Wetensch. publicatieFaculteit der Wiskunde en Natuurwetenschappe

Beenakker, C.W.J.

Foxon, C.T.

Kouwenhoven, L.P.

Molenkamp, L.W.

Staring, A.A.M.

Leiden University Scholary Publications

PHYSICAL REVIEW B VOLUME 41, NUMBER 12156 6715 APRIL 1990-11Magnetoconductance of two point contacts in seriesA. A. M. Staring,* L. W. Molenkamp, and C. W. J. BeenakkerPhilips Research Laboratories, NL-5600JA Eindhoven, The NetherlandsL. P. KouwenhovenDelft University of Technology, NL-2600GA Delft, The NetherlandsC. T. FoxonPhilips Research Laboratories, Redhill, Surrey RH1 5HA, England(Received 6 November 1989)The magnetic-field dependence of the conductance of two opposite point contacts in series is stud-ied experimentally, for magnetic flelds extending into the quantum Hall effect regime. The magne-toconductance has a nonmonotonic, "camel-back" shape, in quantitative agreement with a recenttheory. In addition, Aharonov-Bohm-type periodic magnetoconductance oscillations are observed,which are attributed to circulating edge states in a shallow potential basin between the point con-tacts.The conductance of a constriction in a high-mobilitytwo-dimensional electron gas (2D EG) measures directlythe number of occupied one-dimensional subbands, N, inthe constriction. This was demonstrated in the experi-ments by Van Wees et al.' and Wharam et al.,2 whodiscovered that the two-terminal conductance G of such aquantum point contact is given approximately byG = ̂ -N. ü)A magnetic field B perpendicular to the 2D EG depopu-lates the subbands, leading to a monotonic decrease3 of Gwith B. The series conductance of two opposite pointcontacts in general does not have äs simple a relation äsEq. (1) with the number of occupied subbands. In weakmagnetic field the series conductance was found byWharam et a/.4 and Beton et al.5 to be stronglyenhanced above the value which would follow from Ohm-ic addition of the separate point-contact resistances. Aspointed out by Beenakker and Van Houten,6 this is due tothe direct transmission of ballistic electrons from onepoint contact to the other, without intervening equilibra-tion. The direct transmission probability is enhanced bythe collimation of the electron beam injected into the 2DEG by a horn-shaped point contact. This hörn collima-tion effect6 was recently demonstrated experimentally.7In strenger magnetic fields, where direct transmissionis suppressed, a simple relation between the two-terminalseries conductance Gseries and the number of occupiedsubbands becomes possible. This regime is the subject ofthe present paper. Theoretically, it was predicted in Ref.6 that Gseries has a nonmonotonic ("camel-back" shaped)B dependence, provided that the transmission from onepoint contact to the other occurs with intervening equili-bration of the current-carrying edge states. This curiousbehavior results from the following relation betweenG^HPQ and the number of subbands:6="series rJ_N,(2)Equation (2) results from the additivity of the four-terminal longitudinal magnetoresistance of the individualpoint contacts,6'8 which holds in the case of interveningequilibration. Here, N: and N2 are the number of occu-pied subbands in the two point contacts, and NL is thenumber of Landau levels occupied in the region betweenthe point contacts. At low magnetic fields Gseries first in-creases because the depopulation of the hybrid magne-toelectric subbands in the point contacts is delayed com-pared to the depopulation of Landau levels in the wide2D EG between the point contacts. When the cyclotronradius becomes much smaller than the point-contactwidth, the confinement of the electrons in the point con-tacts is fully magnetic, and Gseries decreases äs a result ofthe overall depopulation of Landau levels. Equation (2) isto be contrasted with the result which holds in the ab-sence of equilibration, i.e., assuming adiabatic transport(a Situation realized at moderate magnetic fields in thegeometry of Ref. 9). One then has2e2(3)which, like the conductance of a single point contact, de-creases monotonically with B. A nonmonotonic B depen-dence is therefore a signature of equilibration among theedge states.In this paper we present measurements of themagnetic-field dependence of the conductance of two op-posite point contacts in series, for magnetic fields extend-ing into the quantum-Hall-effect regime. We find goodagreement with Eq. (2). In addition, we find an oscillato-ry structure superimposed on the magnetoconductancepredicted by Eq. (2). This structure is periodic in themagnetic field and occurs only in transitions between the41 8461 ©1990 The American Physical Society8462 A. A. M. STARING et al 41quantum Hall plateaus. We attribute it to an Aharonov-Bohm-type quantum mterference effecl between edgestates, similar to the effects observed in Refs. 10-12.The measurements were performed on a sample con-sisting of two closely spaced opposite point contacts,defined in the 2D EG of a GaAs-Al^Ga^^As hetero-junction wafer by means of lateral depletion techniques.For sample fabncation we employ electron-beam lithog-raphy in a polymethylmethacrylate double-layer resist(using a Philips EBPG-4 Beamwriter) and lift off to de-posit gold gates on top of a previously fabricated Hall-barstructure. The point contacts have a lithographic open-ing of 0.4 μτη and are separated by a Ι-μιη-wide, 18-μιη-long channel (see inset in Fig. 1). The wide 2D EG hasan electron density of 2.4X10 1 5 m~2 and a mobility ofabout 102 m 2/Vs. The sample was kept at 100 mK in adilution refrigerator and the conductance measurementswere performed using an ac lock-m technique (15 Hz)with a small excitation voltage (20 μ V over the sampleand a series resistance of 3.3 kil) m order to avoid elec-tron heatmg.In Fig. l we show magnetoconductance curves ob-tamed from a two-termmal measurement for the case ofequal gate voltage V„ on all gates defining the point con-tacts, which corresponds to Nl , =Ν. The observedB dependence of Gsenes can be descnbed äs a doublepeaked or "camel-back" shaped magnetoconductance(with peaks around ±0.5 T) with superimposed fine struc-ture. The overall nonmonotonic B dependence is äs pre-dicted by Eq. (2), and will be discussed in more detailbelow. Part of the fine structure consists ofShubnikov-de Haas (SdH) oscillations periodic in l/B,noticeable äs rapid oscillations around +0.5 T, and broaddips in the quantum Hall plateaus at higher magneticfields. These SdH oscillations originate in the wide 2DG00bωO-g-05 05 1 15 2 25Magnetic field (T)FIG 1. Two-termmal magnetoconductance measurementsfor three different values of the gate voltage (solid lines). Forclanty, subsequent curves from bottom to top are offset by0 5X1CT 4 Ω"1, with the Vg = — l.O V curve bemg shown at itsactual value. The dotted lines are obtamed by using Eqs (2), (5),and (6) with the values given m Table I. The inset gives aschematic diagram of the sample (not to scale) The pomt-contact Separation is l μτηEG leads, äs can be deduced from their periodicity,which is a direct measure of the electron density.Around zero field reproducible rapid aperiodic oscilla-tions are seen, remmiscent of the conductance fluctua-tions in disordered conductors. A more unusual finestructure occurs in the transitions between the quantumHall plateaus, and consists of regulär oscillations whichare approximately periodic in B. An enlargement ofsome of these oscillations is given in Fig. 2. We attributethese to the Aharonov-Bohm (AB) eifect in a smgly con-nected geometry.10"12The proposed mechanism for these AB-type oscilla-tions is illustrated in Fig. 3. Electrons in edge state (a)are transmitted without backscattermg, and thus havetransmission probability l. Those in edge state (b) can betransmitted via tunnelmg through the potential barner inthe point contacts to a circulating edge state (c), whichfollows the equipotentials at the guiding center energyEG=EF — (n — \}fiK>c (with Landau-level index n and cy-clotron frequency a>c) in a shallow potential basin be-tween the point contacts. The existence of such a poten-0240170.1016016CooOωuCD4->O13T3OO009(b)24 280170.100.14007(c;24(d)1.7 21Magnetic field (T)FIG. 2. Enlargement of the periodic oscillations observed mthe magnetoconductance for four different values of the gatevoltage (a) Vg = -0 8 V, (b) Kg = -0 9 V, (c) Fg = - l 0 V, (d)Vg = — 1.1 V The V,, = — 0.8 V oscillations are observed in thetransition from quantum Hall plateau ι =3 to / =2, the othersin the ι =2 to / =1 transition These measurements have beenmade in a four-termmal longitudmal configuration m order tosuppress the Shubmkov-de Haas oscillations, onginating in thewide 2D EG leads41 MAGNETOCONDUCTANCE OF TWO POINT CONTACTS IN SERIES 8463B χ(e) v.— (b)-(a)FIG. 3. Illustration of the proposed mechanism for the AB-like oscillations discussed in the text.tial basin is in accord with a calculation of the electro-statics of a similar geometry by Kumar et al.13 The elec-trons can leave this circulating edge state by tunneling ei-ther back to edge state (b) or to edge state (d). An alter-native to this mechanism of resonant transmission isresonant reflection14'15 from edge state (a) to (e) via thecirculating edge state (c). This circulating edge statemust enclose a flux Φ equal to an integer number of fluxquanta h/e, i.e., B A (B) = nh/e, where A (B) is the areaenclosed by the circulating edge state (which depends onB because of the B dependence of the guiding center ener-gy). This condition leads to a transmission and reflectionprobability which close to some field B0 are periodic in Bwith periodicity Δ.Β given byddBBA(B)A(B)+BdA dEfdEr. dB(4)B=B 0The second term in this expression for Δ5 is negativesince dEG/dB <0 and dA/dEG>0 for a typical poten-tial basin, thus the effective area Aeff calculated from theperiodicity AB =h/eA&s is a lower bound on the actualarea A of the basin. The periodicity in Fig. 2 varies be-tween about 25 and 37 mT (depending on gate voltage Vgand magnetic field B0), which corresponds to values ofAeff between about 0.11 X 10"12 and 0.16X 1(T12 m2. Arough estimate of the basin area äs the area of a circle ofl-μηι diameter (the point-contact Separation) givesA «0.8X 10~12 m2, which is consistent with the lowerbound given by AsK, and forms presumably an estimatefor the upper bound of the basin area.In additional measurements of the magnetoconduc-tance of a single point contact (not shown), performed onthe same sample by grounding one pair of gates defining apoint contact, the periodic oscillations in the transitionsbetween the quantum hall plateaus, äs well äs the rapidaperiodic oscillations around zero field, are found to beabsent. This demonstrates that this fine structure doesnot result from the background resistance or from an in-dividual point contact, and is consistent with the mecha-nism proposed above.We now turn to a quantitative analysis of the "camel-back" shaped magnetoconductance, in terms of Eq. (2),which is valid for magnetic fields such that directtransmission from one point contact to the other issuppressed. From measurements of the direct transmis-sion probability, following the method of Ref. 7, wedetermine this to be the case for \B\ >0. l T. We assumea square-well confining potential in the point contacts (ofwidth W and electron density n), which should be areasonable approximation at the low (negative) gate volt-ages considered. The number of occupied hybrid magne-toelectric subbands N is given in this case by (assumingspin degeneracy and ignoring the discreteness of 7V sincewe are interested in the overall behavior only)3'16N = (5)2e B'where ξ= W/2lcyd, and lcyA=fi(2imY/2/eB is the cyclo-tron radius inside the point contacts. The number of oc-cupied Landau levels NL in the region of electron densityns between the point contacts is given byh ns(6)In the following we shall assume that n =ns, which webelieve to be a reasonable approximation at the low (neg-ative) gate voltages considered in Fig. 1. The density inthe region between the point contacts is reduced com-pared to the density of the wide 2D EG areas due tofringing fields from the gates, and is determined indepen-dently from the Hall effect (see Table I). The point-contact width W is treated äs a free parameter in fits withthe measured magnetoconductance.The results of our analysis are given in Fig. l and TableI. The fits (dotted lines) were made using a constantbackground resistance, which is estimated at 2.0 kß (in-dependent of the gate voltage) for low magnetic fields,from measurements of the quantized resistance of the in-dividual point contacts äs a function of the gate voltage.In principle, at higher magnetic fields a different valueshould be used because of the field dependence of thebackground resistance, which is expressed most clearlyby the presence of Shubnikov-de Haas oscillations of thewide 2D EG regions. Because of the large resistance ofthe series point contacts at high magnetic fields, the fit isTABLE I. Values used in Eqs. (2), (5), and (6) for obtainingthe dotted lines in Fig. 1. The point-contact width H^is a fit pa-rameter, the density n in the narrow channel has been deter-mined from the Hall effect, and the low-field background resis-tance Rb is estimated äs the difference between the resistance ofan individual point contact and the quantized value given byEq. (1).V* (V)-0.8-0.9-1.0n (1015 m"2)1.130.990.83W (nm)320260200R g (kß)2.02.02.08464 A. A. M. STARING et al. 41not very sensitive to these variations in the backgroundresistance, and they are ignored for simplicity. Goodagreement between Eq. (2) and the experimental data isobtained with W äs the only free parameter. Note that at5=0, Eq. (2) gives simple Ohmic addition of the point-contact conductances, because direct transmission fromone point contact to the other is neglected in that equa-tion. Enhancement of the conductance above the valuefollowing from Ohmic addition is (for the measurementsshown in Fig. 1) observed clearly at one of the gate volt-ages (Fg = — 0.9 V) only, but may be obscured at the oth-er gate voltages by the aperiodic oscillations which are ofthe same order of magnitude. From the difference be-tween Eq. (2) and the actual measurement (corrected forthe background resistance) we find that Gseries isenhanced by 12% over G/2, which is somewhat smallerthan the enhancement of 20% which one would expectfrom measurements of the direct transmission probabili-ty.7 Clearly, the series resistance experiment is not an ac-curate way to determine the direct transmission probabil-ity.We summarize our main results and conclusions. Wehave performed two-terminal magnetoconductance mea-surements of two opposite point contacts in series. Themagnetoconductance shows the "camel-back" shape pre-dicted previously,6 and is in quantitative agreement withthe theory. In addition, an Aharonov-Bohm-type quan-tum interference effect is observed, which is attributed toresonant tunneling via circulating edge states in a shallowpotential basin in the region between the point contacts.The authors thank C. E. Timmering and M. E. I. Broe-kaart for sample fabrication, and R. Eppenga, H. vanHouten, and J. G. Williamson for valuable discussions.One of us (A.A.M.S.) would like to thank Professor J. H.Wolter (Eindhoven University of Technology) and Pro-fessor M. F. H. Schuurmans (Philips Research Labora-tories) for continuous encouragement and for providingthe opportunity to carry out this work at the PhilipsResearch Laboratories.'Permanent address: Eindhoven University of Technology,5600 MB Eindhoven, The Netherlands.'B. J. van Wees, H. van Houten, C. W. J. Beenakker, J. G. Wil-liamson, L. P. Kouwenhoven, D. van der Marel, and C. T.Foxon, Phys. Rev. Lett. 60, 848 (1988).2D. A. Wharam, T. J. Thornton, R. Newbury, M. Pepper, H.Ahmed, J. E. F. Frost, D. G. Hasko, D. C. Peacock, D. A.Ritchie, and G. A. C. Jones, J. Phys. C 21, L209 (1988).3B. J. van Wees, L. P. Kouwenhoven, H. van Houten, C. W. J.Beenakker, J. E. Mooij, C. T. Foxon, and J. J. Harris, Phys.Rev. 838,3625(1988).4D. A. Wharam, M. Pepper, H. Ahmed, J. E. F. Frost, D. G.Hasko, D. C. Peacock, D. A. Ritchie, and G. A. C. Jones, J.Phys. C 21, L887 (1988).5P. H. Beton, B. R. Snell, P. C. Main, A. Neves, J. R. Owers-Bradley, L. Eaves, M. Henini, O. H. Hughes, S. P. Beaumont,and C. D. W. Wilkinson, J. Phys. Condens. Matter l, 7505(1989).6C. W. J. Beenakker and H. van Houten, Phys. Rev. B 39,10445(1989).7L. W. Molenkamp, A. A. M. Staring, C. W. J. Beenakker, R.Eppenga, C. E. Timmering, J. G. Williamson, C. J. P. M.Harmans, and C. T. Foxon, Phys. Rev. B 41, 1274 (1990).8H. van Houten, C. W. J. Beenakker, P. H. M. van Loosdrecht,T. J. Thornton, H. Ahmed, M. Pepper, C. T. Foxon, and J. J.Harris, Phys. Rev. B 37, 8534 (1988).9L. P. Kouwenhoven, B. J. van Wees, W. Kool, C. J. P. M. Har-mans, A. A. M. Staring, and C. T. Foxon, Phys. Rev. B 40,8083 (1989).10P. H. M. van Loosdrecht, C. W. J. Beenakker, H. van Houten,J. G. Williamson, B. J. van Wees, J. E. Mooij, C. T. Foxon,and J. J. Harris, Phys. Rev. B 38, 10 162 (1988).UB. J. van Wees, L. P. Kouwenhoven, C. J. P. M. Harmans, J.G. Williamson, C. E. Timmering, M. E. I. Broekaart, C. T.Foxon, and J. J. Harris, Phys. Rev. Lett. 62, 2523 (1989).12R. J. Brown, C. G. Smith, M. Pepper, M. J. Kelly, R. New-bury, H. Ahmed, D. G. Hasko, J. E. F. Frost, D. C. Peacock,D. A. Ritchie, and G. A. C. Jones, J. Phys. Condens. Matterl, 6291 (1989).I3A. Kumar, S. E. Laux, and F. Stern, Bull. Am. Phys. Soc. 34,589 (1989) and unpublished.14J. K. Jain and S. A. Kivelson, Phys. Rev. Lett. 60, 1542 (1988).15M. Büttiker, Phys. Rev. B 38, 12 724 (1988).16H. van Houten, C. W. J. Beenakker, J. G. Williamson, M. E. I.Broekaart, P. H. M. van Loosdrecht, B. J. van Wees, J. E.Mooij, C. T. Foxon, and J. J. Harris, Phys. Rev. B 39, 8556(1989).

Magnetoconductance of two point contacts in series

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Magnetoconductance of two point contacts in series

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