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VOLU M E 92, N U M BER 1

PH YSICA L R E VI E W L E T T E RS

week end i ng 9 JA N UARY 20 0 4

Topolog ica l Osci l lat ions of t he Mag net o conducta nce i n Disordere d Ga As Layers
S. S. Mur z i n ,1,4 A. G. M. Ja n sen ,
1 2

2, 4

a nd I. Claus3,

4

I n stit ut e of Sol id St at e Physics R AS, 142 432 C her nogolovka , Moscow Dist r ict , Russia Д ДД ` Ser v ice de Physique St at ist ique , Magnet isme, et Supraconduct iv it e , Depa r t ement de Re cherche Fonda ment a le sur la Mat iere Д Conden see , C E A -G renoble , 38054 G renoble CE DE X 9, France 3 Д Д Cent er for Non l i nea r Phenomena a nd Complex Syst em s, Facult e des Sciences, Un iversit e Libre de Br uxel les, Ca mpus Pla i ne , Co de Post a l 231, B-1050 Br ussels, Belg iu m 4 ? ? G renoble Hig h Magnet ic Field Laborator y, Ma x - Pla nck -I n stit ut f u r Fest kor p er forschung a nd Cent re Nat iona l de la Re cherche Scient ifique , BP 166 , F-380 42 , G renoble C EDE X 9, France ( Re ceived 22 May 20 03; publ ishe d 8 Ja nua r y 20 0 4) Oscil lat or y va r iat ion s of t he d iagona l (Gxx ) a nd Ha l l (Gxy ) mag net o conduct a nces a re discusse d i n v iew of t op olog ica l sca l i ng effe ct s g iv i ng r ise t o t he qua nt u m Ha l l effe ct. T hey o ccu r i n a field ra nge w it hout osci l lat ion s of t he den sit y of stat es due t o L a ndau qua nt i zat ion , a nd a re , t herefore , t ot a l ly d i fferent f rom t he Shubn i kov - de Haa s osci l lat ion s. Such osci l lat ion s a re exper i ment a l ly obser ved i n d isordere d Ga A s layers i n t he ext reme qua nt um l i m it of appl ie d mag net ic field wit h a go od descr ipt ion by t he un ifie d sca l i ng t heor y of t he i nt eger a nd f ract iona l qua nt u m Ha l l effe ct.
DOI: 10.1103/ PhysRevL et t.92. 016802 PACS nu mbers: 73. 50. Jt , 73. 43.Q t , 73.61. Ey

T he i nteger qua nt u m Ha l l effe ct (QH E) is usua l ly obser ved at h igh mag net ic fields, !c 1 (!c eB=m is t he cyclot ron f re quency, is t he t ra n spor t rela xat ion t i me) , a nd it s app ea ra nce develops f rom t he Shubn i kov - de Haa s osci l lat ion s ba sed on t he L a ndau qua nt i zat ion of t he t wo- d i men siona l (2D) ele ct ron system. However, t he sca l i ng t reat ment of t he i nt eger QH E [1] pre d ict s t he exist ence of t he QH E wit hout t he L a ndau qua nt i zat ion of t he ele ct ron spe ct r u m. It could exist even at low mag net ic field !c 1 [2] i n t he absence of mag net o qua nt u m osci l lat ion s of t he den sit y of stat es. T he QH E at low mag net ic fields !c 1 ha s not been obser ve d t hus fa r, probably be cause ext remely low t emp erat u res a re re qui re d [3]. I n add it ion , t he QH E could exist i n a layer whose t h ick ness d is much la rger t ha n t he ele ct ron t ra n sp or t mea n - f ree pat h l, i.e., d l, i n t he ext reme qua nt u m l i m it ( EQL) of appl ie d mag net ic field , where on ly t he lowest L a ndau level is o ccupie d. Such a layer ha s a t h ree d i men siona l (3D) `` ba re'' (non renor ma l i ze d) ele ct ron spe ct r u m w it hout osci l lat ions of t he den sit y of stat es i n t he EQL. I n t h is sit uat ion , t he QH E ha s been obser ve d i n heav i ly Si- dop e d n-type Ga As layers [4, 5]. Here , we add ress t he problem of t he a r isi ng of t he QH E i n t he absence of mag net o qua nt um osci l lat ion s of t he den sit y of st at es. I n t h is ca se, t he va r iat ion w it h t emp era t u re of t he d iagona l conduct a nce p er squa re (Gxx ) a nd Ha l l conduct a nce (Gxy ) is due t o d i ff usive i nter ference effe ct s ( below Gxx and Gxy a re t a ken i n un it s e2 =h), wh ich i n a sca l i ng approach ca n be descr ibe d by t he renor ma l i zat ion -g roup e quat ion s. For compa r ison , accord i ng t o t he convent iona l t heor y, t he t emp erat u re dep endence of t he Shubn i kov - de Haa s osci l lat ion s pre ce d i ng t he QH E is due t o t her ma l broaden i ng of t he Fer m i d ist r ibut ion [6]. At t he moment , t wo t heor ies g ive expl icit expression s for t he renor ma l i zat ion -g roup e qua 016802-1 0 031-9 0 0 7= 04 =92(1)= 016802 (4)$20. 0 0

t ion s. T he first t heor y ha s been der ive d for bot h i nt eger a nd f ract iona l QH E a nd for a ny va lue of Gxx [7]. It is ba sed on t he a ssumpt ion t hat a cer t a i n sym met r y g roup un ifies t he st r uct u re of t he i nteger a nd f ract iona l qua nt u m Ha l l st at es [7 - 9]. T h is so- ca l le d un ifie d sca l i ng ( US) t heor y descr ibes wel l t he shap e of t he sca l i ng flow d ia g ra m depict i ng t he couple d evolut ion of Gxx and Gxy for de crea si ng t emp erat u res i n heav i ly Si- dope d n-type Ga As layers wit h d i fferent t h ick nesses for a wide ra nge of Gxx va lues [10]. T he second t heor y ha s been develop e d i n t he ``d i lut e i n st a nt on ga s'' approx i mat ion ( DIGA), first for non i nt eract i ng [11] a nd t hen for i nt eract i ng ele ct ron s [12]. Bot h t heor ies a re develop e d for a t ot a l ly spi n p ola r i ze d ele ct ron system. For 2Gxx 1, t hey pre d ict a n osci l lat i ng t opolog ica l t er m i n t he sca l i ng  f unct ion wit h t he sa me p er io d icit y. However, t hey d i ffer i n pre d ict ions on t he osci l lat ion a mpl it ude. T he osci l lat i ng t opolog ica l t er m i n t he  f unct ion should lead t o osci l lat ion s i n t he mag net ic-field dep endence of Gxx and Gxy wh ich a re not relat e d t o osci l lat ion s i n t he den sit y of stat es such a s, e.g. , for t he ca se of t he Shubn i kov - de Haa s osci l lat ion s. I n t he present e d work , we der ive expl icit expression s for t he t opolog ica l osci l lat ion s of t he Ha l l conduct iv it y Gxy for bot h t heor ies, a nd compa re t hem wit h exper i ment for t h ick (d l) d isordere d heav i ly Si- dop e d Ga As layers wit h rat her la rge Gxx and Gxy compa re d t o un it y. T he layers st udie d before i n Refs. [4, 5] have a 3D ba re ele ct ron spe ct r u m. However, below 4 K t he cha ract er ist ic d i ff usion leng t h s, L' Dzz ' 1=2 and LT Dzz h= kB T 1=2 , for coherent d i ff usive t ra n spor t i ncrea se t o va l ues la rger t ha n d, a nd t he system be comes 2 D for coherent d i ff usive phenomena (Dzz is t he d i ff usion co efficient of ele ct ron s a long t he mag net ic field , ' is t he pha se brea k i ng t i me). 200 4 T he A mer ica n Physica l Societ y 016802 -1


VOLU M E 92, N U M BER 1

PH YSICA L R E VI E W L E T T E RS

week end i ng 9 JA N UARY 20 0 4

T he US t heor y descr ibes t he renor ma l i zat ion -g roup flow of t he conduct a nces by t he e quat ion [7] s ? s0 ? lnf=f0 ; (1)

for a rea l pa ra met er s monot on ica l ly dep end i ng on t em p erat u re , where G Gxy iGxx , f0 fs0 , a nd P P1 n2 4 n n2 4 1 n?1 q n?1 ?1 q f? ; (2) P1 n1=22 8 2 n0 q wit h q expiG. For jqj2 exp?2Gxx 1, t he f unct ion f ?1=256q2 3=32 Oq2 a nd E q. (1) is re duce d t o s ? s0 i2G ? G0 24ei2
G

a s found by substit ut i ng GT [ E q. (7)] for G1 i n E q. (6). xx xx T h is dep endence is t ot a l ly d i fferent f rom t he exponent ia l va r iat ion wit h t emp erat u re of t he Shubn i kov - de Haa s osci l lat ion s. I n t he d i lut e i n st a nt on ga s approx i mat ion for t he ca se of i nt eract i ng ele ct ron s [12], dGxx ? ? D1 G2 e? xx d lnL dGxy ?D1 G2 e?2 xx d lnL
2G
xx

cos2Gxy ;

(9)

Gxx

sin2Gxy :

(10)

? ei2

G0

:

(3)

I n t he first - order approx i mat ion , by ig nor i ng t he la st osci l lat i ng t er m i n E q. (3) , t h is e quat ion ha s t he solution G
1 xx

Here L hDxx =kB T 1=2 and D1 64=e 74:0. Solvi ng t he quot ient of t hese e quat ion s by ig nor i ng t er m s of order exp?4Gxx , one obt a i n s G
xy

G0 ? xy

G0 ?s ? s0 =2; xx

G

1 xy

G0 : xy

(4)

D1 FGT ? FG0 sin2G0 ; (11) xx xx xy

I n t he second - order approx i mat ion , t he solut ion lo oks li ke G
xx

G1 xx G0 ? xy

12 ? e

2G

1 xx

? e?

2G

0 xx

cos2G0 ; xy sin2G0 : xy

(5) (6)

G

xy

12 ?2 e

G1 xx

? e?2

G0 xx

T h is is a solut ion of E q. (3) for fixe d s. However, for our exper i ment we a re i nt erest e d i n t he solution for fixe d t emp erat u re T . I n t he first - order approx i mat ion , it should coi ncide wit h t he result of t he first - order per t ur bat ion t heor y for t he ele ct ron - ele ct ron i nt eract ion i n coherent d i ff usive t ra n spor t lead i ng t o loga r it h m ic t emp erat u re dep endent cor re ct ion s i n t he d iagona l conduct a nce , G
T xx

G0 =2 lnT=T0 ; xx

(7)

w it hout a ny t emp erat u re dep endence i n t he Ha l l con duct a nce [13]. T herefore , s ? lnT i n t h is approx i mat ion. For a t ot a l ly spi n -p ola r i ze d ele ct ron system 1 [14]. I n second order, s w i l l osci l lat e a s a f unct ion of G0 at xy fixe d t emp erat u re T a nd w i l l g ive a n add it iona l osci l lat i ng t er m i n E q. (5) , but t he relat ion bet ween s and T is un k nown a nd t he a mpl it ude of t he Gxx osci l lat ion s ca n not be found. I n t h is resp e ct , we not e t hat t he la st t er m i n E q. (5) shows ma x i ma at i nt eger G0 a s opp osed t o t he xy expe ct e d m i n i ma for t he i nt eger QH E. T he d i fference bet ween G1 and GT ca n be ignore d i n t he exponent s of xx xx E q. (6). T herefore t he Ha l l conduct iv it y Gxy osci l lat es a s a f unct ion of t he ba re Ha l l conduct a nce G0 a nd, hence , a s a xy f unct ion of t he mag net ic field B, wit h a mpl it ude AUS xy 12 ?2GT 0 xx ? e?2Gxx e 12 ?2G0 xx T =T ? 1; e 0

(8)

where Fx 1=43 22 x2 2x 1 exp?2x. Bot h t heor ies have been develop e d for a t ot a l ly spi n p ola r i ze d ele ct ron system. However, i n a rea l system ele ct ron s ca n have t wo d i fferent spi n proje ct ion s. For t he ca se of non i nt eract i ng ele ct ron s, t he ele ct ron s ca n be descr ibe d i n t er ms of t wo i ndep endent , t ot a l ly spi n p ola r i ze d system s i n t he absence of spin -flip scat t er i ng. T h is approach rema i n s va l id for i nt eract i ng ele ct ron s a s wel l , i f t he t r iplet pa r t of t he con sta nt of i nt eract ion is much sma l ler t ha n t he si nglet one [13,14], be cause on ly t he i nt eract ion bet ween ele ct rons wit h t he sa me spi n leads t o a renor ma l i zat ion of t he conduct a nce i n t h is ca se. For t he sma l l spi n spl it t i ng i n st rongly d isordere d Ga As, t he conduct a nces of t he ele ct ron system s wit h d i fferent spin proje ct ion (G" and G# ) a re approx i mat ely e qua l t o ha l f ij ij t he mea sure d conduct a nce , i. e., G" G# Gij =2. It ij ij a l lows us t o compa re qua nt it at ively t he exper i ment a l result s wit h t he t heor ies. For la rge spi n spl it t i ng, t h is is i mp ossible, be cause G" and G# a re d i fferent , a nd on ly ij ij the sum G" G# ca n be mea sure d. ij ij T he i nvest igat e d heav i ly Si- dope d n-type Ga As layers sa ndwiche d bet ween undope d Ga As were prepa re d by mole cula r-bea m epit a xy. T he nom i na l t h ick ness d e qua ls 10 0 n m for t he layers 2, 3, 6 , a nd 14 0 n m for layer 7. T he Si- donor bul k concent rat ion n e qua ls 1.8, 2. 5, 1.6 , a nd 1:6 1017 cm?3 for sa mples 2 , 3, 6 , a nd 7 a s der ive d f rom t he p er iod of t he Shubn i kov - de Haa s osci l lat ion s at B< 5 T. T he mobi l it ies of t he sa mples at T 4:2 K are 2 40 0, 2500, 2600, a nd 2600 cm2 =Vs, a nd t he ele ct ron den sit ies p er squa re Ns a s der ived f rom t he slope of t he Ha l l resista nce Rxy i n wea k mag net ic fields (0:5-3 T) at T 4:2 K a re 1. 26, 2 , 2. 08, a nd 2:86 1012 cm?2 for sa mples 2 , 3, 6, a nd 7, resp e ct ively. For a l l sa mples, t he ele ct ron t ra n spor t mea n - f ree pat h l is a round 30 n m at zero mag net ic field. T he det a i le d st r uct u re of t he sa mples is descr ibe d i n Ref. [4]. I n Fig. 1, t he mag net ot ra n spor t dat a of t he d iagona l (Rxx , p er squa re) a nd Ha l l (Rxy ) resist a nce ( bot h g iven i n 016802- 2

016802- 2


VOLU M E 92, N U M BER 1
0.6
T = 4.2 K 1 0.28 0.08

PH YSICA L R E VI E W L E T T E RS

week end i ng 9 JA N UARY 20 0 4

Rxy , Rxx (h/e )

0.4

Rxy

0.2

BEQL
0 6

Rxx

4

Gxx
2

Gxy

0

0

5

10

15

20

B(T)
F IG. squa Ha l l p er p 10 0 field 1. Magnet ic field dependence of t he d iagona l (Rxx , p er re) a nd Ha l l (Rxy ) resist a nce a nd of t he d iagona l (Gxx ) a nd (Gxy ) conduct a nce for sa mple 2 i n a mag net ic field end icula r t o t he heav i ly dope d Ga As layer (t h ick ness n m) at d i fferent t emp erat u res. T he a r row i nd icat es t he BEQL of t he ext reme qua nt u m l i m it.

T he size qua nti zat ion could result i n osci l lat or y st r uct u res i n t he mag net ot ra n spor t dat a i n t he EQL i n a pu re layer w it h ba l l ist ic mot ion across t he layer when l=d 1. I n our ca se , however, l=d 0:2 0:3 i n zero mag net ic field , where t he rat io even de crea ses i n t he EQL for t he mea n - f ree pat h a long t he field. T he 3D cha ract er of t he ba re ele ct ron spe ct r u m of t he sa mples ha s been confir me d i n exper i ment s i n a t i lt e d mag net ic field [5]. Not e t hat t he absence of osci l lat ion s at T 4:2 K ca n not be expla i ne d by t emp erat u re broaden i ng of t he osci l lat or y st r uct ures, be cause d isorder broaden i ng dom i nat es la rgely wit h h= kB T (for our sa mples h=kB > 80 K). I n Fig. 2 , we plot t he residua l va r iat ion Gxx T Gxx T ? G0 a s a f unct ion of G0 for sa mple 6 at d i fferxx xy ent t emp erat u res, Gxy Gxy T ? G0 at T 0:46 K xy for sa mple 6 , a nd Gxx at T 0:1 K for t he t h ickest sa mple 7. Here G0 is t he conduct a nce at T 4:2 K t a ken ij a s t he ba re conduct a nce (see below). Bot h Gxx and Gxy osci l lat e wit h compa rable a mpl it udes under t he sa me condit ion s of appl ie d field a nd t emp erat u re. T he m i n i ma of Gxx a re at even - i nt eger va lues of G0 (slig ht ly sh i f t e d xy i n t he ca se of a super i mp osed smoot h va r iat ion of Gxx ) a nd t he m i n i ma of Gxy are sh i f t e d on 0:5 un it i n t he G0 sca le , i n accorda nce wit h t heor y [1,2 ,7,11,12]. xy T he smoot h ly va r y i ng pa r t of Gxx , by ig nor i ng t he osci l lat or y pa r t , de crea ses for de crea sing t emp erat u re wh i le t hat of Gxy does not cha nge. T he t emp erat u re

Gxy , Gxx (e /h)

2

2

0.0
Gxy

0.46 K
sample 6

un it s of h=e2 ), a nd of t he d iagona l (Gxx ) a nd Ha l l (Gxy ) conduct a nce , a re plot t e d for sa mple 2. At 4. 2 K , t he mag net oresist a nce shows t he t y pica l behav ior of bul k mat er ia l w it h wea k Shubn i kov - de Haa s osci l lat ion s for i ncrea si ng field B a nd a st rong monot onous upt u r n i n t he ext reme qua nt u m l i m it ( EQL) where on ly t he lowest L a ndau level is o ccupie d. At lower t emp erat u res, Rxy , Rxx , Gxy , a nd Gxx sta r t t o osci l lat e. Mi n i ma of Gxx and of j@Gxy =@Bj a r ise at mag net ic fields where Gxy at 4 K at t a i n s even - i nt eger va lues, i n accorda nce wit h bot h t he or ies ment ione d above. T hese osci l lat or y st r uct u res de velop i nt o t he QH E at t he lowest t emp erat ures where Rxy and Gxy revea l rema rkable steps nea r t he va lues Rxy 1=2 and 1=4, a nd Gxx 2, a nd 4. I n t he cor resp ondi ng fields pronounce d m i n i ma a re obser ve d i n Rxx and Gxx . Not e t hat , cont ra r y t o t he QH E st r uct u res, t he a mpl it ude of t he wea k Shubn i kov - de Haa s osci l lat ion s below t he EQL does not dep end on t emp erat u re be cause t he t her ma l da mpi ng fact or 22 kB T=h!c sinh22 kB T=h!c 0:994 is close t o 1 for B 5 T at T 1 K. Si m i la r but less pronounce d st r uct u res a re obser ved for t he ot her sa mples i nvest igat e d. Moreover, for sa mples 3 a nd 7, add it iona l m i n i ma of Gxx and of j@Gxy =@Bj a re obser ve d , at fields where Gxy 6 at T 4 K. 016802-3

(e /h)

-0.2

Gxx

0.9 0.67 0.56 0.46

2

Gxx, Gxy

-0.4

-0.6

-0.8

Gxx

0.1

sample 7

2

4
0

6
2

8

Gxy (e /h)
F IG. 2. Residua l va r iat ion for t he d iagona l Gxx a nd for Ha l l conduct a nce Gxy a f t er subt ract ion of t he 4. 2 -K va lues at di fferent t emp erat u res, for sa mples 6 a nd 7. Numbers nea r cu r ves i nd icat e t emp erat u res i n K.

016802 -3


VOLU M E 92, N U M BER 1
US theory DIGA

PH YSICA L R E VI E W L E T T E RS
7

week end i ng 9 JA N UARY 20 0 4

Aij +

U

(e /h)

0.1

3

0.03

sample 2 3 6 7

1.4

1.6
sm

1.8

2.0

Gxx (e /h)
F IG. 3. A Ha l l (solid plus U Gsm of t he xx l i ne shows the unifie d the ``d i lut e and 7. mpl it ude Aij of t he t op olog ica l osci l lat ion of t he symbols) a nd d iagona l (op en symbols) conduct a nce 24= exp?G0 a s a f unct ion of t he smoot h pa r t xx diagona l conduct a nce for fou r sa mples. T he f ul l t he dep endence 24= exp?Gsm fol lowi ng f rom xx sca l i ng t heor y. T he dot t e d l i nes show t he result of i nst a nt on ga s'' approx i mat ion t heor y for sa mples 3

2

I n sum ma r y, due t o t opolog ica l sca l i ng effe ct s, osci l lat ion s of t he d iagona l a nd Ha l l mag net o conduct a nces ca n exist when t here a re no osci l lat ion s of t he den sit y of stat es due t o L a ndau qua nt i zat ion. T he osci l lat ion s obser ved i n t he ext reme qua nt u m l i m it of t he appl ie d magnet ic field i n d isordere d Ga As layers, wit h t h ick ness la rger t ha n t he ele ct ron t ra nspor t mea n - f ree pat h , fa l l i nt o t h is cat egor y. T he osci l lat ion s of Gxy a re qua ntit a t ively wel l descr ibe d by t he un ifie d sca l i ng t heor y for t he i nt eger a nd f ract iona l qua nt u m Ha l l effe ct [7]. T hei r a mpl it ude is much sma l ler t ha n t he di lut e i n st a nt on ga s approx i mat ion [12] pre d ict s. We would l i ke t o t ha n k I. S. Bu r m ist rov for helpf ul d iscussions, a nd N. T. Moshegov, A. I. Torop ov, K. Eberl , a nd B. L em ke for t hei r help i n t he prepa rat ion of t he sa mples. T h is work is suppor t e d by R F BR a nd I N TAS.

2

dep endence of t he smoot h pa r t Gsm of t he d iagona l con xx duct a nce , t a ken a s t he m idp oi nt va lue of t he a r row i n Fig. 2 , is wel l descr ibe d by t he first - order ele ct ron ele ct ron - i nt eract ion cor re ct ion [ E q. (7)] wit h 1:9 for sa mples 2 , 3, a nd 6 , a nd 2 for sa mple 7 i n t he t emp erat u re ra nge f rom 0.15 t o 1 K fol lowe d by a sat urat ion a round 4. 2 K. T hese va lues a re close t o t he t heo ret ica l upper l i m it 2 for a system w it h t wo spi n s [13,14], cor resp ondi ng t o a negl ig ibly sma l l t r iplet pa r t of t he ele ct ron - ele ct ron i nt eract ion. T he choice of t he 4. 2 K va lue for t he ba re conduct a nce G0 ag rees w it h xx t he sat u rat ion of Gsm a round T 4:2 K. xx T he a mpl it udes Aij of t he osci l lat ion s of Gxx and Gxy conduct a nces a re ver y si m i la r a s shown i n Fig. 3, where the sum Aij U is plot t e d a s a f unct ion of t he smoot h pa r t of t he d iagona l conduct a nce Gsm for a l l our sa mples xx wit h U 24= exp?G0 . T he va lues of U xx 0:044, 0. 0 0 9, 0. 02, a nd 0.0 02 for sa mples 2 , 3, 6 , a nd 7, resp e ct ively, a re sma l ler t ha n t he cor resp ondi ng va lues of Aij . T he exper i ment a l dat a a re rat her wel l descr ibe d by t he result of t he US t heor y for Axy [ E q. (8)] appl ie d t o t he t ot a l conduct a nce of t wo i ndep endent ele ct ron system s of opp osite spi n. A lt hough showi ng a ver y si m i la r dep en dence , Axx ca n not be de duce d i n f ra me of t h is t heor y. T he DIGA t heor y pre d ict s much la rger a mpl it udes t ha n exp er i ment a l ly obser ve d , a s show n by t he dot t e d l i nes i n Fig. 3 for ADIGA U accord i ng DIGA t heor y for xy sa mples 3 a nd 7.

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