T HE Q U A NTIZE D H ALL EFFECT

Nobel lect ure, Dece mber 9, 1985 b y KL A US von KLIT ZI N G,

Max- Planck-Instit ut f ür Festkör perforsch ung, D-7000 St uttgart 80

1. I ntrod uctio n Se mico n d uctor researc h a n d t he Nobel Prize i n p hysics see m to be co ntra dic- t or y si n c e o n e m a y c o m e t o t h e c o n cl usi o n t h at s u c h a c o m pli c at e d s yst e m li k e a se mico n uctor is not usef ul for very f u n da me ntal disco veries. I n dee d, most of t he ex peri me ntal data i n soli d state p h ysics are a nal yze d o n t he basis of si m plifie d t he ories, a n d ver y ofte n t he pr o perties of a se mic o n d uct or de vice is describe d by e m pirical for m ulas si nce t he microsco pic details are too co m pli- c at e d. U p t o 1 9 8 0 n o b o d y e x p e ct e d t h at t h er e e xists a n eff e ct li k e t h e Q u a nti z e d Hall Effect, which de pen ds excl usively on f un da mental constants an d is not affecte d b y irre g ularities i n t he se mico n d uctor li ke i m p urities or i nterface eff ects. T he disco ver y of t he Q ua ntize d Hall Effect ( Q H E) was t he res ult of s yste m- atic meas ure me nts o n silico n fiel d effect tra nsistors-t he most i m porta nt de vice i n microelectro nics. S uc h devices are not o nly i m porta nt for a p plicatio ns b ut also for basic research. The pioneering work by Fo wler, Fang, Ho war d an d Stiles [l] has sho wn that ne w q uant u m pheno mena beco me visible if the electro ns of a co n d uctor are co nfi ne d wit hi n a t y pical le n gt h of 10 n m. T heir discoveries o pe ne d t he fiel d of t wo- di me nsio nal electro n syste ms w hic h si nce 1 9 7 5 is t h e s u bj e ct of a c o nf er e n c e s eri es [ 2]. It h as b e e n d e m o nstr at e d t h at t his fi el d is i m p ort a nt f or t h e d es cri pti o n of n e arl y all o pti c al a n d el e ctri c al pr o p er- ties of microelectro nic devices. A t wo- di me nsio nal electro n gas is absol utely necessar y f or t he o bser vati o n of t he Q ua ntize d Hall Effect, a n d t he realizati o n a n d pr o perties of s uc h a s yste m will be disc usse d i n secti o n 2. I n a d diti o n t o the q uant u m pheno mena connecte d with the confine ment of electrons within a t wo- di me nsio nal layer, a not her q ua ntizatio n - t he La n da u q ua ntizatio n of t he el e ctr o n m oti o n i n a str o n g m a g n eti c fi el d - is ess e nti al f or t h e i nt er pr et ati o n of t he Q ua ntize d Hall Effect (sectio n 3). So me ex peri me ntal res ults will be s u m marize d i n sectio n 4 a n d t he a p plicatio n of t he Q H E i n metrology is t he s u bject of secti o n 5.

2 T wo- Di mensional Electron Gas T he f u n da me ntal pro perties of t he Q H E are a co nse q ue nce of t he fact t hat t he e ner g y s pectr u m of t he electro nic s yste m use d for t he ex peri me nts is a discrete energy s pectr u m. Nor mally, the energy E of mobile electrons in a K. von Klitzing 3 1 7 se micon d uctor is q uasicontin uo us an d can be co m pare d with the kinetic e ner g y of free electr o ns wit h wa ve vect or k b ut wit h a n effecti ve mass m*

If t h e e n er g y f or t h e m oti o n i n o n e dir e cti o n ( us u all y z- dir e cti o n) is fi x e d, o n e obtains a q uasi-t wo- di mensional electron gas (2 D E G), an d a strong magnetic fiel d per pe n dic ular to t he t wo- di me nsio nal pla ne will lea d-as disc usse d later- to a f ull y q ua ntize d e ner g y s pectr u m w hic h is necessar y for t he obser vatio n of t h e Q H E. A t wo- di me nsio nal electro n gas ca n be realize d at t he s urface of a se mico n- d uct or li ke silic o n or galli u m arse ni de w here t he s urface is us uall y i n c o ntact wit h a m at eri al w hi c h a cts as a n i ns ul at or ( Si O 2 f or sili c o n fi el d eff e ct tr a nsist ors a n d, e. g. Al x G a l- x As f or heter ostr uct ures). T y pical cr oss secti o ns of s uc h de vices are s h o w n i n Fi g 1. Electr o ns are c o nfi ne d cl ose t o t he s urface of t he se mico n d uctor by a n electrostatic fiel d F z nor mal to t he i nterface, origi nati ng fr o m p ositi ve c har ges (see Fi g. 1) w hic h ca uses a dr o p i n t he electr o n p ote ntial to war ds t he s urface.

Fig. I. A t wo-di mensional electron gas (2 D E G) can be for med at the se miconductor surface if the electrons are fixed close to the surface by an external electric field. Silicon M O S F E Ts (a) and

Ga As- Al x G a l- x As heterostructures (b) are typical structures used for the realization of a 2 D E G.

If t he wi dt h of t his p ote ntial well is s mall c o m pare d t o t he de Br o glie wa vele n gt h of t he electr o ns, t he e ner g y of t he carriers are gr o u pe d i n s o-calle d electric s ubba n ds E i corres po n di ng to q ua ntize d levels for t he motio n i n z- directio n, t he directio n nor mal to t he s urface. I n lo west a p proxi matio n, t he electro nic s ubba n ds ca n be esti mate d by calc ulati ng t he e nergy eige nval ues of a n el e ctr o n i n a tri a n g ul ar p ot e nti al wit h a n i nfi nit e b arri er at t h e s urf a c e ( z = 0) a n d a c o nsta nt electric fiel d F s f or z 0, w hi c h k e e ps t h e el e ctr o ns cl os e t o t h e s urface. T he res ult of s uc h calc ulatio ns ca n be a p proxi mate d by t he eq uatio n 3 1 8 Physics 1985

( 2) j = 0 , 1 , 2 . . .

I n so me materials, li ke silico n, differe nt effecti ve masses m* a n d ma y be prese nt w hic h lea ds t o differe nt series E j a n d E' j. Eq uatio n (2) m ust be i ncorrect if t he e nergy levels Ej are occ u pie d wit h el e ctr o ns, si n c e t h e el e ctri c fi el d F s will be scree ne d b y t he electro nic c har ge. F or a m ore q ua ntitati ve calc ulati o n of t he e ner gies of t he electric s u b ba n ds it is n e c ess ar y t o s ol v e t h e S c hr ö di n g er e q u ati o n f or t h e a ct u al p ot e nti al w hic h c ha n ges wit h t he distrib utio n of t he electro ns i n t he i n versio n la yer. T y pical res ults of s uch calc ulation for both silicon M OSF E Ts an d Ga As-heterostr uc- t ures are s h o w n i n Fi g. 2 [3,4]. Us uall y, t he electr o n c o nce ntrati o n of t he t w o- di m e nsi o n al s yst e m is fi x e d f or a h et er ostr u ct ur e ( Fi g. 1 b) b ut c a n b e v ari e d i n a M OSF E T by changing the gate voltage.

Fi g, 2. C al c ul ati o ns of t h e el e ctri c s u b b a n ds a n d t h e el e ctr o n distri b uti o n wit hi n t h e s urf a c e c h a n n el of a silicon M O S F E T (a) and a Ga As- Al x G a l- x As heter ostr uct ure [ 3, 4].

Ex peri me ntally, t he se paratio n bet wee n electric s ubba n ds, w hic h is of t he or der of 10 me V, ca n be meas ure d by a nalyzi ng t he reso na nce absor ptio n of electro ma g netic wa ves wit h a polarizatio n of t he electric fiel d per pe n dic ular to t he i nterface [5]. At lo w te m perat ures ( T <4 K) a n d s mall carrier de nsities for t he 2 D E G

(Fer mi e nergy E F relati ve to t he lo west electric s ubba n ds E 0 s mall co mpare d with the s ubban d se paration E l- E 0 ) o nl y t he lo west electric s u b ba n d is occ u- pie d wit h electr o ns (electric q ua nt u m li mit), w hic h lea ds t o a strictl y t w o- di mensional electron gas with an energy s pectr u m

w h er e is a w a v e v e ct or wit hi n t h e t w o- di m e nsi o n al pl a n e. 3 1 9

Fi g. 3. T y pi c al s h a p e a n d cr o s s- s e cti o n of a G a A s- Al x G a l- x As hcterostructure used for Hall effect measure ments.

For electrical meas ure ments on a 2 D E G, heavily do pe d n + -c o ntacts at t he se mico n d uctor s urface are use d as c urre nt co ntacts a n d pote ntial probes. T he s ha pe of a ty pical sa m ple use d for Q H E-ex peri me nts ( Ga As- heterostr uct ure) is s h o w n i n Fi g. 3. T h e el e ctri c al c urr e nt is fl o wi n g t hr o u g h t h e s urf a c e c h a n n el, si n c e t h e f ull y d e pl et e d Al x G a l- x As a cts as a n i ns ul at or (t h e s a m e is tr u e f or t h e

Si O 2 of a M OSF E T) a n d t he p-ty pe se mico n d uctor is electrically se parate d fr o m t h e 2 D E G b y a p- n j u n cti o n. It s h o ul d b e n ot e d t h at t h e s a m pl e s h o w n i n Fi g. 3 is b asi c all y i d e nti c al wit h n e w d e vi c es w hi c h m a y b e i m p ort a nt f or t h e next co m p uter generation [6]. Meas ure ments relate d to the Q uantize d Hall Effect w hic h i ncl u de a n a nalysis a n d c haracterizatio n of t he 2 D E G are t here- f or e i m p ort a nt f or t h e d e v el o p m e nt of d e vi c es, t o o.

3. Q ua nt u m Tra nsport of a 2 D E G i n Stro ng Mag netic Fields

A stro n g ma g netic fiel d B wit h a co m po ne nt B z , n or m al t o t h e i nt erf a c e c a us es t h e el e ctr o ns i n t h e t w o- di m e nsi o n al l a y er t o m o v e i n c y cl otr o n or bits p ar all el t o t he s urface. As a c o nse q ue nce of t he or bital q ua ntizati o n t he e ner g y le vels of t he 2 D E G ca n be writte n sc he matically i n t he for m

wit h t he c ycl otr o n e ner g y = *, t he s pi n q ua nt u m n u mbers s = ±1 /2 t he La n dé factor g a n d t he Bo hr mag neto n µ B T h e w a v e f u n cti o n of a 2 D E G i n a str o n g m a g n eti c fi el d m a y b e writt e n i n a for m w here t he y-coor di nate y 0 of t he ce nter of t he c ycl otr o n or bit is a g o o d q ua nt u m n u m ber [7]. ( 5) 3 2 0 Physics 1985

Fi g. 4. S k et c h f or t h e e n er g y d e p e n d e n c e of t h e d e nsit y of st at es ( a), c o n d u cti vit y ( b), a n d H all

resista nce R H (c) at a fi xe d ma g netic fiel d.

Fig. 5. Model of a t wo-di mensional metallic loop used for the derivation of the quantized Hall resistance. K. v o n Klitzi ng 3 2 1 w h er e is t he s ol uti o n of t he har m o nic- oscillat or e q uati o n

( 6)

a n d y 0 is r el at e d t o k b y y = 0 ( 7) T he dege neracy factor for eac h La n da u le vel is gi ve n by t he n u mber of ce nter coor di nates y 0 . wit hi n t h e s a m pl e. F or a gi v e n d e vi c e wit h t h e di m e nsi o n L x . L y , t he ce nter coor di nates y 0 are se parate d by t he a mo u nt

s o t h at t h e d e g e n er a c y f a ct or N = L / ∆ y is i d e nti c al wit h N = L L e B / h t h e 0 y 0 O x y n u mber of fl ux q ua nta wit hi n t he sa m ple. T he dege neracy factor per u nit area is t h er ef or e: ( 9)

It s h o ul d b e n ot e d t h at t his d e g e n er a c y f a ct or f or e a c h L a n d a u l e v el is i n d e p e n- de nt of se mico n d uctor para meters like effecti ve mass. I n a more ge neral wa y o ne ca n s ho w [8] t hat t he co m m utator for t he ce nter c o or di nates of t he c ycl otr o n or bit [x o , y o ] = is fi nit e, w hi c h is e q ui v al e nt t o t h e r es ult t h at e a c h st at e o c c u pi es i n r e al s p a c e t h e ar e a F 0 = h /e B corres pon d- i n g t o t h e ar e a of a fl u x q u a nt u m.

T he classical ex pressio n for t he Hall voltage U H of a 2 D E G wit h a s urface carrier de nsit y n s i s

w h er e I is t h e c urr e nt t hr o u g h t h e s a m pl e. A c al c ul ati o n of t h e H all r esist a n c e

R H = U H /I u n der t he co n ditio n t hat i e ner g y le vels are f ull y occ u pie d = i N), l e a ds t o t h e e x pr essi o n f or t h e q u a nti z e d H all r esist a n c e ( n s

i = 1 , 2 , 3

A q u a nti ze d H all resist a nce is al w a ys e x pecte d if t he c arrier de nsit y n s, a n d t he m a g netic fiel d B are a dj uste d i n s uc h a w a y t h at t he filli n g f act or i of t he e ner g y Le vels ( E q. 4)

is a n i nte ger. U n d er t his c o n diti o n t h e c o n d u cti vit y ( c urr e nt fl o w i n t h e dir e cti o n of t h e electric fiel d) beco mes zero si nce t he electro ns are mo vi n g li ke free particles e x cl usi v el y p er p e n di c ul ar t o t h e el e ctri c fi el d a n d n o diff usi o n ( ori gi n ati n g fr o m 3 2 2 P h ysics 1985 scatteri n g) i n t he directi o n of t he electric fiel d is p ossi ble. Wit hi n t he self- consistent Born a p proxi mation [9] the discrete energy s pectr u m broa dens as sho wn i n Fi g. 4 a. T hi s t heory pre dicts t hat t he co n d uctivity is mai nl y pr o p orti o n al t o t h e s q u ar e of t h e d e n sit y of st at e s at t h e F er mi e n er g y E F w hic h lea ds to a va nis hi ng co n d ucti vity in the q uant u m Hall regi me an d q uan- ti z e d pl at e a us i n t h e H all r esist a n c e R H ( Fi g. 4 c). T he si m ple o ne-electro n pict ure for t he Hall effect of a n ideal t wo-di mensional syste m i n a stro ng mag netic fiel d lea ds alrea dy to t he correct val ue for t he q u a nti z e d H all r esist a n c e ( E q. 1 1) at i nt e g er filli n g f a ct ors of t h e L a n d a u l e v els. Ho wever, a microscopic interpretation of the Q HE has to include the influences of t he fi nite size of t he sa m ple, t he fi nite te m perat ure, t he electr o n-electr o n i nteractio n, i m p urities a n d t he fi nite c urre nt de nsity (i ncl u di ng t he i n ho moge- ni o us c urr e nt distri b uti o n wit hi n t h e s a m pl e) o n t h e e x p eri m e nt al r es ult. U p t o n o w, n o c orrecti o ns t o t he val ue h /ie 2 of t he q ua ntize d Hall resista nce are pre dicte d if t he co n d ucti vit y is z er o. E x p eri m e nt all y, is ne ver exactl y zero i n t he q ua nt u m Hall regi me (see sectio n 4) b ut beco mes u n meas ura bly s mall at hig h mag netic fiel ds a n d lo w te m perat ures. A q ua ntitative t heory of t he Q H E has t o i ncl u de a n a nal ysis of t he l o n git u di nal c o n d ucti vit y u n der real ex peri me ntal co n ditio ns, a n d a large n u m ber of p u blicatio ns are disc ussi ng t he de pen dence of the con d uctivity on the te m perat ure, magnetic fiel d, c urrent d e n sit y, s a m pl e si z e et c. T h e f a ct t h at t h e v al u e of t h e q u a nti z e d H all r esist a n c e see ms t o be exactl y c orrect f or = 0 has le d t o t he c o ncl usi o n t hat t he k n o wle d ge of micr osc o pic details of t he de vice is n ot necessar y f or a calc ulati o n of t he q ua ntize d val ue. Co nseq ue ntly La ug hli n [10] trie d to de d uce t he res ult in a more general way fro m ga uge invariances. He consi dere d the sit uation s h o w n i n Fi g. 5. A ri b b o n of a t w o- di m e n si o n al s y st e m is b e nt i nt o a l o o p a n d pierce d e ver y w here b y a ma g netic fiel d B n or mal t o its s urface. A v olta ge dr o p

U H is a p plie d bet wee n t he t wo e dges of t he ri ng. U n der t he co n ditio n of va nis hi ng co n d uctivity ( no e ner g y dissi patio n), e ner g y is co nser ve d a n d o ne c a n writ e F ar a d a y’s la w of i n d u cti o n i n a f or m w hi c h r el at es t h e c urr e nt I i n t h e l o o p t o t h e a di a b ati c d eri v ati v e of t h e t ot al e n er g y of t h e s yst e m E wit h r es p e ct t o t h e m a g n eti c fl u x t hr e a di n g t h e l o o p

( 1 3)

If the fl ux is varie d by a fl ux q uant u m = h / e, t h e w a v ef u n cti o n e n cl osi n g t h e fl ux m ust c ha n ge b y a p hase fact or 2 π c orres p o n di n g t o a tra nsiti o n of a state wit h wavevector k i nto its neig h bo ur state k + ( 2 π) / ( L x ), w h er e L x i s t h e cir c u m- fere nce of t he ri n g. T he t otal c ha n ge i n e ner g y c orres p o n ds t o a tra ns p ort of states fr o m o ne e d ge t o t he ot her wit h

( 1 4) T he i nteger i corres po n ds to t he n u mber of fille d La n da u le vels if t he free electr o n m o del is use d, b ut ca n be i n pri nci ple a n y p ositi ve or ne gati ve i nte ger nu mber. K. v o n Klitzi ng 3 2 3

Fro m E q. (13) t he relatio n bet wee n t he dissi patio nless Hall c urre nt a n d t he Hall volta ge ca n be de d uce d

( 1 5)

h w hic h lea ds to t he q ua ntize d Hall resista nce R H =

I n t his pi ct ur e t h e m ai n r e as o n f or t h e H all q u a nti z ati o n is t h e fl u x q u a nti z a- tio n h /e a n d t he q ua ntizatio n of c har ge i nto ele me ntar y c har ges e. I n a nalo g y, t h e fr a cti o n al q u a nt u m H all eff e ct, w hi c h will n ot b e dis c uss e d i n t his p a p er, is i nter prete d o n t he basis of ele me ntar y excitatio ns of q uasi particles wit h a c har ge e* =

T h e si m pl e t h e or y pr e di cts t h at t h e r ati o b et w e e n t h e c arri er d e nsit y a n d t h e mag netic fiel d has to be a dj uste d wit h very hig h precisio n i n or der to get e x a ctl y i nt e g er filli n g f a ct ors ( E q. 1 2) a n d t herefore q ua ntize d val ues for t he Hall resista nce. Fort u natel y, t he Hall q ua ntizatio n is o bser ve d not o nl y at s p e ci al m a g n eti c fi el d v al u es b ut i n a wi d e m a g n eti c fi el d r a n g e, s o t h at a n acc urate fixi n g of t he ma g netic fiel d or t he carrier de nsit y f or hi g h precisi o n meas ure me nts of t he q ua ntize d resista nce val ue is not necessary. Ex peri me ntal d at a of s u c h H all pl at e a us ar e s h o w n i n t h e n e xt s e cti o n a n d it is b eli e v e d t h at localize d states are res po nsible for t he obser ve d stabilizatio n of t he Hall resista nce at certai n q ua ntize d val ues. After t he discovery of t he Q H E a large n u mber of t heoretical pa per were p u blis h e d dis c ussi n g t h e i nfl u e n c e of l o c ali z e d st at es o n t h e H all eff e ct [ 1 1 - 1 4] a n d t hese calc ulatio ns de mo nstrate t hat t he Hall platea us ca n be ex plai ne d if localize d states i n t he tails of t he La n da u le vels are ass u me d. T heoretical i n vestigatio ns ha ve s ho w n t hat a mobility e dge exists i n t he tails of La n da u levels se parati ng exte n de d states fro m localize d states [15-18]. T he mobility e d ges are l ocate d cl ose t o t he ce nter of a La n da u le vel f or l o n g-ra n ge p ote ntial fl uct uatio ns. Co ntrary to t he co ncl usio n reac he d by Abra ha ms, et al [19] t hat all st at es of a t w o- di m e nsi o n al s yst e m ar e l o c ali z e d, o n e h as t o ass u m e t h at i n a stro n g ma g netic fiel d at least o ne state of eac h La n da u le vel is exte n de d i n or der to observe a q ua ntize d Hall resista nce. So me calc ulatio ns i n dicate t hat t he exte n de d states are co n necte d wit h e d ge states [17]. I n pri n ci pl e, a n e x pl a n ati o n of t h e H all pl at e a us wit h o ut i n cl u di n g l o c ali z e d states i n t he tails of t he La n da u le vels is p ossi ble if a reser v oir of states is prese nt o utsi de t he t wo- di me nsio nal syste m [20, 21]. S uc h a reser voir for electro ns, w hic h s ho ul d be i n eq uilibri u m wit h t he 2 D E G, fixes t he Fer mi e n er g y wit hi n t h e e n er g y g a p b et w e e n t h e L a n d a u le v els if t h e m a g n eti c fi el d or the n u mber of electrons is change d. Ho wever, this mechanis m see ms to be m or e u nli k el y t h a n l o c ali z ati o n i n t h e t h e t ails of t h e L a n d a u l e v els d u e t o disor der. The follo wing disc ussion ass u mes therefore a mo del with exten de d a n d l ocalize d states wit hi n o ne La n da u le vel a n d a de nsit y of states as s ketc he d i n Fi g. 6. 3 2 4 Physics 1985

Fi g. 6. M o d el f or t h e br o a d e n e d d e nsit y of st at es of a 2 D E G i n a str o n g m a g n eti c fi el d. M o bilit y e d g es cl os e t o t h e c e nt er of t h e L a n d a u l e v els s e p ar at e e xt e n d e d st at es fr o m l o c ali z e d st at es.

4. Experi me ntal Data Magnetoquantu m transport measure ments on t wo-di mensional syste ms are k no w n a n d p u blis he d for more t ha n 20 years. T he first data were o btai ne d wit h silico n M OSF E Ts a n d at t he begi n ni ng mai nly res ults for t he co n d uctivity as a f u nctio n of t he carrier de nsity (gate voltage) were a nalyze d. A ty pical c ur v e is s h o w n i n Fi g. 7. T h e c o n d u cti vit y os cill at es as a f u n cti o n of t h e filli n g of t he La n da u le vels a n d beco mes zero at certai n gate voltages V g . I n str o n g m a g n eti c fi el ds v a nis h es n ot o nl y at a fi x e d v al u e V g b ut i n a r a n g e a n d K a w aji w as t h e first o n e w h o p oi nt e d o ut t h at s o m e ki n d of i m m o bil e el e ctr o ns m ust be i ntro d uce d [22], si nce t he co n d ucti vity r e m ai n s z er o e v e n if t h e carrier de nsity is c ha nge d. Ho wever, no reliable t heory was available for a disc ussio n of localize d electro ns, w hereas t he pea k val ue of was well ex plai ne d by calc ulatio ns base d o n t he self-co nsiste nt Bor n a p proxi matio n a n d s hort-ra nge scatterers w hic h pre dict ~ ( n + l /2) i n de pe n de nt of t he m a g n eti c fi el d. T he t heory for t he Hall co n d ucti vity is m uc h more co m plicate d, a n d i n t he lo west a p proxi matio n o ne ex pects t hat t he Hall co n d uctivity de viates fro m

t he classical c ur ve = - w er h e n s is t he t otal n u m ber of electr o ns i n t he t wo- di me nsio nal syste m per u nit area) by a n a mo u nt which depen ds mai nl y o n t he t hir d p o wer of t he de nsit y of states at t he Fer mi e ner g y [23]. K. v o n Klitzi ng 3 2 5

Fi g. 7. C o n d u cti vit y of a sili c o n M O S F E T at diff er e nt m a g n eti c fi el ds B as a f u n cti o n of t h e g at e v o lta g e V g .

Ho wever, no agree me nt bet wee n t heory a n d ex peri me nt was obtai ne d. To day, it is belie ve d, t hat is mai nly i nfl ue nce d by localize d states, w hic h ca n e x pl ai n t h e f a ct t h at n ot o nl y a p ositi v e b ut al s o a n e g ati v e si g n f or is observe d. U p to 1980 all ex peri me ntal Hall effect data were a nalyze d o n t he basis of a n i ncorrect mo del so t hat t he q ua ntize d Hall resista nce, w hic h is alr e a d y vi si bl e i n t h e d at a p u bli s h e d i n 1 9 7 8 [ 2 4] r e m ai n e d u n e x pl ai n e d. W hereas t he co n d ucti vity ca n be meas ure d directly by usi ng a Corbi no dis k g e o m etr y f or t h e s a m pl e, t h e H all c o n d u cti vit y is n ot dir e ctl y a c c essi bl e i n a n ex peri me nt b ut ca n be calc ulate d fro m t he lo ngit u di nal resisti vity a n d t h e H all r esisti vit y meas ure d o n sa m ples wit h Hall geo metry (see Fig. 3):

Fig. 8 s ho ws meas ure me nts for a n d of a silic o n M OS F E T as a f u nctio n of t he gate volta ge at a fixe d ma g netic fiel d. T he corres po n di n g a n d - d at a ar e c al c ul at e d o n t h e b asis of E q. ( 1 6).

T he classical c ur ve = - i n Fi g. 8 is dr a w n o n t h e b asis of t h e i n c orr e ct mo del, t hat t he ex peri me ntal data s ho ul d lie al ways belo w t he classical c ur ve ( = fi x e d si g n f or A s o t h at t h e pl at e a u v al u e = c o nst. ( o bs er v a bl e i n t h e g at e v olt a g e r e gi o n w h er e beco mes zero) should change with the width of the pl at e a u. Wi d er pl at e a us s h o ul d gi v e s m all er v al u es f or T he mai n disco very i n 1 9 8 0 w as [ 2 3] t h at t h e v al u e of t h e H all r esist a n c e i n t h e pl at e a u r e gi o n is n ot i nfl u e n c e d b y t h e pl at e a u wi dt h as s h o w n i n Fi g. 9. E v e n t h e as p e ct r ati o L / W ( L = l e n gt h, W = wi dt h of t h e s a m pl e), w hi c h i nfl u e n c es n or m all y t h e a c c ur a c y i n Hall effect meas ure me nts, beco mes u ni m p ort a nt as s h o w n i n Fi g. 1 0. Us u all y,

e x p t he meas ure d Hall resista nce R H is al ways s maller t ha n t he t heoretical val ue t h e o r = p [ 2 6 , 2 7 ] R H x y 3 2 6 Physics 1985

Fig. 8. Measured a n d of a silic o n M O S F E T as a f u ncti o n of t he gate v olta ge at B = 1 4. 2 t o get her wit h t he calc ulate d a n d

Fig. 9. Measure ments of the Hall resistance R H a n d t he resisti vit y R x as a f u ncti o n of t he gate 2 voltage at different magnetic field values. The plateau values R H = h/ 4 e are independent of the wi dt h of t he platea us.

Ho we ver, as s ho w n i n Fi g. 11, t he correctio n 1- G beco mes zero (i n de pe n de nt of

t h e as p e ct r ati o) if 0 or t h e H all a n gl e θ a p proac hes 90º (ta n θ =

T his mea ns t hat a ny s ha pe of t he sa m ple ca n be use d i n Q H E-ex peri me nts as l o n g a s t h e H all a n gl e is 9 0º ( or = 0). Ho we ver, o utsi de t he platea u regio n

e x p is in dee d al ways 0) t he meas ure d Hall resista nce R H = 3 2 7

6454

Fi g. 1 0. Hall resista nce R H f or t w o diff er e nt s a m pl es wit h diff er e nt as p e ct r ati os L/ W as a f u n cti o n of t h e g at e v olt a g e ( B = 1 3. 9 T). s maller t ha n t he t heoretical pxy val ue [28]. T his lea ds to t he ex peri me ntal res ult

e X P t hat a n a d ditio nal mi ni m u m i n R H beco mes visi ble o utsi de t he platea u regio n as s h o w n i n Fi g. 9, w hic h disa p pears if t he c orrecti o n d ue t o t he fi nite le ngt h of t he sa m ple is i ncl u de d ( See Fig. 12). T he first hig h- precisio n meas ure me nts i n 1980 of t he platea u val ue i n R H ( Vg) sho wed already that t hese resista nce val ues are q ua ntize d i n i nteger parts of h/e 2 = 2 5 8 1 2. 8 wit hi n t he ex peri me ntal u ncertai nty of 3 p p m. T he Hall platea us are m uc h more pro no u nce d i n meas ure me nts o n Ga As-

Al x Gal-x As heter ostr uct ures, si nce t he s mall effecti ve mass m* of t he electr o ns i n Ga As ( m*( Si) / m* ( Ga As) > 3) lea ds to a relatively large e nergy s plitti ng bet wee n La n da u levels ( E q. a n d t he hi g h q ualit y of t he i nterface ( nearly no s urface ro ug h ness) lea ds to a hig h mo bility µ of t he electro ns, so t hat t he co n ditio n µ B > 1 for La n da u q ua ntizatio ns is f ulfille d alrea dy at relatively lo w mag netic fiel ds. Fig. 13 s ho ws t hat well- develo pe d Hall platea us are visi ble f or t his material alrea d y at a ma g netic fiel d stre n gt h of 4 tesla. Si nce a fi nite carrier de nsit y is us uall y prese nt i n heter ostr uct ures, e ve n 3 2 8 Physics 1985

r

Fig. 11. Calculations of the correction ter m G in Hall resistance measure ments due to the finite l e n gt h t o wi dt h r ati o L/ W of t h e d e vi c e (l/ L = 0, 5).

at a gate v olta ge V g = 0 V, most of t he p ublis he d tra ns port data are base d o n meas ure me nts wit ho ut a p plie d gate volta ge as a f u nctio n of t he ma g netic fiel d.

A t y pical res ult is s h o w n i n Fi g. 14. T he Hall resista nce R H = i ncreases ste plike wit h platea us i n t he mag netic fiel d regio n w here t he lo ngit u di nal resista nce v a nis h es. T h e wi dt h of t h e i n t h e li mit of z er o t e m p er a- t ure ca n be use d for a deter mi natio n of t he a mo u nt of exte n de d states a n d t he a n al ysis [ 2 9] s h o ws, t h at o nl y fe w p er c e nt of t h e st at es of a L a n d a u le v el ar e n ot localize d. T he fractio n of exte n de d states wit hi n o ne La n da u level decreases wit h i ncreasi n g ma g netic fiel d ( Fi g. 15) b ut t he n u m ber of exte n de d states wit hi n eac h le vel re mai ns a p proxi mately co nsta nt si nce t he dege neracy of eac h L a n d a u l e v el i n cr e as es pr o p orti o n all y t o t h e m a g n eti c fi el d. At fi nite te m perat ures is n e v er e x a ctl y z er o a n d t h e s a m e is tr u e f or t h e sl o p e of t h e i n t h e pl at e a u r e gi o n. B ut i n r e alit y, t h e sl o p e at T <2 K a n d ma g netic fiel ds abo ve 8 Tesla is so s mall t hat t he st a ys co nsta nt wit hi n t he ex peri me ntal u ncertai nty of 6 ⋅ 1 0 - 8 e ve n if t he ma g netic fi el d is c h a n g e d b y 5 %. Si m ult a n e o usl y t h e r esisti vit y i s u s u all y s m all er t h a n Ho wever, at hig her te m perat ures or lo wer mag netic fiel ds a fi nite r esisti vit y a n d a fi nit e sl o p e ( or ca n be meas ure d. T he data are well descri be d wit hi n t he m o del of exte n de d states at t he e ner g y p ositi o n of t he u n dist urbe d La n da u level E n a n d a fi nite de nsit y of l ocalize d states bet wee n t he La n da u levels ( mobility ga p). Like i n a mor p ho us syste ms, t he te m perat ure de pe n de nce of t he co n d uctivity ( or r esisti vit y is t her mall y acti vate d 3 2 9

6,:

Fig. 12. Co mparison bet ween the measured quantities R H a n d R x and the corresponding resistivity co mponents Q x y a n d respectively.

wit h a n acti vatio n e ner g y E a corres pon ding to the energy difference bet ween t he Fer mi e nergy E F a n d t he mobility e dge. T he largest acti vatio n e nergy wit h a v al u e E a = (if t h e s pi n s plitti n g is n e gli gi bl y s m all a n d t h e m o bilit y e d ge is l ocate d at t he ce nter E n of a La n da u le vel) is ex pecte d if t he Fer mi e n er g y is l o c at e d e x a ctl y at t h e mi d p oi nt b et w e e n t w o L a n d a u l e v els. Ex peri me ntally, a n acti vate d resisti vity ( 1 8) is observe d i n a wi de te m perat ure ra nge for differe nt t wo- di me nsio nal syste ms ( deviations fro m this behavio ur, which a p pear mainly at te m perat ures belo w 1 K, will be disc usse d se paratel y) a n d a res ult is s ho w n i n Fi g. 16. T he acti vatio n e ner gies ( de d uce d fro m t hese data) arc plotte d i n Fi g. 17 for bot h, silicon M OSFETs and Ga As- Al x G a l- x As heterostr uct ures as a f u nctio n of t he 3 3 0 Physics 1985

Fi g. 1 3. Measured curves for the Hall resistance R H and the longitudinal resistance R x of a Ga As-

A l x G a l- x As heterostructure as a function of the gate voltage at different magnetic fields.

ma g netic fiel d a n d t he data a gree fairl y well wit h t he ex pecte d c ur ve E a = U p to no w, it is not clear w het her t he s mall s yste matic s hift of t he meas ure d activatio n e nergies to hig her val ues origi nates fro m a te m perat ure de pe n de nt prefact or i n E q. (18) or is a res ult of t he e n ha nce me nt of t he e ner g y ga p d ue to ma n y bo d y effects. T he ass u m ptio n, t hat t he mo bilit y e d ge is locate d close to t he ce nter of a

La n da u le vel E n is s u p p orte d b y t he fact t hat f or t he sa m ples use d i n t he ex peri me nts o nl y fe w perce nt of t he states of a La n da u le vel are exte n de d [29]. Fr o m a s yst e m ati c a n al ysis of t h e a cti v ati o n e n er g y as a f u n cti o n of t h e tilli n g fact or of a La n da u le vel it is p ossi ble t o deter mi ne t he de nsit y of states D( E) [30]. T he s ur prisi n g res ult is, t hat t he de nsit y of states ( D OS) is fi nite a n d a p proxi mately co nsta nt wit hi n 60 % of t he mobility ga p as s ho w n i n Fig. (18). T his backgro u n d D OS de pe n ds o n t he electro n mobility as s u m marize d i n Fig. ( 1 9). A n a c c ur at e d et er mi n ati o n of t h e D O S cl os e t o t h e c e nt er of t h e L a n d a u l e v el is not possible by this metho d since the Fer mi energy beco mes te m perat ure de pe n de nt if t he D OS c ha nges drastically wit hi n t he e nergy ra nge of 3k T. Ho wever, fro m a n a nalysis of t he ca pacita nce C as a f u nctio n of t he Fer mi e n er g y t h e p e a k v al u e of t h e D O S a n d its s h a p e cl os e t o E n ca n be de d uce d [31, 3 2]. K. von Klitzing

MAGNETIC FIELD (T)

Fig. 14. Experi mental curves for the Hall resistance R H = a n d t he resisti vit y of a heterostructure as a function of the magnetic field at a fixed carrier density corresponding to a gate voltage V g , = 0 V. The te mperature is about 8 m K.

T hi s a n al y si s i s b a s e d o n t h e e q u ati o n

The co mbination of the different metho ds for the deter mination of the D OS l e a ds t o a r es ult as s h o w n i n Fi g. ( 2 0). Si mil ar r es ults ar e o bt ai n e d fr o m ot h er ex peri me nts, too [33, 34] b ut no t heoretical ex pla natio n is a vaila ble. If o ne ass u mes t hat o nly t he occ u patio n of exte n de d states i nfl ue nces t he H all eff e ct, t h a n t h e sl o p e i n t h e pl at e a u r e gi o n s h o ul d b e d o mi n at e d 3 3 2 Physics 1985

Fig. 15. Fraction of extended states relative to the nu mber of states of one Landau level as a function of t h e m a g n eti c fi el d.

b y t h e s a m e a cti v ati o n e n er g y as f o u n d f or Ex peri me ntall y [35], a o ne t o o n e r el ati o n b et w e e n t h e mi ni m al r esisti vit y at i nt e g er filli n g f a ct ors a n d t h e sl o p e of t h e H all pl at e a u h as b e e n f o u n d ( Fi g. 2 1) s o t h at t h e fl at n ess of t he platea u i ncreases wit h decreasi ng resistivity, w hic h mea ns lo wer te m pera- t ur e or hi g h er m a g n eti c fi el ds. T he te m perat ure de pe n de nce of t he resisti vity for Fer mi e nergies wit hi n t he mobility ga p deviates fro m a n activate d be havio ur at lo w te m perat ures, ty pi- cally at T

Ther mally activate d r esisti vit y at a filli n g f a ct or i = 4 for a silico n M OS F E T at differe nt m a g n eti c fi el d v al u es. 3 3 4 P h ysics 1985

6

Fi g. 1 7. Meas ure d acti vati o n e ner gies at filli n g fact ors i = 2 ( Ga As heter ostr uct ure) or i = 4 ( Si- M O S F E T) as a function of the magnetic field. The data are co mpared with the energy

syste ms, too, a n d are i nter prete d as variable ra nge ho p pi ng. For a t wo- di me n- sio nal syste m wit h ex po ne ntially localize d states a be ha vio ur

~ e x p ( 2 0) is ex pecte d. For a Ga ussia n localizatio n t he follo wi ng de pe n de nce is pre dicte d [36, 37]

T he a nal ysis of t he ex peri me ntal data de mo nstrates ( Fi g. 22) t hat t he meas ure- me nts are best descri be d o n t he basis of E q. (21). T he sa me be ha vi o ur has bee n found in measure ments on another t wo-di mensional syste m, on InP-In Ga As heter ostr uct ures [38]. T he co ntrib utio n of t he variable ra nge ho p pi ng ( V R H) process to t he Hall eff e ct is n e gli gi bl y s m all [ 3 9] so that ex peri mentally the te m perat ure de pen d- e n c e of re mai ns t her mall y acti vate d e ve n if t he resisti vit y is do minated by V R H. K. von Klitzing 3 3 5

I -

Fig. 18. Measured density of states (deduced fro m an analysis of the activated resisitivity) as a function of the energy relative to the center bet ween t wo Landau levels ( Ga As-heterostructure).

T he Q H E breaks do w n if t he Hall fiel d beco mes larger t ha n abo ut

E H = 60 V /c m at ma g netic fiel ds of 5 Tesla. T his c orres p o n ds t o a classical drift vel ocit y v = 1200 m /s. At t he D B criti c al H all fi el d E H ( or c urre nt de nsit y j) t he resisti vit y i ncreases a br u ptl y b y or ders of magnit u de an d the Hall platea u disa p pears. This pheno menon has bee n observe d by differe nt a ut hors for differe nt materials [40-47]. A ty pical r es ult is s h o w n i n Fi g. 2 3. At a c urr e nt d e nsit y of j C = 0, 5 A / m t h e r esisti vit y at t he ce nter of t he platea u (filli n g fact or i = 2) i ncreases drasticall y. T his i nstability, w hic h de velo ps wit hi n a ti me scale of less t ha n 100 ns see ms to origi nate fro m a r u na way i n t he electro n te m perat ure b ut also ot her mec ha n- is m li ke electric fiel d de pe n de nt delocalizatio n, Ze ner t u n neli n g or e missio n of 3 3 6 Physics 1985

Fi g. 1 9. Bac k gr o u n d de nsit y of states as a f u ncti o n of t he m o bilit y of t he de vice.

aco ustic p ho no ns, if t he drift velocit y excee ds t he so u n d velocit y, ca n be use d for a n ex pla natio n [48-50]. Fi g. 2 3 s h o w s t h at i ncreases alrea d y at c urre nt de nsities well belo w t he criti c al v al u e w hic h ma y be ex plai ne d b y a broa de ni n g of t he exte n de d state r e gi o n a n d t h er ef or e a r e d u cti o n i n t h e m o bilit y g a p A E. If t h e r esisti vit y i s t her mally acti vate d a n d t he mobility ga p c ha nges li nearly wit h t he Hall fiel d ( w hi c h is pr o p orti o n al t o t h e c urr e nt d e nsit y j) t h e n a v ari ati o n

is ex pecte d. S uc h a de pe n de nce is see n i n Fi g. 24 b ut a q ua ntitati ve a nal ysis is diffi c ult si n c e t h e c urr e nt distri b uti o n wit hi n t h e s a m pl e is us u all y i n h o m o g e n- i o us a n d t h e H all fi el d, c al c ul at e d fr o m t h e H all v olt a g e a n d t h e wi dt h of t h e sa m ple, re presents only a mean val ue. Even for an i deal t wo- di mensional syste m a n i n ho moge nio us Hall pote ntial distrib utio n across t he wi dt h of t he sa m ple is ex pecte d [51-53] wit h a n e n ha nce me nt of t he c urre nt de nsit y close t o t h e b o u n d ari es of t h e s a m pl e. T h e e x p eri m e nt al sit u ati o n is still m or e c o m pli c at e d as s h o w n i n Fi g. 2 5. T h e pote ntial distrib utio n de pe n ds stro ngly o n t he mag netic fiel d. Wit hi n t he pl at e a u r e gi o n t h e c urr e nt p at h m o v es wit h i n cr e asi n g m a g n eti c fi el d a cr oss t h e 3 3 7

Fi g. 2 0. E x p eri m e nt al 1 y d e d uce d density of states of a Ga As heterostructure at B = 4 T co mpare d wit h t he calc ulate d res ult base d o n t he self-co nsiste nt Bor n a p proxi matio n (S C B A). wi dt h of t h e s a m pl e fr o m o n e e d g e t o t h e ot h er o n e. A gr a di e nt i n t h e c arri er density within the t wo- di mensional syste m see ms to be the most pla usible ex planation b ut in a d dition an inho mogeneity pro d uce d by the c urrent itself may play a role. U p to no w, not e no ug h microsco pic details abo ut t he t wo- di me nsio nal syste m are k no w n so t hat at prese nt a microsco pic t heory, w hic h describes t he Q H E u n der real ex peri me ntal co n ditio ns, is not available. Ho w- e ver, all ex peri me nts a n d t heories i n dicate t hat i n t he li mit of va nis hi n g r esisti vit y t he val ue of t he q ua ntize d Hall resista nce de pe n ds excl usi vel y o n f u n da me ntal co nsta nts. T his lea ds to a direct a p plicatio n of t he Q H E i n metr ol o g y. 3 3 8 Physics 1985

Fi g. 2 1. R el ati o n b et w e e n t h e sl o p e of t h e H all pl at e a us and the corresponding at i nt e g er filli n g f a ct ors.

5. A p plic ati o n of t he Q u a nt u m H all Effect i n Metr ol o g y T he a p plicatio ns of t he Q ua nt u m Hall Effect are very si milar to t he a p plica- tio ns of t he Jose p hso n- Effect w hic h ca n be use d for t he deter mi natio n of t he f u n da me ntal co nsta nt h /e or for t he realizatio n of a voltage sta n dar d. I n a nalogy, t he Q H E ca n be use d for a deter mi natio n of h /e 2 or as a resista nce sta n dar d. [54]. Si nce t he i n verse fi ne str uct ure co nsta nt is m or e or l ess i d e nti c al wit h h / e 2 (t he pro portio nal co nsta nt is a fixe d n u m ber w hic h i ncl u des t he velocit y of lig ht), hig h precisio n meas ure me nts of t he q ua ntize d Hall resista nce are i m por- ta nt for all areas i n p hysics w hic h are co n necte d wit h t he fi nestr uct ure co n- st a nt. Ex peri mentally, the precision meas ure ment of a is r e d u c e d t o t h e pr o bl e m of meas uri ng a n electrical resista nce wit h hig h acc uracy a n d t he differe nt met h- o ds a n d res ults are s u m marize d i n t he Procee di ngs of t he 1984 Co nfere nce o n Precision Electro magnetic Measure ments ( CP E M 84) [55]. The mean value of meas ure me nts at la boratories i n t hree differe nt co u ntries is

The internationally reco m men de d val ue (1973) is 3 3 9

a n d t he preli mi nar y val ue for t he ti nestr uct ure co nsta nt base d o n a ne w least sq uare a dj ust me nt of f u n da me ntal co nsta nts (1985) is

Differe nt gro u ps ha ve de mo nstrate d t hat t he ex peri me ntal res ult is wit hi n t he ex peri me ntal u ncertai nt y of less t ha n 3.7 · 1 0 - 8 in de pen dent of the material ( Si, G a As, a n d of t he gro wi ng tec h niq ue of t he devices ( M B E or M O C V D) [56]. The main proble m in high precision meas ure ments of is- at prese nt-t he cali brati o n a n d sta bilit y of t he refere nce resist or. Fi g. 26 s h o ws t he drift of t he mai ntai ne d at differe nt natio nal la boratories. T he 3 4 0 Physics 1985

Fi g. 23. C urre nt- v olta ge c haracteristic of a heter ostr uct ure at a filli n g fact or i = 2 ( T = 1,4 K). The device geo metry an d the are s ho w n i n t he i nserts.

v er y first a p pli c ati o n of t h e Q H E is t h e d et er mi n ati o n of t h e drift c o effi ci e nt of t he sta n dar d resistors si nce t he q ua ntize d Hall resista nce is more stable a n d more re pro d ucible t ha n a ny wire resistor. A nice de mo nstratio n of s uc h a n a p pli c ati o n is s h o w n i n Fi g. 2 7. I n t his e x p eri m e nt t h e q u a nti z e d H all r esis- ta nce R H has been measure d at the “Physikalisch Technische Bun desanstalt” relati ve t o a refere nce resist or R R as a f u ncti o n of ti me. T he rati o R H / R R c ha nges a p proxi mately li nearly wit h ti me b ut t he res ult is i n de pe n de nt of t he Q H E-sa m ple. T his de mo nstrates t hat t he refere nce resistor c ha nges its val ue wit h ti me. T he o ne sta n dar d de viatio n of t he ex peri me ntal data fro m t he mea n v al u e is o nl y 2. 4 · 1 0 - 8 s o t h at t h e Q H E c a n b e us e d alr e a d y t o d a y as a r el ati v e sta n dar d to mai ntai n a laboratory u nit of resista nce base d o n wire- wo u n d resistors. T here exists a n agree me nt t hat t he Q H E s ho ul d be use d as a n absol ute resista nce sta n dar d if t hree i n de pe n de nt laboratories meas ure t he sa me val ue for t he q ua ntize d Hall resista nce (i n SI- u nits) wit h a n u ncertai nty of l ess t h a n 2· 1 0 - 7 . It is ex pecte d t hat t hese meas ure me nts will be fi nis he d u ntil t h e e n d of 1 9 8 6. Fig. 24. Nonoh mic conductivity of a Ga As heterostructure at different te mperatures T L (filli n g f a ct or i = 2). A n i nst a bilit y is o bs er v e d at s o ur c e- dr ai n fi el ds l ar g er t h a n 4 0 V/ c m.

Ackno wledge ments T he p ublicity of t he Nobel Prize has ma de clear t hat t he researc h work co n necte d wit h t he Q ua nt u m Hall Effect was so s uccessf ul beca use a tre me n- do us large n u mber of i nstit utio ns a n d i n divi d uals s u p porte d t his activity. I w o ul d li k e t o t h a n k all of t h e m a n d I will m e nti o n b y n a m e o nl y t h os e s ci e ntists w ho s u p porte d my researc h work at t he ti me of t he disco very of t he Q H E i n 3 4 2 Physics 1985

Fig. 25. Measured Hall potential distribution of a Ga As heterostructure as a function of the m a g n eti c fi el d.

1980. Pri marily, I wo ul d like to thank G. Dor da (Sie mens Forsch ungslabo- ratorien) an d M. Pe p per ( Caven dish Laboratory, Ca mbri dge) for provi ding me with high q uality M OS- devices. The contin uo us s u p port of my research work by G. La n d we hr a n d t he fr uitf ul disc ussio ns wit h my co worker, T h. E nglert, were esse ntial for t he disco very of t he Q ua nt u m Hall Effect a n d are greatf ully ackno wle dge d. Fi g. 2 6. Ti me de pe n de nce of t he I Ω standard resistors maintained at the differcnt national laboratories. Physics 1985

Fi g. 2 7. Rati o R H / R R bet ween the quantized Hall resistance R H a n d a wir e r e si st or R R as a f u ncti o n of ti me, The result is ti me dependent but indcpcndent of the Hall device used in the experi ment. K. von Klitzing 3 4 5

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