wh e r e th e e l ec t r o n

m o t io n in th e f i e ld c r ea t ed b y th e o th e r e l ec t r o n si s q u an t i sed , an d th e f i e ld E f is n o n - u n i f o r m o v e rth e e l ec t r o n wav e len g th .

A c k n o w l e d g e m e n t sT h i s w o r k w a s s u p p o r t e d b y t h e E P S R C , U K ,a n d E U C o n t r a c t C H R X C T 9 3 0 3 7 4 .

R e f e r e n c e s[1 ] M. I . Dyirm An; M J. Lea , P . Foz oon i and J . F ros t , Phys .

Rev. Lett . 70 (1993) 3975.[ 2 ] M J . L e a , P . F o z o o n i , P J . R i c h a r d s o n a n d A . B l a c k b u rn ,Phys. Rev. Left. 73 (1994) 1142.[33 Yu .Z. Ko vdry a e t a l. , J . Low Tem p. Phys . 91 (1993) 371 ;

J . Neuenschwander , W. Joss and P . Wyder , He i r . Phys .Ac ta 65 (1992) 32 5; Phy sica B 194 (1994) 1231.14] Y . Iye , J . Low Tem p. Phys . 40 (1980) 441 .[ 5 ] I L M e h r o t r a , C J . G u o , Y . Z . R u a n , D . B . M a s t a n dA.J. Dahm; Phys. Rev. B 29 (1984) 5239.[6 ] M. S a i toh , J . Phys . Soc . Jpn . 42 (1977) 201 ; So l id S ta teCom m un. 52 (1984) 63 ;V .A . Bun ta r ' e t aL , J . Low Tem p. Phys . 79 (1990) 323 .17] M.L Dylrm a, and L .S . Khazan , Zh . Eksp . Teor . F iz . 77(1979) 1488 [Soy . Phys . JET P 50 (1979) 747 ;M.I . Dykm an, J . Phys . C 15 (1982) 7397 ;M.I . Dykm an e t a l . , to be pub l ished .1,8"1 M I Dy km an a nd C . Fang-Yen , to be pub l ished .191 L. W'tlen and R. Giannetta, J . Phys. Soc. Jpn. (1988).[ 1 0 ] K . S h i r a h a m a a n d I C K o n o , P h y s . R e v . L e t t. 7 4

(1995) 781.I '11"1 M. S aitoh, J . Ph ys. Soe. Jpt~ 56 (1987) 706.