We study the visibility of Aharonov-Bohm interference in an electronic
Mach-Zehnder interferometer (MZI) in the integer quantum Hall regime. The
visibility is controlled by the filling factor $\nu$ and is observed only
between $\nu \approx 2.0$ and 1.0, with an unexpected maximum near $\nu=1.5$.
Three energy scales extracted from the temperature and voltage dependences of
the visibility change in a very similar way with the filling factor, indicating
that the different aspects of the interference depend sensitively on the local
structure of the compressible and incompressible strips forming the quantum
Hall edge channels.Comment: 5 pages, 5 figures, final version accepted for publication in Phys.
  Rev. 

A. Helzel

C. Strunk

H.-P. Tranitz

L. V. Litvin

W. Wegscheider

English

arXiv

Crossref

Edge-channel interference controlled by Landau level filling

We study the visibility of Aharonov-Bohm interference in an electronic Mach-Zehnder interferometer in the integer quantum Hall regime. The visibility is controlled by the filling factor v and is observed only between nu approximate to 2.0 and 1.0, with an unexpected maximum near nu=1.5. Three energy scales extracted from the temperature and voltage dependences of the visibility change in a very similar way with the filling factor, indicating that the different aspects of the interference depend sensitively on the local structure of the compressible and incompressible strips forming the quantum Hall edge channels

Litvin, Leonid

Helzel, Andreas

Tranitz, H.-P.

Wegscheider, Werner

Strunk, Christoph

University of Regensburg Publication Server

Edge-channel interference controlled by Landau level fillingL. V. Litvin, A. Helzel, H.-P. Tranitz, W. Wegscheider, and C. StrunkInstitut für experimentelle und angewandte Physik, Universität Regensburg, D-93040 Regensburg, GermanyReceived 9 May 2008; published 5 August 2008We study the visibility of Aharonov-Bohm interference in an electronic Mach-Zehnder interferometer in theinteger quantum Hall regime. The visibility is controlled by the filling factor  and is observed only between2.0 and 1.0, with an unexpected maximum near =1.5. Three energy scales extracted from the temperatureand voltage dependences of the visibility change in a very similar way with the filling factor, indicating that thedifferent aspects of the interference depend sensitively on the local structure of the compressible and incom-pressible strips forming the quantum Hall edge channels.DOI: 10.1103/PhysRevB.78.075303 PACS numbers: 73.23.Ad, 73.63.Nm, 71.10.PmI. INTRODUCTIONThe electronic Mach-Zehnder interferometer1 MZI wasproposed to study the decoherence1,2 and orbital entangle-ment effects3,4 by using edge channels in the regime of thequantum Hall effect QHE. Its high interference contrast,observed at temperature of about 20 mK, is the consequenceof ballistic transport through the quasi-one-dimensional edgechannels. The effect of temperature,1,5,6 bias voltage,1,5,7 andthe interferometer size6,8 on the interference contrast are cur-rently under intense investigation. The magnetic field B isanother important parameter, which controls the structure ofthe edge state consisting of compressible and incompressiblestripes.9 In previous experiments both a monotonic growth ofvisibility with increasing magnetic field8 and a localmaximum6 were observed within the =2 plateau. In theseworks only the number of edge channels rather than the pre-cise value of the filling factor =nh /eB was specified. Here,n is the electron density, h the Planck constant, and e theelementary charge. In this paper, we systematically study thebehavior of the MZI visibility in a broad range of fillingfactors. We found that the interference for the lowest Landaulevel appears at a filling factor of =2.0, reaches a maximumof visibility of about 50% at =1.5, and then decays to zeronear =1.0. Although the interference occurs only in theouter edge channel, the visibility is strongly affected by thepresence of the inner edge channel and its evolution, when is varied between 1 and 2.II. EXPERIMENTAL DETAILSThe interferometer see Fig. 1 was fabricated on the basisof a modulation doped GaAs /GaxAl1−xAs heterostructurecontaining a two-dimensional electron gas 2DEG 90 nmbelow the surface. At 4 K, the unpatterned 2DEG density andmobility were n=2.01015 m−2 and =206 m2 / Vs, re-spectively. Photolithography was employed to define Hallbars with large contacts connected to leads S, D2, and allgates at Fig. 1. The ring-shaped interferometer mesa, con-tact D1, the quantum point contacts with air bridges, andthe modulation gate labeled as MG in Fig. 1 were patternedby means of electron-beam lithography. The quantum pointcontact QPC No. 0 was used to select the outer edge chan-nel for the interference experiment. We studied two MZIswith an arm length of 14 and 9 m and an arm width of2.5–3 m 1.7 m, respectively see Fig. 1. The QPCs ofthe larger MZI had 120 nm gap between sharp tips tip radius50 nm; for the smaller MZI a 400 nm gap was used. Thearea between two interfering paths, determined from the pe-riod of the AB oscillations in a magnetic field, was 48 and25 m2 for these MZIs. A standard lock-in technique f300 Hz with 1 V excitation at terminal S and detectionat terminal D2 was employed see Fig. 1. Most of the mea-surements were performed at a temperature of 25 mK.III. MAGNETIC-FIELD DEPENDENCEOF THE VISIBILITYIn Fig. 2 inset we show a typical trace of current IVMGin detector D2 vs the voltage at the modulation gate VMG.The measured interference contrast is quantified by the vis-ibility which is defined as I= Imax− Imin / Imax+ Imin. Whenchanging magnetic field B the QPC transmission has to bereadjusted to 1/2, because it sensitivitely depends on B. Insome cases resonances in the QPC transmission characteris-tics occurred. In this case the half transmission point with thehighest I was selected.10 In order to relate the magnetic fieldto the filling factor in the MZI arms, the two-point conduc-tance of the interferometer between terminal S and D2 withall QPCs opened was measured Fig. 2a. The electron den-sity of the narrowest section of the interferometer can beFIG. 1. SEM image of smaller Mach-Zehnder interferometerwith the scheme of edge states for filling factor 2. The transmissionsof QPC1 and QPC2 are set to 0.5. QPC0 reflects the inner edgechannel dashed line and transmits the outer one solid line. MGshifts the phase.PHYSICAL REVIEW B 78, 075303 20081098-0121/2008/787/0753034 ©2008 The American Physical Society075303-1probed in this way. For the larger interferometer the transi-tion between the =2 and =1 Hall plateaus is located atB=5 T and corresponds to the filling factor =1.5. For thesmaller interferometer the transition point shifts to 4.8 T.These reference points were used to determine the fillingfactor in the range of 3 TB8 T. The visibility of thelarger MZI shown in Fig. 2b with full squares emerges atB4.4 T 1.7, reaches a maximum of 17% at =1.5,and then nonmonotonically decreases to 6% at B=7.5 T =1.0. For the smaller interferometer, a measurable visibilityfull circles and open squares at Fig. 2b emerges near B=3.5 T 2.0, reaches maximum around B=5 T, andthen decreases to zero at about B=8 T 0.9. Right afterthe magnetic field is ramped to the next data point, the vis-ibility drops significantly about 50%. However, after somewaiting time it recovers and approaches a saturation value.The saturation takes up to 10 h. As a reasonable compromisebetween optimum visibility and reasonable data-acquisitiontime, we have chosen 1 h full circles in Fig. 2b and 4 hopen squares in Fig. 2b for the waiting time. A measur-able visibility was observed in the  interval 1.02.0with a maximum in IB near =1.5.To assure that it is the filling factor which controls thevisibility and not the absolute magnetic-field magnitude, wechanged the electron density in the interferometer by a backgate. A decrease in the electron density shifts the entire IBcurve to smaller magnetic fields Fig. 3a and inset in Fig.3b. As an independent check of the determination of fillingfactor within the interferometer arm, we looked at the back-scattering properties. The quantum point contact QPC0 is setto transmit only the outer channel, QPC1 is completelyclosed, and QPC2 is completely opened. In absence of back-scattering within the lower interferometer arm, QPC1 redi-rects all current from S to the detector D1, while D2 seeszero signal. However, when the filling factor is close to 1.5,backscattering occurs between the counter-propagating lowerarm edge channels which appears as signal in D2 Fig. 3b.The maximum of this signal corresponds to the point =1.5 where the backscattering is strongest. We see that aback gate voltage of −28 V shifts the peak, in other words,the point of maximum scattering, from 4.8 to 4.1 T, i.e., thedensity decreases by 15%. The maximum of visibility wasearlier reported to occur on the upper end of the =2plateau,6 while we observed it at =1.5. This discrepancymay be caused by an unequal filling factor in the two MZIarms in Ref. 6. According to Ref. 9 a mesa width which isnot much larger than the depletion length results in differentelectron densities and therefore different filling factors. Toavoid this problem we used the same width for both interfer-ometer arms as well as for the input and output leads.IV. CHARACTERISTIC ENERGY SCALESThere are three fundamental reasons for a reduction of thevisibility: i genuine decoherence by inelastic scattering; iiphase averaging, due to a finite-energy window, imposed bytemperature and voltage;11 and iii phase averaging due tofluctuations of charges trapped nearby.2 The last possibilitycan probably be excluded, since fluctuations in the environ-ment are not expected to strongly depend on B. A recentexperimental study of IT ,B showed that it changes expo-nentially with T:I = I0 exp− T/T0 = I0 exp− 2L/l , 1where the characteristic temperature T0 is inversely propor-tional to the length of the interferometer arm L, i.e., T0 lT /2L or l2LT0 /T.6 The electromagnetic environmentof the interferometer is expected to give rise to a l1 /Tdependence.12 On the other hand, it is again unclear whysuch an environmental effect should strongly vary with B.As demonstrated by the solid lines in Fig. 4, our data alsovary exponentially with T above 45 mK. However, at lowerFIG. 2. Color online a Two terminal magnetoconductance ofthe large interferometer with all QPCs opened. Inset shows an ex-ample of interference pattern at B=4.7 T. b Visibility versus mag-netic field for both interferometers. Full and open squares corre-spond to data acquired with at least 4 h waiting time, full circleswith 1 h.FIG. 3. Color online a Visibility versus magnetic field fortwo back gate voltages, i.e., electron densities. b Determination ofthe =1.5 point. Current recorded at detector D2, when QPC1 isclosed and QPC2 is opened. The peak in each curve corresponds tomaximum scattering between edges in the lower MZI arm and cor-respond to =1.5. Inset: Data from a plotted vs filling factor.LITVIN et al. PHYSICAL REVIEW B 78, 075303 2008075303-2temperatures a crossover to a weaker temperature depen-dence is observed. The presence of such a crossover is re-flected by the extrapolated values of I0, which significantlyexceed the allowed maximum of 100%, and the fact that thefit lines do not cross at T=0, but rather at 7 mK. Although itis notoriously hard to exclude that electron heating contrib-utes to the apparent saturation of I at T45 mK Fig. 4,the latter two facts refer to the high-temperature regime andindicate that the behavior of IT may be more complexthan a simple exponential.Despite these differences, our data confirm a clear corre-lation between the extracted values of T0 and the visibility,when the magnetic field is varied. Like the visibility, T0 hasa maximum around =1.5 see the inset of Fig. 4.13 Thus, itis clear that the phase coherence length is magnetic fielddependent.The visibility can also be measured as a function of a dcvoltage added to the small ac bias. This differential visibilityshows a lobe structure7,8 vs Vdc. In Fig. 5a we show theevolution of the lobe structure with filling factor. The ob-served change of the lobe characteristics resembles thechange of visibility with  see Fig. 2b, i.e., the largestdistance between the zeros in I is found near =1.5 andreduces, when  moves from=1.5 to =1.0 or 2.0 Fig.5b, open squares. For 	1.5 more than one pair ofside lobes can be observed Fig. 5a, B=3.9 T. The datain Fig. 5a can be well approximated by a product of anoscillatory function and a Gaussian envelope: I=I0coseV /Lexp−eV2 /202 which contains the pa-rameters L as period of the cosine term and 0 as character-istic width of the envelope. For a direct comparison, we plotthese characteristic energies together with the characteristictemperature kBT0 extracted from the temperature dependenceof I. The energies L and 0 agree rather well for 2		1.5, while 0 is slightly smaller than L for 1.5. On theother hand, kBT0 is about 15 times smaller. Despite the dif-ference in numbers, the overall  dependence of all energyscales is quite similar. Hence, it appears that the B depen-dence of the characteristic energies is closely related to theevolution of the structure of edge states with .V. DISCUSSIONFor the following, one has to keep in mind that QPC0 istuned such that current is injected exclusively into the outerincompressible strip with =1. According to acceptedtheory,14 at small bias the current flows close to the interfacebetween compressible and incompressible stripes. Already inRef. 8 it was realized that the lobe pattern cannot be under-stood in terms of a single electron picture, nor within asimple mean-field approach. Nevertheless, the charges in-jected into the interferometer by the dc component of thebias appear to induce an overall phase shift, i.e., they actsimilar to the modulation gate. Youn et al.15 proposed thatthe average phase shift 	 is determined by the averagenumber N of nonequilibrium electrons and the intrachannelCoulomb interaction constant U0:	 = NU0tfl=eVtflU0tfl, 2where tfl=L /vD is the traversal time of the electrons throughthe interferometer and vD the drift velocity along the edge.Using vD3·104 m /s and L=9 m, one obtains N1 atVdc=10 V. With increasing NVdc the number of possiblecharge-density distributions in the interferometer and hencethe fluctuations of  increases, which according to the nu-merical calculations in Ref. 15 leads to a suppression of thevisibility at higher Vdc. For smaller U0 a larger Vdc is neededto reach the first zero of I.If one now assumes that the interaction parameter U0 isaffected by the screening properties of the environment ofthe outer edge channel, the changes of the structure of the2DEG enclosed by the outer edge between =2 and =1would indeed suggest an nonmonotonic variation of U0B,FIG. 4. a Temperature dependencies of visibilities for largeopen symbols and small full symbols MZIs at different magneticfields. The curves “a” and “b” for B=4.8 T correspond to differentQPC half transmission points. Inset: Characteristic temperatures T0extracted from the exponential fits according to Eq. 1. The data forthe large interferometer open circles have been multiplied by 2.FIG. 5. Differential visibility of the small MZI. a Lobe struc-tures at different magnetic fields at VBG=−25 V. b Characteristicenergies L and 0 see text extracted from IV together withkBT0 as a function of magnetic field B.EDGE-CHANNEL INTERFERENCE CONTROLLED BY… PHYSICAL REVIEW B 78, 075303 2008075303-3because the screening is most effective at =1.5, where theentire bulk of the 2DEG is compressible. On the other hand,at =1 and 2, the bulk of the 2DEG is incompressible, im-plying reduced screening and correspondingly larger valuesof U0. This scenario is consistent with the observed non-monotonic variation of L. The energy scales L and 0 mustthen be related to  / tfl and U0B.Very recently, another theory based on the chiral Luttingerliquid approach to the QHE has been suggested.16 In thistheory, the excitations are dipolar neutral and charged edgemagnetoplasmon modes with different group velocities v andu. The model allows to calculate lT=uv /kBTu−v,which is consistent with the observed T dependence of l. Inaddition it predicts for the ratio between L /kBT0=2219.7. From our data we deduce a similar experimentalvalue of L /kBT015. This model requires two well definededge channels and is valid at 1.5. Hence, it remains to beexplained, why the experimental data show a visibility maxi-mum for 1.5.Another theory for =1 results in a lT−3 for screenedand lT−1 ln2 T for unscreened Coulomb interactions.17 Forthe transition between =1 and =2, so far no theory exists.Qualitatively, one may expect that a fermionic picture ismore appropriate here, since the screening by the compress-ible interior of the interior of the 2DEG tends to suppress theLuttinger liquid effects.For a better microscopic understanding of the effect, it isessential to relate the various phenomenological energyscales of the different theories to a realistic model of thestructure of the edge channels, i.e., the distribution of com-pressible and incompressible strips at the mesa edge.18 Thisdistribution considerably changes between =1 and 2. At =1.5 and above the outer incompressible strip is well local-ized at the mesa edge. Depending on the steepness of theconfining potential, it starts to spread out for 1.5 and fillsthe whole mesa, once =1 is reached. The precise reason forthe decay of the interference for 1.5 is still an open ques-tion. The measured temperature dependence of the visibilitysuggests that l is suppressed in this regime again. On theother hand, an edge strip that spreads out over 100 nm andmore cannot be considered as a quasi-one-dimensional objectanymore. In particular, it does not enclose a well definedmagnetic flux. The integration of such QHE-specific featuresinto models considering simple one-dimensional conductionchannels provides new challenges for the theory.VI. CONCLUSIONSIn summary we investigated the effect of the filling factoron the visibility in a Mach-Zehnder interferometer. Surpris-ingly, the visibility was found to be highest around =1.5and decreases to zero when the adjacent integer filling fac-tors are reached. This dependence originates from an evolu-tion of the structure of edge channels with magnetic field,which strongly affects three energy scales of the interferencein a very similar way: kBT0, which determines the tempera-ture dependence of the visibility in the linear regime, and Land 0, which determine the size and damping of the lobestructure of I in the nonlinear regime. This observation sug-gests that the linear and the nonlinear regimes are governedby the same energy scale.ACKNOWLEDGMENTSWe thank M. Heiblum and I. Neder for support in theframework of EU-Transnational Access Program, ContractNo. RITA-CT-2003–506095. We are grateful to A. Siddiki, S.Ludwig, E.V. Devyatov, E. Sukhorukov, and M. Büttiker forhelpful discussions. The work was funded by the DFG withinthe SFB631 “Solid state quantum information processing”and the BMBF via Project No. 01BM465 within the program“Nanoquit.”1 Y. Ji, Y. Chung, D. Sprinzak, M. Heiblum, D. Mahalu, and H.Shtrikman, Nature London 422, 415 2003.2 F. Marquardt and C. Bruder, Phys. Rev. B 70, 125305 2004.3 P. Samuelsson, E. V. Sukhorukov, and M. Büttiker, Phys. Rev.Lett. 92, 026805 2004.4 I. Neder, N. Ofek, Y. Chung, M. Heiblum, D. Mahalu, and V.Umansky, Nature London 448, 333 2007.5 L. V. Litvin, H.-P. Tranitz, W. Wegscheider, and C. Strunk, Phys.Rev. B 75, 033315 2007.6 P. Roulleau, F. Portier, D. C. Glattli, P. Roche, A. Cavanna, G.Faini, U. Gennser, and D. Mailly, Phys. Rev. Lett. 100, 1268022008.7 P. Roulleau, F. Portier, D. C. Glattli, P. Roche, A. Cavanna, G.Faini, U. Gennser, and D. Mailly, Phys. Rev. B 76, 161309R2007.8 I. Neder, M. Heiblum, Y. Levinson, D. Mahalu, and V. Umansky,Phys. Rev. Lett. 96, 016804 2006.9 D. B. Chklovskii, K. A. Matveev, and B. I. Shklovskii, Phys.Rev. B 47, 12605 1993.10 L. V. Litvin, A. Helzel, H.-P. Tranitz, W. Wegscheider, and C.Strunk, Physica E Amsterdam 40, 1706 2008.11 V. S.-W. Chung, P. Samuelsson, and M. Buttiker, Phys. Rev. B72, 125320 2005.12 G. Seelig and M. Buttiker, Phys. Rev. B 64, 245313 2001.13 The uppermost trace in Fig. 4 was measured with back gatevoltage of −25 V, corresponding to a reduced electron density.14 See, e.g., the reviews in The Quantum Hall Effect, edited by R.E. Prange and S. M. Girvin Springer-Verlag, New York, 1987;E. V. Devyatov, Phys. Usp. 50, 197 2007.15 S.-C. Youn, H.-W. Lee, and H.-S. Sim, Phys. Rev. Lett. 100,196807 2008.16 I. P. Levkivskyi and E. V. Sukhorukov, Phys. Rev. B 78, 0453222008.17 J. T. Chalker, Yuval Gefen, and M. Y. Veillette, Phys. Rev. B 76,085320 2007.18 A. Siddiki and F. Marquardt, Phys. Rev. B 75, 045325 2007.LITVIN et al. PHYSICAL REVIEW B 78, 075303 2008075303-4

We study the visibility of Aharonov-Bohm interference in an electronic Mach-Zehnder interferometer in the integer quantum Hall regime. The visibility is controlled by the filling factor nu and is observed only between nu[approximate]2.0 and 1.0, with an unexpected maximum near nu=1.5. Three energy scales extracted from the temperature and voltage dependences of the visibility change in a very similar way with the filling factor, indicating that the different aspects of the interference depend sensitively on the local structure of the compressible and incompressible strips forming the quantum Hall edge channels

Edge Channel Interference Controlled by Landau Level Filling

Edge Channel Interference Controlled by Landau Level Filling

Abstract

Similar works

Full text

Available Versions

Crossref

Crossref

University of Regensburg Publication Server

University of Regensburg Publication Server