Mapping an electron wave function by a local electron scattering probe

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Phys. 17 (2015)113048 doi:10.1088/1367-2630/17/11/113048 PAPER Mapping an elec on wa e unc ion by a local elec on sca e ing p obe C Reichl 1 , W Die sche 1,2 , T Tschi ky 1 , T Hya 3 and W Wegscheide 1 1 Solid S a e Physics Labo a o y, ETH Zu ich, O o-S e n-Weg 1, 8093 Zü ich, Swi ze land 2 Max Planck Ins i u ü Fes kö pe o schung, Heisenbe gs asse 1, D-70569 S u ga , Ge many 3 Uni e si y o Jy äskylä, Depa men o Physics and Nanoscience Cen e , PO Box 35 (YFL), FI-40014, Finland E-mail: [email p o ec ed] Keywo ds: 2DEG, he e os uc u es, elec on wa e unc ion, GaAs/AlGaAs, elec on sca e ing Abs ac A echnique is de eloped which allows o he de ailed mapping o he elec onic wa e unc ion in wo-dimensional elec on gases wi h low- empe a u e mobili ies up o 1 510cmVs 6211 ´-- . Thin (‘del a’)laye s o aluminium a e placed in o he egions whe e he elec ons eside. This causes elec on sca e ing which depends e y locally on he ampli ude o he elec on wa e unc ion a he posi ion o he Al δ-laye . By changing he dis ance o his laye om he in e ace we map he shape o he wa e unc ion pe pendicula o he in e ace. Despi e ha ing a p o ound e ec on he elec on mobiliy, he δ-laye s do no cause a widening o he quan um Hall pla eaus. 1. In oduc ion The en elope wa e unc ion ( )Y o localized elec ons in semiconduc o s is de e mined by he laws o quan um-mechanics and elec os a ics. Al hough he shape o ( )Y de e mines many physical p ope ies, i s p ecise o m is expe imen ally only accessible unde e y a ou able condi ions and wi h subs an ial e o , o example using an UHV-STM [1]. In his wo k we u ilize he ex emely sho in e ac ion leng h o neu al impu i ies in high quali y GaAs/AlGaAs he e os uc u es, syn hesized by molecula beam epi axy (MBE) o map ou he squa e o he elec on wa e unc ion pe pendicula o he in e ace. This equi es o place e y hin (‘del a’)laye s o Al a oms a a ying posi ions and measu e he elec on mobili ies, om which he elec on sca e ing a es a e de e mined. These sca e ing a es eflec he ampli ude o he wa e unc ion a he posi ion o he δ-laye . Elec ons in wo-dimensional elec on gases (2DEGs)in he e os uc u es a e ee o p opaga e along he in e ace bu a e localized pe pendicula o i [2]. The eigens a es and eigenene gies in he absence o a sca e ing po en ial a e z L zE E k m k,1e, 2,1 kk 2 i0 22 () ()() () · *  yc==+ whe e xy ,,()=kkk,, xy ()= z()c is he no malized wa e unc ion o he lowes ene gy ans e se mode. The ac o L 2is a no maliza ion, E 0 is he g ound s a e eigenene gy and m m0.067 e *=is he e ec i e mass. The unc ion z()c can be calcula ed sel -consis en ly by combining he Sch ödinge and he Poisson equa ion. This equi es assump ions abou he ma e ial pa ame e s o he semiconduc o s uc u es, pa icula ly he bounda y condi ions, band o se s a he in e ace and he inco po a ion o doping a oms. Se e al so wa e packages a e a ailable o nume ical solu ions [3–5], which howe e su e o example om he lack o p ecise alues o he band o se s [6]. Consequen ly, while he heo e ical model desc ibing he wa e unc ion is well es ablished, z()c is ypically ob ained only app oxima ely by means o simula ion. Neu al impu i ies, e.g. a oms like Al wi h he same ou e elec on shell as Ga, a e known o ha e e y sho in e ac ion leng hs [7,8], al hough de ails o he sca e ing mechanism ha e no ye been esol ed. Adding OPEN ACCESS RECEIVED 22 July 2015 REVISED 8 Oc obe 2015 ACCEPTED FOR PUBLICATION 29 Oc obe 2015 PUBLISHED 19 No embe 2015 Con en om his wo k may be used unde he e ms o he C ea i e Commons A ibu ion 3.0 licence. Any u he dis ibu ion o his wo k mus main ain a ibu ion o he au ho (s)and he i le o he wo k, jou nal ci a ion and DOI. © 2015 IOP Publishing L d and Deu sche Physikalische Gesellscha δ-laye s o Al o he GaAs in he egion whe e he 2DEG esides should allow o es he ampli ude o z()c a he δ-laye posi ion. This is done by analyzing he elec on mobili y μ. To u ilize he Al δ-laye as local sca e ing cen e, i is o pa amoun impo ance o educe all o he sca e ing p ocesses as much as possible. These p ocesses include sca e ing by cha ged ionized dono s, phonons, in e ace oughness and backg ound impu i ies (see e.g. [7]). The la e s em om esiduals in he MBE chambe ha a e ine i ably inco po a ed du ing he g ow h p ocess. The phonon sca e ing can be e ec i ely emo ed by cooling he sample o low empe a u es. The ole o he ionized dono s is minimized by la ge se back dis ances be ween doping and 2DEG, and he e ec o in e ace oughness appea s o be negligible unde op imized g ow h condi ions. The backg ound impu i ies can only be educed i he he e os uc u es a e syn hesized unde ex eme pu i y in specialized MBE se ups. The ‘quali y’o a gi en MBE se up is gene ally measu ed by he maximum elec on mobili y which has been achie ed in quan um-well s uc u es [9–13]. Mobili ies exceeding 2.5 10 cm V s 721 1 ´-- (measu ed a 300 mK)ha e been achie ed wi h he MBE se up used by us o g owing he Al-doped samples [14]. 2. Expe imen al de ails As he basic sample design we use single-sidedly doped he e os uc u es (figu e 1)g own in he ollowing sequence: we s a wi h a supe la ice consis ing o 100 pe iods o 7 nm AlGaAs and 3 nm GaAs. This is ollowed by 1000 nm GaAs hos ing he 2DEG a he in e ace o an adjacen 310 nm hick AlGaAs laye . This egion con ains a hin doping laye o silicon, placed a a se back dis ance o 70 nm om he in e ace. The whole s uc u e is capped by 10 nm o GaAs. The Al con en o he AlGaAs is 25% h oughou . These alues a e based on a g ow h a e calib a ion ha is pe o med on a daily basis and secu es ha a es and wi h ha laye hicknesses a e p ecise wi hin a ma gin o less han 2% (see [14] o a de ailled desc ip ion). A se ies o di e en samples a e g own con aining Al impu i ies which eplace he Ga a oms in he GaAs c ys al s uc u e. This is done by adding 0.28 nm (one monolaye )AlGaAs and wi h ha 1 .5 10 cm 14 2 ´ -Al a oms o he GaAs a dis ances a om he in e ace a ying om a 5nm= o a 30 nm=. The a e age dis ance be ween he Al a oms in his laye is 0 .8 nm. The dispe sion o he AlGaAs del a laye due o mig a ion du ing he g ow h p ocess can be conside ed negligible. TEM analysis o s uc u es p oduced unde simila g ow h condi ions shows sha p in e aces o an AlAs laye s o 2 nm wid h. The same is ue o a bu e supe la ice as desc ibed abo e. A TEM image o such a supe la ice wi h compa able in e ace quali y is shown in [13]. T anspo p ope ies we e measu ed by he an-de -Pauw echnique, bo h in he da k and a e illumina ion wi h a ed (710 nm)LED. Magne o anspo da a we e ob ained a 1.3 K a magne ic fields up o 6 T. The elec on densi ies and mobili ies a 1.3 K o he sample wi hou any Al impu i ies a e 1 .5 10 cm 11 2 ´-and 8.0 10 cm V s 6211 ´ -- in he da k and 2.0 10 cm 11 2 ´ - and 1 410cmVs 621 1 ´ -- a e illumina ion, espec i ely. This s uc u e se es as he e e ence o he se ies wi h Al δ-laye s a a ying dis ances and will u he on be e e ed o as a=0. 3. The sca e ing e sus al-doping dep h The ables 1and 2summa ize he esul s o measu emen s made in he da k and a e illumina ion a 1 .3 K, espec i ely. As usual, he illumina ed s uc u es ha e la ge elec on densi ies compa ed o hose measu ed in he da k. Adding he Al δ-laye s has a significan e ec on elec on mobili y: μd ops by an o de o magni ude Figu e 1. Schema ic o he he e os uc u es. The black lineillus a es he conduc ion band along he g ow h di ec ion (wi h he sample su ace owa ds he le ), he unc ion z 2 ()cis he squa ed en elope wa e unc ion. The g ey a ea deno es he AlGaAs egion, ma ked in yellow is he Si doping laye . The black do ed line illus a es he band shape wi h an included laye o neu al impu i ies (aluminium), leading o a small de ia ion in he shape o he wa e unc ion (ligh blue line). 2 New J. Phys. 17 (2015)113048 C Reichl e al om 1 410cmVs 621 1 ´ -- o 1 .1 10 cm V s 6211 ´ -- a e illumina ion, and om 8.0 10 cm V s 6211 ´ -- o 0 .8 10 cm V s 621 1 ´-- in he da k), i he Al is placed 1 0nm away om he in e ace (which is he mobili y ange o wha Ga dne e al epo ed o a compa able, homogeneously dis ibu ed amoun o Al a oms [15]). We no e ha we a e able o ep oduce he magne o anspo cha ac e is ics o nominally iden ical samples, o igina ing om di e en g ow h uns wi hin a ma gin o 2% o elec on densi y and 4% o mobili y [16]. I is use ul o compa e he anspo sca e ing a es 1 a he han he mobili ies o disc imina e he in insic sca e ing p ocesses—caused by backg ound impu i ies in he g ow h chambe , emo e ionized dono po en ial diso de , in e ace oughness and phonon sca e ing— om he ones induced exclusi ely by he Al impu i ies. The o al sca e ing a e 1 o should be he sum o he in insic a e 1 in and he one due o he Al impu i ies 1 : Al 111 2 o in Al () =+ 1 o is calcula ed om he ela ion em , o ()· * m = whe e eand m * a e he elemen a y cha ge and he e ec i e elec on mass, espec i ely. The esul ing sca e ing a es a e shown in ables 1and 2and a e plo ed in figu e 2. Unexpec edly, he elec on densi y is educed by up o 10%, i he Al δ-laye is loca ed in he 5–15 nm ange. This sys ema ic change is oo la ge o be accus omed o unce ain ies in he g ow h p ocess ( hose may accoun o a densi y a ia ion o no mo e han 1%)o he e o ma gin o he cha ac e iza ion. The sca e ing a es ha e a maximum a a dis ance o 1 0nm om he in e ace whe e hey exceed he e e ence alues by a ac o o abou 10. I is no ewo hy ha no only he e e ence sca e ing a e bu also he one due o he Al a oms dec ease a e illumina ion. 4. Sca e ing by neu al impu i ies— heo y Al hough a fi s p inciple calcula ion o he sca e ing is di ficul , a simple app oxima ion can be ob ained by modi ying he app oach used in [17,18]in such a way ha he sca e ing si es now exis only in he Al δ-laye . In his app oach one conside s he Ga a oms being eplaced by Al a oms andomly in some si es i (possibly also clus e ing a ound hese si es, such ha i a e he cen es o hese clus e s). The a e age ( V z , a () )and he andom ( V z , and ( ) )pa o he po en ial a e Table 1. Elec on densi ies and mobili ies o he samples wi h di e en dis- ances o he Al δ-laye s om he in e ace. Also shown a e he sca e ing a es calcula ed om he mobili ies. All da a a e ob ained in he da k. Dis ance (nm) Densi y (10 11 cm −2 ) Mobili y (10 6 cm 2 V −1 s −1 ) Sca e ing a e (ns −1 ) 0 1.52 8.036 3.27 5 1.40 1.293 20.30 10 1.39 0.793 33.1 15 1.416 1.813 14.48 20 1.461 3.746 7.01 25 1.561 7.331 3.58 30 1.574 8.281 3.17 Table 2. Cha ac e iza ion da a as in able 1bu a e illumina ion wi h a ed LED. Dis ance (nm) Densi y (10 11 cm −2 ) Mobili y (10 6 cm 2 V −1 s −1 ) Sca e ing a e (ns −1 ) 0 1.98 13.75 1.91 5 1.79 1.98 13.255 10 1.76 1.14 22.98 15 1.81 2.309 11.37 20 1.96 5.21 5.04 25 1.99 9.53 2.75 30 2.04 12.77 2.05 3 New J. Phys. 17 (2015)113048 C Reichl e al Vz zaW xV xV VzzaWCV V ,1 ,,3 i ii i ii i a Ga Al and Ga Al () ( ) [( )( ) ( )] () ( ) [( ) ( )] () å å d d =- - -+ - =- -- - whe e W 0.28 nm= and aa e he app oxima e hickness and he posi ion o he Al δ-laye , espec i ely, and each C i is a andom a iable which is xwi h a p obabili y o 1−xand x−1 wi h p obabili y x. He e x=0.25 is he Al concen a ion in he δ-laye . We assume ha C i in di e en si es a e unco ela ed so ha he expec a ion alue o e he diso de ealiza ions sa isfies CC x x1 ij ij ()d á ñ= - . The homogeneous a e age po en ial V a ( ) does no cause sca e ing, so ha he sca e ing a e is comple ely de e mined by he andom po en ial V and () . We pa ame ize he po en ial a ound each si e i as VV EH ,4 ii iGa Al 0 () () (∣∣) ()-- -=D -- whe e H(x)is he Hea iside s ep unc ion, and E D and 0 desc ibe he magni ude and he ange o he sca e ing po en ial caused by each clus e . The sca e ing a e can be calcula ed using Fe mi’s golden ule k LkkM EEkk k k 12 2dd , 1cos, 5 2 2 2 () ( ) ∣ ( )∣ ( ( ) ( ))( ) ( )  òò p pqd q=¢¢ á¢ñ- ¢- whe e θis he angle be ween k and k ¢ and M z zV z zkk ,dd, ,, kk 2 and ( ) () ()( ) * òòyy ¢=¢is he ma ix elemen caused by he andom alloy sca e ing po en ial. By assuming ha he Fe mi wa e ec o k F sa isfies k 1, F0  we ob ain aW E m LxxN 11, 6 242 20 4 32 () () () () *  pc=D- whe e Nis he numbe o sca e ing si es. This o mula shows ha he sca e ing a e is p opo ional o he ou h powe o he wa e unc ion a z=a. This beha iou is used o he wa e unc ion mapping. The dependence o he sca e ing a e on he ou h powe o z()c leads o he apid a ia ion o he sca e ing a es wi h he dis ance om he in e ace as seen in he da a p esen ed in figu e 2. Fo a quan i a i e es ima e o he sca e ing a e one has o make assump ions abou 0 and he he sca e ing po en ial E D . We assume ha he ange 0 and he spacing be ween he sca e ing si es LN 2a e on he same o de LN 1nm 02 »» . Then he only ee pa ame e is he magni ude o he sca e ing po en ial E, D which is expec ed o be on he o de o E0.1 1 D ~- eV co esponding o he conduc ion band a ia ions i Al a oms a e alloyed o he GaAs. Wi h a0.06 nm 21 () ( )c» - a a 10 nm=(see figu e 3)and a E0.2 e VD » we find a sca e ing a e 28 ns 11 () » -- which is e y close o he numbe s measu ed. Al hough he alue o E D deduced om his analysis is conside able smalle han he one ound by Li e al [8] o GaAs homogeneously doped wi h Al, we belie e ha hisdisc epancy a ises mainly om model- specific assump ions ha influence he alue ob ained o E D . In pa icula he pa ame e s 0 and L N 2 ha e significan unce ain ies due o he possible clus e ing o he a oms, and in his wo k we ha e made di e en assump ions o hese pa ame e s han in [8]. Despi e hese unce ain ies in he ela i e magni udes o he pa ame e s, his heo e ical calcula ion illus a es ha he expe imen ally measu ed sca e ing a es a e Figu e 2. Sca e ing a es 1 o as a unc ion o he dis ance ao he Al δ-laye om he in e ace. The black squa es ep esen da a measu ed in he da k, he ed ci cles a e ob ained a e illumina ion. 4 New J. Phys. 17 (2015)113048 C Reichl e al consis en wi h he alloy sca e ing mechanisms since quan i a i e ag eemen can be ob ained wi h a easonable choice o he pa ame e s E D , 0 and L N 2 . 5. De e mining he wa e- unc ion shape Based upon he da a p esen ed in figu e 2, we use equa ion (6) o deduce he shape o he squa ed en elope wa e unc ion a 2()c. Fi s , one needs o subs ac an es ima e o 1 in (3.1 and 1 .8 ns 1- o he da k and illumina ed s a e, espec i ely). The squa e oo o he esul ing 1 A l is plo ed as do s in figu es 3(a)and (b) o he illumina ed and he non-illumina ed case espec i ely. These da a poin s can be compa ed wi h heo e ically expec ed wa e unc ions 4 a 2()co he 2DEGs, ob ained om he 8-band Sch ödinge –Poisson-sol e so wa e Nex nano[3]which uses pa ame e s om [19], including a conduc ion band o se o 250 meV o an Al- ac ion o 25%. The simula ed s uc u e is iden ical o he ac ual samples, including a silicon doping laye wi h a densi y o 3 10 cm 12 2 ´ -. Since he simula ion neglec s he o ma ion (and ac ion)o DX-cen es, he esul ing wa e unc ion is only applicable o he illumina ed case, when almos all DX-cen es a e ionized. The esul ing wa e unc ion is shown as he dashed ed cu e in figu e 3(a), i s calcula ed elec on densi y is highe (2.25 10 cm 11 2 ´ - ) han wha was obse ed expe imen ally (1.9 10 cm 11 2 ~´ -); howe e , i s ag eemen wi h he expe imen al da a is al eady e y good and gi es us in he mapping echnique used he e. The fi can e en be imp o ed by adjus ing he densi y o ac i e dono s in he simula ion o find a 2DEG densi y ha ma ches he measu ed one. This app oach leads o he ed solid line in figu e 3(a), which ag ees excellen ly wi h he da a poin s. Figu e 3. Squa e oo o he sca e ing a es 1 Al (do s)as a unc ion o he dis ance o he Al δ-laye om he in e ace. Da a ob ained a e illumina ion and in he da k a e shown in (a)and (b), espec i ely. The solidlines co espond o he espec i e z 2 ()cas ob ained om he Sch ödinge –Poisson sol e wi h adap ed Si doping densi y. The ag eemen be ween expe imen al da a and heo e ical cu e is e y good, especially o he illumina ed case. The dashed line in (a)is he esul o a calcula ion whe e he ac ual Si doping densi y was used. In (b) he dashed line ep esen s simula ion da a ha includes an impu i y backg ound in he ini ial AlGaAs laye s a he beginning o he g ow h p ocess. 4 Fo his compa ison be ween measu emen and simula ion da a we use as a fi s o de app oxima ion wa e unc ion as ob ained by simula ion wi hou he Al δ-laye . This appea s o be a easonable app oxima ion, since including he Al laye in o he simula ion leads o a densi y a ia ion o less han 0.5% and a maximum change in a 2 ( ) c o less han 10%. 5 New J. Phys. 17 (2015)113048 C Reichl e al Figu e 3(b)plo s he da a ob ained in he da k. Using he he p ocedu e as in he illumina ed case, including an adjus men in he ac i e dono densi y ( ep esen ed by he black solid line), leads o a less good ag eemen wi h he da a, pa icula ly on he wa e unc ion’(s)flank a om he in e ace. This hin s ha he 2DEG is mo e s ongly confined han an icipa ed by he simula ion so wa e. Such an enhanced confinemen could be he esul o deep le el p- ype impu i es ge e ed by he highly eac i e aluminium in he AlGaAs/GaAs supe la ice loca ed a below he ac ual he e os uc u e. The dashed line in figu e 3(b)exempla ily shows he esul ing wa e unc ion o a backg ound impu i y densi y o 1 0cm 15 3-in he AlGaAs bu e laye s 5 . Using his scena io, he expe imen ally obse ed wa e unc ion can be ep oduced e y well o he da k case also. By means o illumina ion, he backg ound impu i ies in he bu e laye s migh be compensa ed, leading back o he si ua ion desc ibed abo e o he illumina ed case. O e all, he ag eemen o he fi and he expe imen al da a is su p isingly good, om which we conclude ha he sca e ing po en ial o he Al a oms ac s e y locally on he elec on wa e unc ion. I is no ewo hy ha no only he in insic sca e ing a es bu also 1 A l a e educed a e illumina ion. The in insic sca e ing is p obably due o cha ged impu i ies, bo h om he Si-doping and in he 2DEG egion. In bo h cases sc eening has always been conside ed o be e y e ec i e. Ou da a indica e ha o he sca e ing by neu al impu i ies, a densi y dependence exis s, which also canno be explained by he shi o he wa e unc ion due o he illumina ion. Such a dependency has, howe e , been neglec ed in p e ious heo ies [7]and is also no pa o ou analysis in sec ion 4. 6. The e ec on magne o anspo In high pe pendicula magne ic fields, he elec onic anspo p ope ies show he in ege quan um Hall e ec (IQHE). Gene ally, he wid hs o he pla eaus and he accompanying minima in he esis ance depend on he densi y o localized s a es be ween he Landau le els [20]con aining he ex ended s a es. Inc easing he sca e ing a e is he e o e expec ed o inc ease he densi y o localized s a es a he expense o he ex ended ones and o lead o a widening o he SdH minima in he ange o he IQHE pla eaus. This beha iou is demons a ed by he ace co esponding o he sample ‘low μ’(g ey line in figu e 4)which has been g own in an MBE sys em ha was in a poo s a e a he ime o g ow h, i.e. which con ains a high numbe o esidual cha ged and neu al impu i ies. I ’s elec on mobili y o 0 .7 10 cm V s 621 1 ´-- is simila o he one o he a 10 nm=sample ( ep esen ed by he ed line). One migh expec a simila widening o minima om samples wi h an Al δ-laye ha ing a compa able mobili y. Figu e 4. Longi udinal esis ance in he magne ic field ange co esponding o filling ac o 2n= o 6. No widening o he minima is obse ed o he di e en posi ions o he Al δ-laye s (colou -coded). Fo compa ison, he g ey line labelled as ‘low μ’ ep esen s he R XX ace o a sample wi h low mobili y— e y simila o he a 10 nm=sample—wi hou any Al-doping; he minima he e a e significan ly b oade . The inc ease in esis ance be ween he minima seems o be mo e ela ed o he absolu e sca e ing a e a he han o he posi ion o he Al δ-laye . 5 No e ha a backg ound impu i y densi y his high is assumed only o he ini al s ages o he g ow h un and migh o example s em om he oxide laye p o ec ing he subs a e su ace be o e g ow h. We u he assume ha he backg ound impu i y le el is no cons an du ing he g ow h un bu is con inuously educed by ge e ing/pumping. A mobili y-densi y analysis as desc ibed in [24]was pe o med by he Ri chie g oup on a compa able s uc u e g own by us and sugges s a cha ged backg ound impu i y o 410cm 13 3 »´ - in he 2DEG egion. 6 New J. Phys. 17 (2015)113048 C Reichl e al We ha e measu ed he magne o anspo cha ac e is ics a 1.3 K up o 6 T o ou samples. The esul ing longi udinal esis ances as unc ion o filling ac o a e shown in figu e 4. Clea ly, no significan widening o he minima a in ege filling wi h he sca e ing a e is obse ed, al hough he sca e ing a es a y by a ull o de o magni ude. In con as , he maxima be ween he in ege filling inc ease conside ably wi h he sca e ing a es. The dis ance o he δ-laye om he in e ace seems o be mo e ele an o he shape o he cu es in he egions be ween he in ege fillings. I would be o in e es o s udy his beha iou as unc ion o (lowe ) empe a u e and compa e he esul s wi h he scaling s udy o Li e al [21]. This is howe e beyond he scope o his wo k. I is no ewo hy ha also ac ional quan um Hall e ec gaps, measu ed by Deng e al [22], showed su p isingly li le change om mode a e bu homogeneous Al doping which may be ela ed o he lack o he localized-s a es backg ound. 7. Conclusions Placing δ-laye s o Al impu i ies in o GaAs in he egions o he 2DEG leads o subs an ially enhanced elec on sca e ing a es. The dependence o hese sca e ing a es p ecisely images he shape o he wa e unc ion z,()c e i ying ha he sca e ing po en ial ac s e y locally on he elec on wa e unc ion. This beha iou makes his simple echnique a unique way o map ou he spa ial dis ibu ion o 2DEG wa e unc ions. Al hough he sca e ing a e due o he Al a oms was enhanced by a ac o o 10 compa ed o he e e ence sample, i does no influence he wid h o he IQHE pla eaus. This indica es ha his sca e ing p ocess does no con ibu e o he backg ound o localized s a es be ween he Landau le els. The Al a oms do howe e enhance he esis ance maxima be ween he in ege filling ac o s. This indica es ha he Al a oms cause a pu ely elas ic sca e ing p ocess. The missing o an inc ease o he localized backg ound may also be ele an o he obse a ion by Deng e al ha neu al backg ound impu i ies—in he o m o a homogeneous Al-doping—do no ha e a significan impac on he ac i a ion ene gy o he 52 n =FQHS [22]. Using his echnique i will be possible o map ou wa e unc ions o ec angula quan um wells which a e o special in e es o highe mobili ies. Such s uc u es a e he es bed o in es iga ions on he exo ic ν=5/2 s a e, whose quali y is cu en ly limi ed by he influence o emo e ionized dono s [11,23]. Thei e ec would be minimal on a symme ic wa e unc ion. Cu en ly such a symme y can only be aimed a by calcula ing he equi ed uppe and lowe doping densi y, bu is e y di ficul o e i y. Fu he on, he echnique can be used o wide quan um wells and double-quan um well sys ems. In such sys ems, he local elec on densi y dis ibu ion de elops wo maxima ha need o be balanced. Again, ca e ully placed Al δ-laye s would be help ul as a senso o op imize he g ow h pa ame e s o achie e a balancing be ween wo (pa ial)wa e unc ions. Acknowledgmen s We acknowledge s imula ing discussions wi h Yongqing Li, Fabian Schläp e and La s Tiemann. 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