VOLUME 79, NUMBER 2 PHYSICAL REVIEW LETTERS 14JULY 1997Channel Interference in a Quasiballistic Aharonov-Bohm ExperimentG. Cernicchiaro,1,4T. Martin,2,5K. Hasselbach,1D. Mailly,3and A. Benoit11CRTBT-CNRS, 25 av. des Martyrs, 38042 Grenoble, France2CPT-Université d’Aix-Marseille II, Case 907, 13288 Marseille, France*3LMM-CNRS, 196 av. H. Ravera, 92220 Bagneux, France4CBPF-CNPq, 150 av. Xavier Sigaud, 22000 Rio de Janeiro, Brazil5Institut Laue–Langevin, B.P. 156x, 38042 Grenoble, France(Received 20 December 1996)New experiments are presented on the transmission of electron waves through a two-dimensionalelectron gas ring with a gate on top of one of the branches. Magnetoconductance oscillations areobserved, and the phase of the Aharonov-Bohm signal alternates between 0 and p as the gatevoltage is scanned. A Fourier transform of the data reveals a dominant period in the voltage,interpreted as the energy spacing between successive transverse modes. A theoretical model includingrandom phase shifts between successive modes reproduces the essential features of the experiment.[S0031-9007(97)03531-X]PACS numbers: 73.40.–cThe Aharonov-Bohm (AB) effect has proven to be aninvaluable tool for quantifying interference phenomena inmesoscopic physics. Early experiments on long metalcylinders [1] revealed that an electron accumulates a phaseRA ? dl as it is scattered elastically by impurities whiletraveling around the loop: when the magnetic flux is varied,an oscillatory pattern with periodicity hy2e results fromthe interference of an electron wave with its time reversedpath [2]. Experiments on gold loops [3] confirmed that fornormal metals which are laterally confined, the expectedperiodicity [4] is that of the single flux quantum f0 hye.The amplitude of the magnetoresistance background can beunderstood within the framework of universal conductancefluctuations (UCF) [5]. The two-dimensional electrongas (2DEG) formed at the heterojunction between twosemiconductors is used for experiments in the ballistictransport regime. Recently, oscillations associated withthe modulation of the electron wavelength under the gatewere observed in two experiments on gated rings in thediffusive [6] and the ballistic [7] regimes.In the present Letter, results on a new AB transportmeasurement in the ballistic regime are reported. Thenumber of lateral channels in one branch of the ring isadjusted by means of an electrostatic gate. Unlike previ-ously, data are analyzed over the whole significant volt-age range: from zero voltage to full depletion. A periodicpattern appears when the number of modes is modulatedusing the gate. Phase shifting and period halving in theAB pattern is monitored as the confinement is varied. Theessential features of the data are interpreted using the scat-tering formulation of quantum transport [4]. The inclu-sion of disorder is necessary to explain the alternation ofAB phases.The 2D electron gas was created at the interface ofa GaAlAsyGaAs heterojunction. A single loop device[8] of width 1.2 mm with inner diameter 4 mm con-nected to measurement leads was fabricated lithographi-cally [Fig. 1(a)]. In this etched structure, the width of thewire constituting the ring is further reduced by a deple-tion of 0.27 mm at each edge. The 2DEG had a mobilityof 1.14 3 106cm2yV s and an electron density of ns3.6 3 1011cm22. The coherence length lf. 20 mm andthe mean free path le 11, 3 mm indicate the ballisticregime. A metallic gate was deposited over one branchof the ring (the “upper” branch) allowing a controlled de-pletion of the 2DEG underneath. The number of electronchannels N in the wires defining the ring was estimated as-suming parabolic confinement in the transverse direction.The width W of the channel roughly equals the ratio of the1D to the 2D electron density [9]. For W 600 nm, anda Fermi wave length lF 40 nm, N s3py4dWylFø30 channels. The lateral dimensions of the wire are compa-rable to those of the conductance quantization experiments[10], where the number of transmitted channels was shownto scale linearly with the depletion voltage.FIG. 1. (a) Atomic force microscope image: detail of asample with the gate (white regions) over the upper branch andthe gates for the leads. (b) A schematic diagram of the ring,beam splitters (triangles), and gate (see text).0031-9007y97y79(2)y273(4)$10.00 © 1997 The American Physical Society 273VOLUME 79, NUMBER 2 PHYSICAL REVIEW LETTERS 14JULY 1997The conductance was determined using standard syn-chronous detection measurements. A low-frequency accurrent of 10 nA was injected and measurements weretaken at 15 mK. An external magnetic field variation of1.2 mT was applied corresponding to 4 flux quanta in themean radius of the ring. While lowering the gate voltagefrom 0 to 2300 mV by 1 mV steps, the complete con-ductance pattern was measured over a period of 4 hours.Digital filtering routines were applied to reduce base-linevariations due to UCFs.In Fig. 2, a “landscape plot” shows that the periodic-ity of the AB signal survives until a voltage of about2250 mV where the electrons underneath the gate arecompletely depleted and the ring is effectively cut off.When both arms transmit, shaded and clear areas alternatein the vertical direction, indicating oscillatory behavior asa function of the gate voltage. Attention is focused on thealternating contrast and the phase reversals when the gatevoltage is increased. The smoothly changing backgroundis identified as a residue of the total UCF signal. The ABphase of the pattern takes only values close to 0 or p [11].The symmetry of the conductance under field reversal isassociated with the two-terminal nature [12] of this mea-surement, as the spacing between the pair of current andvoltage terminals on each side of the ring is not small com-pared to lf.In Fig. 3, a magnetoresistance trace (inset) is displayedfor a ring with one (both) branch(es) conducting: Vg2300 mV (Vg 0); the oscillatory signal appears to beFIG. 2. AB component of the conductance, measured as afunction of applied flux f (horizontal axis) and gate voltage(vertical axis). Dark (clear) areas indicate minima (maxima) inthe AB signal.an even function of flux. The corresponding Fourier signalshows a dominant component at the single flux quantumhye. Higher harmonics have a much reduced amplitude.In Fig. 4, the modulus of the hye Fourier component ofthe ring resistance is plotted for each value of Vg. Thelocation of the
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