Observation of quantum asymmetry in an Aharonov-Bohm ringS. Pedersen, A. E. Hansen, A. Kristensen, C. B. So”rensen, and P. E. LindelofThe Niels Bohr Institute, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen, Denmark共Received 12 April 1999兲We have investigated the Aharonov-Bohm effect in a one-dimensional GaAs/Ga0.7Al0.3As ring at low-magnetic fields. The oscillatory magnetoconductance of these systems is systematically studied as a function ofdensity. We observe phase shifts of␲in the magnetoconductance oscillations, and halving of the fundamentalh/e period, as the density is varied. Theoretically we find agreement with the experiment, by introducing anasymmetry between the two arms of the ring.I. INTRODUCTIONThe Aharonov-Bohm effect, first proposed in 1959,1wasexperimentally realized in a normal metal cylinder in 1982 asan oscillatory magnetoresistance with period h/2e 共Ref.2兲, a periodicity originating from the interference between anelectron path and its time-reversed path.3The fundamentalh/e period was later observed in normal metal loops4andsoon after in mesoscopic semiconductor loops.5TheAharonov-Bohm effect quickly became a very fruitful re-search area in experimental mesoscopic semiconductorphysics.6–8All these early investigations, except one,9how-ever focus their attention on the Aharonov-Bohm effect atrelatively high-magnetic fields (␻c␶⬃1). From a theoreticalpoint of view the behavior of these high-field investigationshas been treated in Refs. 10 and 11.Recently, due to the perfection of device fabrication, theAharonov-Bohm effect has gained renewed interest.Aharonov-Bohm rings are now used to perform phase sensi-tive measurement on, e.g., quantum dots.12Theoretically thisexperiment has attracted much interest13,14—see also refer-ences therein. Also experiments where a local gate only af-fects the properties in one of the branches of the Aharonov-Bohm device has been performed.15Both these reports usethe idea, that by locally changing the properties of one of thearms of the ring, and studying the Aharonov-Bohm effect asa function of this perturbation, information about the changesin the phase can be extracted from the measurements. Fur-thermore, a realization of the electronic double-slit interfer-ence experiment presented surprising results,16which has re-cently been considered from a theoretical point of view.17Especially the observation of a period halving from h/e toh/2e and phase shifts of␲has attracted large interest in thesereports.All these investigations, are as in contrast to the priorones, performed at relative low magnetic fields and the per-turbation enforced on the ring is regarded as local. Further-more, they are all performed in the multimode regime.Hence, we find it of importance to study the Aharonov-Bohm effect in the single-mode regime at low-magneticfields and as a function of a global perturbation.II. EXPERIMENTOur starting point in the fabrication of the Aharonov-Bohm structures is a standard two-dimensional electron gas共2DEG兲 realized in a GaAs/Ga0.7Al0.3As heterostructure. Thetwo-dimensional electron density is n⫽ 2.0⫻ 1015m⫺ 2andthe mobility of the heterostructure is␮⫽ 90 T⫺ 1. This cor-responds to a mean free path of approximately 6␮m. The2DEG is made by conventional molecular-beam epitaxy andis situated 90 nm below the surface of the wafer. For furtherdetails regarding the heterostrucure, contacts, etc., we referto Ref. 18.Using standard e-beam lithography 共EBL兲 a 100-nm-thickPMMA etch mask is defined on the surface of the hetero-structure. The pattern written in the PMMA was transferredto the 2DEG by a 50-nm shallow wet etch inH3PO4:H2O2:H2O. The dimensions of the etchedAharonov-Bohm structure is given by a ring radius r⫽ 0.65␮m and a width of the arms w⫽ 200 nm as can be seen onFig. 1. In a second EBL step we define a PMMA lift-offmask for a 50-nm-thick and 30-␮m-wide gold gate, whichcovers the entire Aharonov-Bohm ring. This allows us toglobally control the electron density in the Aharonov-Bohmring during the measurements. Due to depletion from theedges, the structure is initially pinched off. By applying apositive voltage Vgon the global gate, electrons are accumu-lated in the structure and the structure begins to conduct.The sample was emerged in a3He cryostat equipped witha copper electromagnet. All measurements were performedat T⫽ 0.3 K if nothing else is mentioned. The measurementswere done by conventional voltage biased lock-in techniqueswith an excitation voltage of Vpp⫽ 7.7␮V at a frequency of131Hz. In this paper, we focus on measurements performedon one device, almost identical results have been obtainedwith another device in a total of six different cool downs.III. RESULTS AND DISCUSSIONFigure 1 presents a measurement of the magnetoconduc-tance of the device displayed in the left insert. As expectedthe magnetoconductance show large Aharonov-Bohm oscil-lations. Due to the long distance between the voltage probes,the measurement is an effective two-terminal measurement;hence the Aharonov-Bohm magnetoconductance is as ob-served forced to be symmetrical as a consequence of theOnsager relations.A Fourier transform of the magnetoconductance displaysa large peak corresponding to a period of 33 Gs. This is infull agreement with the dimensions of the ring obtained fromthe scanning electron microscopy 共SEM兲 picture.PHYSICAL REVIEW B 15 FEBRUARY 2000-IIVOLUME 61, NUMBER 8PRB 610163-1829/2000/61共8兲/5457共4兲/$15.00 5457 ©2000 The American Physical SocietyThe right inset in Fig. 1 displays the conductance as func-tion of gate voltage at T⫽ 4.2 K. Steps are observed at ap-proximate integer values of e2/h. Five steps are seen as thevoltage is increased with 0.18 V from pinch-off. Such stepshave previously been reported in Aharonov-Bohm rings8andcan be interpreted using classical addition of conductance,which is reasonable at these relatively high temperatures.19In any case, this indicates that our system, in the gate-voltageregime used here, only has a few propagating modes. Whenthe temperature is lowered a fluctuating signal is superposedon the conductance curve. At T⫽ 0.3 K the steps becomescompletely washed out by these fluctuations. The fluctua-tions, which we ascribe to resonance’s, appear at the sametemperatures where the Aharonov-Bohm magnetoconduc-tance oscillations become visible and are the signature of aphase coherent device.Figure 2