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The Quantized Hall Effect 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 .
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