Coherent electron focusing

Festkörperprobleme 29 (1989) Coherent electron focusing

C. W. J. Beenakker

Philips Research Laboratories, 5600 JA Eindhoven, The Netherlands H. van Houten

Philips Laboratories, Briarcliff Manor, NY 10510, USA B. J. van Wees

Applied Physics, Delft University of Technology, 2600 GA Delft, The Netherlands

Summnry: Thcory and cxpcrimcnl arc rcvicwcd οΓ Ihc clussical and q u a n t u m mc-chanical focusing by a niagncüc fickl of ballislic clcctrons injcctcd tlirough a point contact in a Uvo-dimcnsional electron gas. Two a l t c r n a l i v c poinLs of vicw arc cm-phasizcd. On thc onc b a n d , thc cxpcrimcnL i s n rcalizntion ofclcclron oplics in Ihc solid stalc. Thc thrcc basic b u i l d i n g blocks arc a cohcicnt and monochromalic point sourcc/dctcctor, an elcctrostalic mirror wilh littlc diffuse scatlcring, and a magnclic Icns. On thc othcr b a n d , cohcrent clcclron focusing is a rcsislancc mcasurcmcnl in thc q u a n t u m ballistic transport rcgime, which cxhibits Ihc charactcristic featurcs of this rcgime in a most extreme way. For cxamplc, largc magnclorcsi.stancc oscil-lations occur (up to 9.5% amplitudc modulalion is obscrvcd), w i t h a pcriodicily which is non-locally dclcrmincd by thc Separation bctwccn currcnl and vollage p o i n t contacLs. Λ W K B calculatio n of Ihc liansmissio n probabililics shows llia t Ihi s cffect is thc rcsult of thc intcrfcrcncc of cohcrcnlly cxcitcd magnctic edgc slalcs al the electron gas boundary, Anothcr cxamplc is thc abscncc of local c q u i l i b r i u m : Thc mcasurcmcnts shovv that thc point conlacts can sclectivcly populatc (and clelcct) specific Land au Icvcls, and that Ihis highly non-cquilibrium population is main-laincd ovcr dislances of microns.

I Introdnclion

mng-netic ficld. In metals, electron focnsing is essentially a clasvical phe-nomenon because of (he s m a l l Fermi w a v c length λ\ (typically 0.5 nm, 011 t h e order of the inter-atomic Separation).

The Fermi wave length is 100 times äs lange in the two-dimensional electron gas (2DEG) which is present at the inteiface of a GaAs-AlGaAs heterostrticture. This l e n g f h scale is w i t h i n reach of electron-beam lithography, w h i l e r e m a i n i n g well below die mean fiee path in high-mobility m a t e n a l ( 1 0 / i m can be icalizecl at lovv temper-atures in lieterostriictnies grown by molecular-beam e p i t a x y ) . For these two reasons the quantum halliitic t r a n s p o i t tegime lias become accessi-ble in a 2DEG [4]. In the present paper theory and experiment are re-viewed of electron focusing in this reg i m e [5 — S], which t u r n s out to be strikingly different from the classical legime f a m i l i ä r Pro m metals. This has motivated the new name: cohctcnt clecironfoaniug.

The geometiy oP the experiment (Fig. I) is the transverse Pocusing ge-ometry of Tsoi [2], and consists of two point contacts on the same b o u n d a r y in a p e r p e n d i c u l a r magnetic fiele! B. [ In m e t a l s one can also use the geometry of Sharvin [l], witli opposite p o i n t contacts in a lon-g i t u d i n a l field. This is not possible in two climensions. ] Because the electron gas is confined to the i n t e r i o r of the heterostructuie, one can not just use a metal needle to fabricate a p o i n t contacl to a 2DFG. In-stead, the point contacts are created electrostatically by d e p o s i t i n g an electrocle of a suitable shape on top of the h e t e r o s t r u c t u r e |9J. On ap-plying a negative voltagc to the split-gate electiodc shown m Fig. l the electron gas u n d e r n e a t h the gale s t r u c t u r e is depletecl, crealing two 2DEG regions (i and c) electiically isolated from the rest of the 2DEG — apart from the two n a r r o w and s h o i t constrictions ( p o i n t contacts)

under the 250 n m wide openings in the split-ga(e. Tlie clevices stuclied had point contact separalions L of 1.5 and 3.0 //m, bolh values heing below the mean free path of 9 /im estimated from the m o b i l i l y . Because the depletion potential cxtends laterally beyoncl the gate p a t t e r n for high (negative) gate voltages, one can force the constrictions to become progressively narrowcr (at the same time reducing the electron gas density in the constrictions) — until they are fully pinchecl off. By ( h i s technique it is possible to create point contacts of variable w i d t h W, something which is not realizable in a metal. N o t e t h a t W is comparable in magnitude to λ\· (which was 40 nm in f h e devices stuclied). These are ψιαηΐιιιη point contacts, äs evidenced by t h e i r conductance which was

discovered to be approximately quanti'/.ed in u n i t s of 2c~/// 1 1 0 , 1 1], Electron focusing can be seen äs a t r a n s m i s s i o n experiment in electron optics. The classical regime then corresponds to geometrical optics, the q u a n t u m regime to wave optics. The optical analogy is useftil, both (o linderstand the experiments and lo inspire new ones [12]. An a l t e r n a t i v e point of view is t h a t coherent electron focusing is a prototype of a non-Iocal resistance measurement in tlie q u a n t u m b a l l i s t i c t r a n s p o r t re-gime, such äs studied extensivcly in n a r r o w - c h a n n e l geometries [13]. L o n g i t u d i n a l resistanccs which are negative, not + B Symmetrie, and dependent on the properties of the current and vollagc contacts äs well äs on their S e p a r a t i o n ; periodic and a p e r i o d i c magnetoresistance oscil-lations; absence of local e q u i l i b r i u m — these are all characterislic fea-tures of this t r a n s p o r t regime w h i c h appear in a most extreme and bare form in the electron focusing geometry. One reason for the s i m p l i f i -cation offerecl by this geometry is t h a t the c u r r e n t and v o l f a g e contacts, being point contacts, are not nearly äs invasive äs the widc Icads in a M a l l bar geometry [14]. A n o t h e r reason is t h a t the electrons interact with only one b o u n d a r y (insteacl of two in a n a r r o w channel).

The outline of this paper is äs follows. In See. 2 the e x p e r i m e n l a l results on electron focusing [5,8] are describccl äs a t r a n s m i s s i o n e x p e r i m e n t in a 2DEG. A theoretical description |6,8j is given in See. 3, in tenns of mode interference in (he wave guide form cd by the magnetic field al the 2DEG b o u n d a r y . In See. 4 we discuss the q u a n t u m H a l l effcct in the electron focusing geometry [7,K] äs a non-Iocal resistance measurement. The (heoretical framework uscd to r c l a t e these two a l t e r n a t i v e de-scriptions is the L a n d a u e r - ß ü ü i k e r f o r m a l i s m [ 1 5 , 1 6 ] , w h i c h treats a resistance measurement äs a t r a n s m i s s i o n e x p e r i m e n t . We concludc in See. 5.

2 IVlirror, Icns, and point source

Θ Β

Fig.2

Top: Skipping orbits along thc 2DEG bounclnry. Thc trojcclorics are drnwn u p (o thc (liird specular rcflcction. Bottoni: Plol of die causlics, which nrc (lic collcction of focal poinls of Ihc trajcctorics. [ Pro m Rcf 8. ]

2DEG. The injected electrons all have the same Fcrmi velocity v,,, b u t in different directions. Electrons are detected if (hey reach the adjacent collector (c), aftcr one or more specular reflections at the b o u n d a r y connecting i and c. These skipping nrhils are composed of Iranslated circular arcs of identical radius /cyd =///cr/<?/?, which is the cyclotron raclius in a perpendicular magnetic field B (k\< = mv\://i is the Fermi wave vector). The focusing action of (he magnetic field is evident in Fig. 2 (top) from the black lines of high density of (rajectories. These lines are known in optics äs caitxticx, and are plotled separately in Fig. 2 (bo(tom). The caustics intersect (he 2DEG b o u n d a r y a( m u l t i p l e s of the cyclotron d i a m e t e r from t h e injector. As the m a g n e t i c field is in-creased, a series of (hese focal p o i n t s shifts past (he collector. The eleciron flux i n c i d e n t on the collector tluis reaches a m a x i m t i m when-ever its Separation L from the injector is an integer m u l t i p l e of 2/cyc, .

This occurs when B = }>BrociK , p = l ,2,..., w i t h

focii s — ^ I' l ' \ /

For a given injected c u r r e n t 7, the voltage Vc on the collector is pro-p o r t i o n a l to the incident f l u x . The classical pro-p i c l u r e I h u s pro-predicts a series of e q u i d i s t a n t peaks in thc collector voltage äs a f u n c l i o n of m a g n e l i c field.

In Fig. 3 (top) we show such a classical focusing s p e c t r u m , calculaled for p a r a m e l e r s corresponding to (he e x p e r i m e n t discussed below

Fig.3

Botlom: Πχ per! mental clcctron focusing spcctrum (7'= 50 mK, L— 3.0 μ m) in (hc gcncralized Hall rcsistancc configuraüon dc-pictcd in Ihc insct. The Iwo Iraccs a and h arc measurcd wilh intcrchangcd currcnt and vollagc Icads, and demonslratc thc injcctor-collcclor rcciprocily äs well äs (hc rcpioducibility ο Γ Ihc finc slructiirc. Top: Calculalcd classical focusing spcctrum cor-rcsponding to Ihc cxpcrimcntal tracc i? (50 n m wiclc poinl con-lacls wcic assumcd). Thc dashcd linc is thc cxlrapolalion of thc classical Hall icsislancc sccn in rcvcrsc ficlds. [ From Rcf 8. ]

-0.3 -'0.2 -0.1 0 0.1 B (T)

0.2

height of subsequent peak.s bccause of partially diffuse scallering nL the metal surface. Note that the peaks occur in one fielcl direction only; In reverse fields the focal points are at the vvrong siele of the injector for detection, and the n o r m a l Hall resistancc is obtaincd. The experimental result for a 2DEG is shovvn in the bottom half of Fig. 3 (trace a ; trace /; is discussed below). Λ series of five focusing peaks is evidenl at the expectecl positions. This observation by itself has (wo i m p o r t a n t im-plicalions:

• A point contact acts äs a monochromatic point source of ballistic eleclrons wilh a well-definecl energy;

• The electrostatically defined 2DEG b o u n d a r y is a good m i r r o r w i l h little diffuse scattering.

Fig. 3 is obtained in a m e a s u r i n g c o n f i g u r a t i o n (inset) in which an im-a g i n im-a r y line connecting the voltim-age probes crosses thim-at between the current source and d r a i n . This is the c o n f i g u r a t i o n for a generalized Hall resistance measurement. Alternatively, one can measure a general-ized l o n g i t u d i n a l resistance, in the c o n f i g u r a t i o n shown in the inset of Fig. 4. One then measures the focusing peaks w i t h o u t a superimposed H a l l slope. Note that the experimental l o n g i t u d i n a l resistance (Fig. 4, bottom) becomes negative. This is a classical result of magnetic focusing, äs demonstrated by the calculation shown in the top h a l f of Fig. 4. B ü t t i k e r [18] has stuclied negative l o n g i t u d i n a l resistances in a different ( H a l l bar) geometry.

1.0

a 0.5

0-a 0.5

As Fig. 3, buf in thc longitudinn! tc.sistnncc configuralion. f From Rcf S. l

-0.3 -0.2 -0.1 0 0.1 0.2 0.3 B (T)

4), bul sample dependent. H is only resolved at Iow lemperatures (hclow l K.) and small injeclion voltages (l!ie measurcments shown are (aken at 50 mK and a few μλ? AC voitage over (he injeclor). Λ nice demon-stration of the reproducibility of the fine structure i.s obtained npon inlerchanging current and voitage leads, so (hat Ihe injector hecomes the collector and vice versa. The resulting focusing spcctnim shown in Fig. 3 (trace b) is almost the precise mirror image of tlie original one (trace

a) — although this particular device had a strong asymmefry in the

widths of injector and collector. The symmetry in Ihe focusing spectra is a consequence of the fundamental reciprocity relation derived hy

Büttiker [16], which generalizes the familiär On.sager-Casimir symmetry relation for the resislivity icnxor to resistanccs (see See. 4).

The fine structure on the focusing peaks in Figs. 3 and 4 is the first in-dication that electron focusing in a 2DEG is qualitatively different from the corresponding experiment in metals. A t higher magnetic fields the resemblance to the classical focusing spectrum is lost, sce Fig. 5. A Fourier transform of the spectrum for Π > 0.8 T (inset in Fig. 5) shows

tha( the large-amplitude high-ficlcl oscillatinns have a dominant periodicity of 0.1 T, which is approximately thc same äs (he periodicity 7?(OCi„ of the much smaller focusing peaks at Iow magnetic fields (/?coci,s

in Fig. 5 differs from Fig. 3 because of a smaller L= 1.5 /im ). This dominant periodicity is the result of quantum interference between the different trajectories in Fig. 2 which take an electron from injector to collector, [ In See. 3 we show this in a mode picture, which in the WKB approximation is equivalent to calculating the interferences of the (complex) probability amplitude along classical trajectories. The lalter ray picture is treated extensively in Ref. 8. ] The theoretical analysis implies for the experiment that:

10 G 04 08 12 16 B (Tesla) 24 Fig.5

Expciimcnlal clcclron foctising spcctrimi ovcr n l arger fielt! ränge and Tor vcry narrow point contacls (cstimnlcd widlh 20 - 40 nm; T= 50 m K , L = l .5/mi). The inset givcs thc Fouiicr transform for R > 0.8 T. The high-fickl os-cillalions havc Ihc samc domi-n a domi-n t pciiodicity äs Ihc lovv-fickl focusing pcaks — bul with a mucli l arge r amplitudc. [ Pro m R c f R . l

3 Edge slalcs and skipping orbits

Magnetic edge states [19,20] are transverse inodes of a wave guide of w i d t h ~ /cyc] formed by thc m a g n e l i c fielcl at tlie 2DEG b o u n d a r y . The

edge states at tlie Fermi level are labclled by a q u a n t u m n u m b e r n = 1,2 ... N, with W = /cr.· /cyd /2 tlie total n u m b e r of p r o p a g a t i n g modes

or edge channels (for simplicity we ignore here tlie discreleness of N). An injector of w i d t h below X\: cxcites a coherent superposition of these propagating modes (plus evanescent modes, which using the ray trcat-ment of Ref. 8 are fonnd to give only a s m a l l c o n t r i b u t i o n for (he large k]. L considered, and will be neglected here). The wave function M' is of the form

N

(2)

Here k„ is the wave n u m b e r for p r o p a g a t i o n of mocle /; in (he y — direc-tion (along (he 2DEG b o u n d a r y , see Fig. 2 for o u r choice of axes), f„(x) is the transverse a m p l i t u d e profile of mocle n, and a„ its excitation factor. For /cr L > l the phase factors exp(i/c„ L) vary rapidly äs a

Consider again the classical s k i p p i n g orbits (Hg. 2). The po.sition („x, j') of the electron on the circle w i t h center coordinatcs (X, Y) can be expressed in terms of il.s velocity v by

x — X + vy l<oc , y = V- νΛ. /coc , (3)

w i t h o)c ~ eB/m the cyclotron frequency. Note ( h a t the Separation X of the center from tlie b o u n d a r y is c o n s t a n t on a s k i p p i n g o r b i t , only the center c o o r d i n a t e Y p a r a l l e l to the houndary changes at each specular reflection. The c a n o n i c a l m o m e n t u m of (he elecfron is p = mv —eA. In the L a n c l a u gauge Λ = (0,/?.v,0) we have

Px= mvx - Py= -eß X . (4)

The wave n u m b e r k corresponds classically to the canonic.il i n o m e n f u m component />y= h/i, so t h n t in view of Eq. (4) we have the

corrcsponcl-ence /c = — (cßjh) X. Since the m o l i o n projected on (he .v — a x i s is peri-odic, one can apply the Bohr-Sommerfeld q u a n t i z a t i o n r u l e [21J

~ $/>xdx + y = 2πη . (5)

The integral is over one period of the m o l i o n , n is an integer, and y is the sum of the phase s h i f t s acquired at the two l u r n i n g p o i n l s of tlie m o t i o n . The phase shift upon reflection at the b o u n d a r y is π (for an i n f i n i t e barrier potential, to ensure that inciclent and reflccted waves cancel); The other t u r n i n g point is a caustic of the s k i p p i n g orbits w i t h constant X, leading to a phase shift of — π / 2 [22]. This (otals to

y — π/2. Using also Eqs. (3) and (4) we may t h u s w r i t e Eq. (5) in (he

fo rm

n= 1,2,... N. (6)

This quantization rule h a s (he simple geomelrical i n t e r p r e f a l i o n [20] (hat the f l u x enclosed by one arc of the s k i p p i n g o r b i t and the b o u n d a r y equals (n— 1/4) times the flux q u a n t u m h\e (see insets in Fig. 6). Eq. (6) detcrmines, for a given magnetic fielcl, the energy E— niv2 /2 äs a f u n c t i o n of the q u a n t u m n u m b e r n and the wave n u m b e r

k — — (cBlti) X. To carry out the I n t e g r a t i o n in Eq. (6) we express y in

terms of χ by means of Eq. (3). The resultin g energy spectru m E„ (/<.) is given by

arccosξ - ξ (l - ξ2 )'/ 2 ) = 2π (η - ~), ξ = hk (2mΕ} Ι / 2 , (7)

ho) v > 4

C

10

n-1

-3 -1

Fig.6 Bncrgy spcctium F.n (/<) oP magnctic cclgc slnlcs ;it an i n f i n i t e b a n i c r polcnlial

boundnry. Note t h a l klm = —.V//,,,, willi X Uic Separation of Ihc oibital ccnlcr Pro m

tlic b o u n d n r y and /„, = (/i/cß)1'2 Uic mngnclic Icngth. The inscts show classic;il skipping orbits for positive and negative k. In thc semi-classical approximalion thc magnetic flux Ihrough tlic shadcd arcas is q u a n ü / c d . Tlic icsull Pro m Rq. (1) (solid curvcs) is indistinguishablc Pro m thc cxact solulion (daslicd curvcs, Proni RcP. 23), unIcss k is within !//„, oP thc transition Pro m s k i p p i n g lo cyclotron orbits (dotlccl curvc). [Prom ReP. 8.]

+kFL

0)

(Λ

CÖ

•a

slope -2nO/Drocua

Fig.7

Phase k„ L of thc cclgc channcls nt (hc colleclor, calculntcd from Eq. (7). Note Ihc clomain of np-proximatcly linear /; — dc-pcndcncc of thc phasc, rcsponsihlc for Ihc oscil-lalions vvith fytKm —

pciioclicity. [ From RcfX. ]

n —

To determine the a m p l i t u d e of the oscilliitions in the collcctor vol(<ige, we need to k n o w the e x c i t a t i o n factors of (hc modes by thc injector and the transmission a m p l i t u d e t h r o u g h the collector. In Ref. 8 we calcu-lated these q u a n t i t i e s using a point-dipole injector and a transmission a m p l i t u d e p r o p o r t i o n a l to the derivative ff^'ldx of the unperturbcd wavc function at the collector — thereby neglecting the finite wiclth of the injector and collector point contacts. The result obiained there can be w r i t t e n in the form

le1

N

J_ V

N LJ

In Fig. 8 we liave plotted the focusing spectrum from Eq. (9), corre-sponding to the experimental Fig. 5. The inset shows the Fourier transform for B > 0.8 T. There is no detailed one-to-one correspond-ence between the e x p e r i m e n t a l and theoretical spectra. No such corre-spondence was to be expectecl in view of the sensitivity of (he experimental spectrum to s m a l l v a r i a t i o n s in gate voltage (which defines (he point contacts and the 2DEG b o u n d a r y ) . Those features of the ex-perimental spectrum which are insensitive to the precise measurement conditions are, however, well reproduced by the calculation: We recog-nize in Fig. 8 the Iow-fielcl focusing peaks and (he l a r g e - a m p l i t u d e high-field oscillations w i t h the same p e r i o d i c i f y . [ Thc reason ( h a t the periodicity 5rocil, in Fig. X is somewhat langer l h a n in Fig. 5 is most likely

c: 10 o zo Γι equency (l/T) 04 08 12 13 (T) 16 24

Fig.8 Focusing spccüum cnlculatcd fiom Eq. (9), Γη r paiamctcts concspond'mg to thc cxpcrimenlal Fig. 5. The insct sliows tlic Fouricr Unnsform for B > 0.8 T.

I n f i n i t c s i m a l l y s m a l l p o i n t conlacl widllis nrc assumcd in llie calculalion.

This m a x i m u m a m p l i t u d e is not far below the tlieoretical u p per bound of h/2e2 « 13 Ι(Ω, which follovvs from Eq. (9) if we assume that all the

modes interfere constructively. This indicates that a maximal phase co-hcrence is realized in the experiment, and implies that:

• The experimentell injector and collecfor point contacts resemhle tlie idealized point source/cletcctor in the c a l c u l a t i o n ;

• Scattering events other than specular scattering on thc boundary can be largely ignored (since any other inelastic äs well äs elastic scattering events would scramble tlie pliases and recluce the o s c i l l a t i o n s vvilh

•ßfocm- periodicity).

It follows from Eq. (9) t h a t if inlerference of thc modcs is ignored, the n o r m a l q u a n t u m I l a l l resistancc h/2Nc2 is o b t a i n c d . This is not a general result, but depends spccifically on thc properlies of (he i n j e c t o r and collector point contacts — äs we w i l l discuss in the f o l l o w i n g section.

4 Quantum point contncls ns Landau Icvcl sclcctors

Fig.9

SchemaLic polcntial landscapc, sliowing thc 2DEG hounclniy and thc saddlc-shaped injcctor and collcclor point contacts. In a strong magnetic ficld thc cdgc channcls arc cxtcndcd along cquipotcntials (Cq. (10)), äs indi-catcd lieic for /; = l ,2 (Ihc airows point in thc diicclion o f m o t i o n ) . In this casc a Hall condiictancc of (2c-//i)N wilh N= l would bc mcasurcd by thc point con-tacts — in spitc of thc picscncc of 2 occupicd Lanclau Icvcls in thc bulk 2DEG.

electrostatic potential V(x,y) d e f i n i n g the p o i n t c o n t a c t s becomes im-portant, and the p o i n t injector/detector model tised in (he previous section — while adequate at Iower magnetic fiekls — is insufficient. Schematically, V(x,y) is represented in Fig. 9. F r i n g i n g fields from (he split-gate creale a potential b a r r i e r in the point contacts, so t h a t V has a saddle form äs sliown. The heights of the barriers E·, , Ec in (he injector and collector are separately a d j u s t a b l e by means of the voltages on the split-gates, and can be determined from (he condtictances of the i n d i v i d u a l point contacls [24]. The w i d t h of the p o i n t contacts does not play a role, because it is larger t h a n /cyc( . The a d i a b a t i c ( r a n s p o r t is

along equipotentials äs indicated in Fig. 9 (arrovvs p o i n l in (he direction of motion, determined by the p o t e n t i a l gradient). The energy of (he equipotential is the guiding rcnler energy EG , w h i c h is given for edge channel n by

ECj = £,,- - (n - — ) h(oc (10)

(Zeeman s p i n - s p l i t t i n g of the energy levels should be included at large magnetic fields, but is ignored here for simplicity). The edge channels can only be t r a n s m i t t e d t h r o n g h a p o i n t contact if EG exceeds (he po-t e n po-t i a l barrier heighpo-t (disregarding po-t u n n e l i n g po-t h r o u g h (he barrier). The injector thus injects N, ~ (Ep — E\)lho)c edge c h a n n e l s i n t o the 2DEG, while the collector is capable of detecting Nc Ä; (E\; — /ic)//icoc channels.

Along the b o u n d a r y of the 2DEG, however, a larger n u m b e r of N χ Ef lho)c edge channels, equal to the number of b u l k L a n d a u levels

in the 2DEG, are available for the current transport. The selective population, and detection, of Lanclau levels leads to cleviations from the normal Hall resistance.

to Iransmission p r o b a b i l i t i e s i n t o current and voltage probes. Consicler the geometry in Fig. l o f a t h r e e - t e r m i n a l c o n c l u c t o r w i t h p o i n t contacls in two of the probes. The probes are connectecl by perfect leacls to res-ervoirs which have a c o n s t a n t electro-chemical p o t e n t i a l . We denote by / f , and //c the chemical polentials of the two rcservoirs connectecl,

re-spectively, to the injector and collector p o i n t c o n l a c t , and by //,, the chemical potential of the thircl reservoir (the c u r r e n t d r a i n ) . F o l l o w i n g B ü t t i k e r [16], we can relate t h e currents 4 (a = i,c,d) in the three leads to these chemical potentials via the t r a n s m i s s i o n p r o b a b i l i t i e s T«_ß (from reservoir α to reservoir ß ) and reflection p r o b a b i l i t i e s Rx (Pro m

reservoir α back to the same reservoir), h \~^

~

/« = (jv

- κ

κ - 2^ η ->« /'/» - oo

/M«

N« being the number of occupied modes in the leacl a. We now impose

the condition t h a t the collector draws no net current, which implies 4 = 0 and 7d = — 7, , and choose our zero of energy such that //<: = 0. One then finds from Eq. (l 1) the two e q u a t i o n s

rr*

/'i ' - /i = (Wi-

ÄO/M-

T

_

j / i

, (12)

c

'

_

,

and obtains for the ratio of collector voltage Vc = /;c j e (measured

rela-tive to the voltage of the c u r r e n t d r a i n ) to injected c u r r e n t 1\ the result

, /i h G; Gc - δ

Here δ = (2e2 /hf T·^ CTC^·, , and G] = (2e2lh}(N-,-R-() and Gc ΞΞ (2e2 l/i)(Nc — Rc) denote the conductances of the injector and collector point contact, rcspectively. The injector-collector reciprocity in electron focusing, demonstrated in Fig. 3, is manifest in Eq. (13), since G, and Gc are even in D and [16] η_ c (B) = 7'c^ ·, ( -#).

In the electron focusing geometry the term δ in Eq. (13) can be neg-lected, since Τ,,^-,χΟ. An a d d i t i o n a l s i m p l i f i c a t i o n is possible in the acliabatic transport regime. We consider the casc that the barrier in one of the two p o i n t contacts is sufficienlly higher than in the other, (o en-sure that eleclrons which are transmitted over the highest barrier w i l l have a negligible p r o b a b i l i t y of being reflccted al the lowest barrier. Then 7"j_( c is dominated by (he transmission p r o b a b i l i t y over the highest barrier, Τ|_ c « min (N\ — R-t , Nc — Rc ) . S u b s t i t u t i o n into Eq. (l 1) gives the remarkable result that (he Hall condnctancc G\\ = L, JVC measured in the electron focusing geometry can be expressed entirely in terms of the

contact conductances G\ and Gc ,

O 1 o GH GH - 2 2 -18 GATE VOLTAGE (V) Fig. 10.

nxpcrimcntal conclation bclwccn tlic coiuliicüinccs G,, Gc of injcctor and

collector, and (hc Hall concluctancc

GH s /, / Kc, shown to d c m o n s t i a l c Ihc

valiclily of F.q. (14) (T= 1.3 K, L = 1.5 /im). The magncüc ficld was kcpt fixcd (top: B = 2.5 T, boüom:

B = 3.8 T ). By incicasing tlic galc

voltagc on onc h a l f of tlic splil-galc dcfining Ihc injcctor, (7, was varicd at conslanl Gc . [ Fiom Rcf 7. ]

Eq. (14) teils us t h a t q u a n t i z e d values of Gn occur not at (2f2 //;) /V, äs

one woulcl expcct from the N L a n d a u Icvel.s in (he 2 D E G — Im t a( t h e smaller valne of (2e2 //;) max (N, , Nc) . Moreover, thcre is no quantized

H a l l conductance unless the largest of the two contact conductances is q u a n t i z e d . As shown in Fig. 10, t h i s is incleccl obscrved e x p e r i m e n t a l l y . Notice in particular hovv any d e v i a t i o n from q u a n t i / a t i o n in max (Gj , Gc) is f a i t h f u l l y reproduced in Gn . The i m p l i c a t i o n of this

e x p e r i m e n t is that:

• Point contacts can be usecl to sclectivcly p o p u l a l e and detect L a n d a u levels at the 2DEG b o u n c l a r y ;

• A d i a b a t i c t r a n s p o r t (i.e. t r a n s p o r t in the absence of i n l c r - L a n c l a u level scaltering) has been realized ovcr a d i s t a n c e of 1.5/mi a l o n g the 2DEG b o u n d a r y .

As discnssecl by Büttiker [25], the f u n d a m e n t a l origin for clcviations from the n o r m a l q u a n t u m H a l l effect is the absence of local e q u i l i b r i u m among the edge c h a n n e l s . Selective p o p u l a t i o n is indeed an extreme ex-ample of a non-equilibrium population. Reccnt relatcd experiments [26,27J have demonstrated t h a t a n o n - e q u i l i b r i i i m p o p u l a t i o n of edge channels can be m a i n t a i n e d on even longer l e n g t h scales, possibly äs large äs several h u n d r e d microns. It remains a ( h e o r e t i c a l c h a l l e n g e to e x p l a i n these s u r p r i s i n g l y long r e l a x a t i o n lengths.

5 Conclusion

rela-tively lange value of tlie Fermi wave length. The adiabatic transport discussed in See. 4 is also made possible by the large Ap , sinee now 4yd = ΙιΙεΒλρ can become comparable to W at magnetic fields of a few Tesla. To achieve the same in a metal would require fields over 100 T. The difference in cncrgy scale between a metal and a 2DEG manifests itself in the dependence of the focnsing spectrum on the voltage drop over the injector. In metals, electrons are injected al energies above E\s which are generally much less than Εγ ~ 5 eV [28]. In contrast,

E\- ~ l O m e V in a 2DEG, and DC-biasing the s m a l l AC injection

voltage used in the electron focnsing experiment slioulcl lead to a no-ticeable shift in the focusing peaks, in analogy with a β —spectrometer. In the simplest model one would have (cf. Eq. (1)) #ibcm «: (E\> + fFijc)"2 > so t-hat f°r a DC bias Vnc — \ mV one would expect a 5% shift in the focusing periodicity — provided the hot electrons remain ballistic. This is incleed observed [29J, a l t h o u g h devi-ations from this simple behavior are found for langer DC biases (possi-bly related to the non-linear current-vollage characteristics of the point contacts IhemselvespO]). The observation of hot-electron transport over several microns is remarkable, and unexpected from related work in different Systems [31].

The main result of the theoretical analysis of coherent electron focusing in See. 3 is (he demonstration of high-fiekl oscillations w i t h

Bfocm —periodicity, but much langer amplitude t h a n the Iow-field

focus-ing peaks. This is also the feature of the experimental focusfocus-ing spectra which is insensitive to small changes in gale voltage and which is found in both the devices studied. The theory can be improved in sevenal ways. This w i l l affect the detailed fonm of the spectra, but probably not the fundamental periodicity. Since the exact wave functions of the edge states are known (Weber functions), one could go beyond the WICB approximation. This w i l l become important at large magnetic fields, when the relevant edge states have small q u a n t u m numbers. In this regime one vvould also have to take into account a possible B— de-pendence of Ef relative to the conduction band bottom (due to p i n n i n g of the Fermi energy at the Lanclau levels). It would be intenesting to find out to w h a t extenl this bulk effect is reduced at the 2DEG bouncl-any by the presence of edge states to fill the gap between the Landau levels.

Energy averaging due lo a finite temperature is not (he reason for this difference (lemperatures on the order of 10 K are necessary (o smear out the rapid oscillations). We surmise (hat the rapid oscillations are recluced by the collimation effect proposed o r i g i n a l l y [32] to explain (he n o n - a d d i t i v i t y of the resistance of (wo opposile poinl contacts in series [33] (and more recently invoked [34] (o e x p l a i n the q u e n c h i n g of the H a l l resistance in a n a r r o w - c h a n n e l geometry [13]). Both the flaring of a point contact to form a h ö r n , and the presence of a potential b a r r i e r in the point contact region tend lo collimate the injecfed electron beam [32], so that electrons are p r e d o m i n a n t l y injected at right angles to (he bonndary. The q u a n t u m mechanical correspondence discussed in See. 3 then implies that such a point contact excites (and detects) predomi-nantly the edge channels witli q u a n t u m number n close to N/2, at the expense of channels with smaller or larger /;. Since the former edge channels are responsible for the oscillations w i t h /?|-oclls — p e r i o d i c i l y ,

while the latter give rise to rapid aperiodic oscillations (see Fig. 7 and the accompanying discussion), the collimation effect provides one mechanism for the absence of rapid oscillations in the experimental fo-cusing spectrum.

Acknowledgemenl

The authors havc grcatly bcncfittccl from thcir collahorntion with M.R.I. Brockaarl, C.T. Foxon, C.J.P.M. Harmans, J.J. Harris, L.P. Kouwcnhovcn, P . I I . M . van Loosdrccht, D. van der Marcl, .1.13. Mooij, M . F . I I . Scluiurmans, .Ι.Λ. Pah, Π.Μ.Μ. Willcms, and J.G. W i l l i a m s o n .

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Festkörperprobleme 29 Advances in Solid State Physics