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Aharonov Bohm effect in 2D topological insulator. G.M.Gusev,1 Z.D.Kvon,2,3 O.A.Shegai,2 N.N.Mikhailov,2 and S.A.Dvoretsky,2 1Instituto de F´ısica da Universidade de São Paulo, 135960-170, São Paulo, SP, Brazil 2Institute of Semiconductor Physics, Novosibirsk 630090, Russia and 3Novosibirsk State University, Novosibirsk, 630090, Russia (Dated: January 16, 2015) WepresentmagnetotransportmeasurementsinHgTequantumwellwithinvertedbandstructure, 5 whichexpectedtobeatwo-dimensionaltopologicalinsulatorhavingthebulkgapwithhelicalgapless 1 states at theedge. The negative magnetoresistance is observed in thelocal and nonlocal resistance 0 configuration followed by the periodic oscillations damping with magnetic field. We attribute such 2 behaviourtoAharonov-Bohmeffectduetomagneticfluxthroughthechargecarrierpuddlescoupled tothe helical edge states. The characteristic size of these puddlesis about 100 nm. n a J I. INTRODUCTION tive spin-orbit coupling is weak, and one can expect the 5 1 conventional WL behaviour14. When the Fermi energy becomes larger than the gap width, the energy disper- Theinvestigationofthequantuminterferencephenom- ] sionislinear,andtheorydescribescrossoverfromWLto ll ena,suchasAharonov-Bohm(AB)oscillationsandweak WAL behaviour14–17. The magnetoresistance of 2D TI a localizations, provides an important information about with a dominant edge state contribution is described by h fundamental properties of various electronic systems1. - two mechanisms: first scenario relies to the frequent de- s Recently new class of materials with interesting prop- viationsoftheedgeelectronsintothedisorderedABflux me erties have emerged, called topological insulators (TI), threaded 2D bulk18; and second scenario to the localiza- whichareinsulatinginthebulkandcharacterizedbythe tion of the helical edge sates due to the collective action . existenceofrobustgaplessexcitationsattheirsurface2–5. t oftherandommagneticfluxthroughtheloops,naturally a ManifestationofABoscillationsintopologicalinsulators, formed by rough edges of the sample19. Both models m can be utilized to probe the surface nature in TI and its predict quasi-linear positive magnetoresistance MR. Ac- - properties. The time-reversal-symmetric 2D topological d cording to this theoretical predictions positive MR was n insulatorisinducedbyastrongspin-orbitinteraction6–10 observed in 8 nm HgTe quantum wells20. and characterized by edge modes with opposite spins o c propagating in opposite directions. The 2D TI have [ been realizedin HgTe quantumwells with invertedband structure11,12. Aharonov Bohm oscillations is intimately 1 v related to the weak localization (WL) corrections. For 2 example,WLphenomenaisdiscussedintermsofthecol- 5 lective action of the magnetic flux through the random * 6 loops ofthe pair oftime reversedtrajectories1. The con- ) EF k 3 structiveinterferenceofthesepathinreturnpointissup- E( E 0 F pressedby AB flux,whichleadto the resistanceincrease . 1 and negative magnetoresistance13. The spin orbit cou- 0 plingstronglymodifiesthe quantumcorrectionsthe con- 5 ductivity, it leads to a destructive interference between 1 clockwiseandcounter-clockwisetrajectoriesandchanges : v the WL to weak anti-localization behavior (WAL). i X In 2D topological system WL is quite different from K r conventional 2D metals and strongly affected by the a Dirac spectrum of massive fermions14,15 with the mass FIG. 1: (Color online)Schematic of the energy spectrum of proportional to the band gap. The electronic spectrum 2DTIneark=0. Solidlines-spectrumforthebulkelectrons, is shown in Figure 1. The band structure has a sim- dashes- helical edge state spectrum. Positions of the Fermi ple parabolic formnear the bottom of the band (marked energyinthebulkgapandnearthebottomoftheconductive ∗ by E in Figure 1 and linear Dirac-like dispersion re- bandsare indicated. F lation at high Fermi energy E . When the Fermi en- F ergyliesinthe bulk gapnearthe chargeneutralitypoint Recently interaction of the helical states with multi- (CNP), spectrum described by pair of the helical edge ple puddles of charge carriers formed by fluctuations in states (Figure 1). More over, the edge state must have the donor density has been considered in 2D topological linear Dirac like dispersion. The localization corrections insulators21. This leads to significant inelastic backscat- arestronglygovernedbysuchremarkablepropertyofthe tering due to the dwelling of carriers in the puddles and spectrum. When the Fermi energy is small, the effec- to a large resistance, which depends weakly on temper- 2 ature. Moreover, carriers in the puddles describes loops trajectories,whicharesensitivetothemagneticfieldflux. therefore, one could expect that the magnetoconductiv- 2 3 2 3 4 ity is mainly contributed by the quantum interference of such electrons. 1 4 1 5 8 m 32 m 6 m In the present paper we investigate transport prop- 9 8 7 6 6 5 erties of the HgTe quantum wells with the width d of 8-8.3nm. Slightly abovethe the chargeneutralitypoint, m)100000 (a) whenthesystemisexpectedtobetwo-dimensionaltopo- h O logical insulator, we observe the negative magnetoresis- e ( tance measured in the local and nonlocal configurations, anc 10000 RI=1,4;V=2,3 followed by periodic oscillations damping with magnetic st si field. When the applied field is larger than 1 T, the MR Re holes electrons becomespositive. Fromthesignofthemagnetoresistance 1000 RI=2,6;V=3,5 wespecifiedtwocontributions: electronsfromedgestates and bulk electrons, which are localized in the metallic 0 3 puddles formed by potential fluctuations. Vg-VCNP(V) 1m)000000 (b) RI=1,5;V=7,6 h II. EXPERIMENT e(O holes electrons nc100000 RI=1,5;V=8,7 a st The Cd0.65Hg0.35Te/HgTe/Cd0.65Hg0.35Te quantum esi wells with (013) surface orientations and a width d of 8- R 10000 RI=9,2;V=3,8 RI=1,5;V=9,8 8.3 nm were preparedby molecular beam epitaxy. A de- taileddescriptionof the sample structurehas been given RI=4,7;V=2,9 1000 in22,23. Device A is six-probe Hall bar, while device B is -2 0 2 Vg-VCNP(V) designed for multiterminal measurements. The device A wasfabricatedwithalithographiclength6µmandwidth FIG. 2: (Color online) Color online) (a) a) The local and 5µm (Figure 2, top panel). The device B consists of nonlocalresistancesRasafunctionofthegatevoltageatzero three 4µm wide consecutive segments of different length magneticfieldmeasuredbyvariousvoltageprobesforsamples (2,8,32µm), and 7 voltage probes. Device C (figure 5) A (a) and B (b), T=4.2 K. Top panel-shows schematic view is a structure with large gate area for identifying nonlo- of the samples. The perimeter of the gate is shown by blue cal transport over macroscopic distances24. The lengths rectangle. of the edge states are determined by the perimeter of the sample part covered by metallic gate (mostly side branches)ratherthanbythelengthofthebaritself. The showninfigure 2a andb, whenthe gate voltageinduces ohmic contacts to the two-dimensional gas were formed an additional charge density, altering the quantum wells byin-burningofindium. Topreparethegate,adielectric fromann-typeconductortoap-typeconductorviaa2D layer containing 100 nm SiO and 200 nm Si Ni was TI state. It has been shown11,12 that the 4-probe resis- 2 3 4 first grown on the structure using the plasmochemical tance in an HgTe/CdTe micrometer-sized ballistic Hall method. Then, the TiAu gate with sizes of 18×10µm2 bar demonstrated a quantized plateaux R ≃ h/2e2. 14,23 was deposited. The ungated HgTe well was initially n- Itisexpectedthatthescatteringbetweenthehelicaledge dopedwithdensityn =1.8×1011cm−2. Severaldevices states in the topological insulator is unaffected by the s withthesameconfigurationhavebeenstudied. Theden- presenceof a weakdisorder2,6,7. Note, however,that the sity variationwith gate voltagewas1.09×1015m−2V−1. resistance of samples longer than 1µm might be much The magnetotransport measurements in the structures higherthanh/2e2duetothepresenceoftheelectronspin described were performed in the temperature range 1.4- flip backscattering on each boundary. Mechanism of the 25 K and in magnetic fields up to 12 T using a stan- back scattering is not clear and appealing task for theo- dard four point circuit with a 3-13 Hz ac current of 0.1- reticiansandamatterofongoingdebate21,25,26. TheHall 10 nA through the sample, which is sufficiently low to effectreversesitssignandR ≈0(notsown)whenlon- xy avoid overheating effects. The density of the carriers in gitudinal resistance R approaches its maximum value, xx the HgTe quantum wells can be electrically manipulated which can be identified as the charge neutrality point with local gate voltage V . The typical dependence of (CNP). These behaviour is similar to those described in g the four-terminal resistance of two of the representative graphene27. An unambiguous way to prove the presence samples A and B as a function of V is shown in Figure of edge state transport mechanism in 2D TI with strong g 2. The resistance R =R of the sample A backscattering on the boundary are the nonlocal electri- 14,23 I=1,4;V=2,3 and resistances for sample B, measured by various volt- cal measurements. The nonlocal response always exists age probes in a zero magnetic field reveala sharp peaks, because of the presence of the two counter-propagating 3 edgestates,whichflowssidewaysandmayreachanycon- ear negative magnetoconductance in HgTe-based quan- tacts in the device24,28. Figures 2 a and b show the tum wells in the 2DTI regime near CNP, when the edge nonlocal resistances corresponding to the different con- statetransportprevails. Ourobservationagreeswiththe figurations. Forexample,nonlocalresistanceinfigure2a model25 which describes the effects of WAL (see discus- corresponds to the contact configuration, when the cur- sion section below for more details). The model predicts rent flows between contacts 2 and 6 and the voltage is almost linear positive magnetoresistance ∆R = A|B|, e2/h measured between contacts 3 and 5. One can see that where parameter A strongly depends on the disorder the nonlocal resistance near CNP has a peak of a com- strength W in comparison with the energy gap E : the g parable amplitude, though less wide, and approximately fluctuations W > E result in a large B-slope corre- g in the same position as the local resistance. Outside of sponding to a strong disorder regime and fluctuations the peak the nonlocal resistance is negligibly small. The W <E lead a small B-slope. Figure 3 shows the evolu- g apparent residual nonlocal resistance in figure 2b above tion of the resistance R with magnetic field and den- xx Vg − VCNP > 1V and below Vg −VCNP < −1V is is sity, when the chemical potential crosses the bulk gap. relatedtothe noiseinthefulllinearscalemeasurements. The magnetoresistance(MR) demonstratesastrikingV- shapedependenceinmagneticfieldsbelow1TnearCNP, whichconfirmsourpreviousobservations. Notehowever, that the V-shaped magnetoresistance is strongly trans- formed, when Fermi energy moves away from CNP to 300000 the electronic side of the resistance peak. In this region (a) MR inverses the signnear zero magnetic field and shows ) m triangular-shapedpeakaccompaniedbytwosatellitefea- h O tures or damped oscillations. The width of the negative ( 9,8 magnetoresistancespikeandtheperiodoftheoscillations = V areslightlyvariedfromsampletosamples,asonecansee =1,5; in figures 3a and b. Both samples are almost identical RI with approximately equal mobility. Note that the Fermi levelstillliesinthe bulkgapandtransportisdominated 0 by edge states, because we see nonlocal effects. In the -4 -2 0 2 4 hole side of the peak we do not find neither small neg- B(T) ative magnetoresistance near B=0, nor the oscillations on the side branches of the the positive magnetoresis- (b) 300000 tance. In magnetic fields above 3 T the magnetoresis- m) tancefalls offrapidlymarkinga pronouncedcrossoverto h the quantum Hall effect regime in accordance with pre- O (9,8 vious observations20. = V The figure 4 shows the low-fled part of the relative =1,5; magnetoresistanceforthe twovaluesofthe gatevoltages RI andtwotemperatures. Onecanseethatforthisparticu- larsampletheMRvarieslinearlywithmagneticfield. As 0 voltageVg increases,theB-slopeoftheMRdecreasesand -4 -2 0 2 4 additional oscillation emerges. The MR profile does not B(T) showanysignificanttemperaturedependence. Figure4b displaystracesoftheresistanceinnonlocalconfiguration FIG. 3: (Color online) Color online) The local resistance R for device B (R = R ). One can see simi- fortwosamples withmultiterminal configuration (devicesB) NL I=2,6;V=3,5 lartriangular-shapedMRpeak,asinthelocalgeometry, as a function of the magnetic field for the different values though less wide, with two satellite peaks. Coexistence of the gate voltages, when the Fermi energy passes through CNP from the hole to electronic side of the resistance peak. of the low-field negative MR peak in nonlocal configu- (a) Vg −VCNP (V): -0.15 (cyan), -0.02 (blue), 0.1 (green), ration exclude the possibility that this effect has a bulk 0.35(red),0.47(black). (b)Vg−VCNP (V):-0.18(cyan),0.05 origin. It is worth noting , however, that around B=0 (green), 0.25 ( blue),0.37 (red), 0.45 (black). the magnetoresistance is parabolic rather than linear in nature. It is expected that a magnetic field perpendicular to Finally we present the results for a device C with a the quantum well breaks time reversal symmetry (TRS) large gate. We expect that the length of the edge states and thereby enables elastic scattering between counter- in this structure is determined by the perimeter of the propagating chiral edge modes. However, a number sample covered by the metallic gate. Figure 5 shows of the different theoretical models has previously been schematic view of the sample. One can see that the cur- proposed11,15,25,29,30 withsubstantiallydifferentphysical rent flows mostly along the edges of the side branches. scenarios. Inourpreviousstudy20wehaveobservedalin- The resistance reveals saw-tooth oscillations, shown in 4 rapidly and not gradually, as in devices A and B. III. DISCUSSION (a) 0) 1,0 R(L We now proceed to an analysis of the data described. B)/ We will focus on the two models that can explain the R(L T=4.2 K magnetoresistance of one dimensional edge electrons. Both models considered alternative paths for the edge 0,9 states due to bulk disorder or rough edges of the realis- T=1.4 K tic sample. Figure 6 illustrates the helical edge states in -1 0 1 a disordered 2D TI in an uniform magnetic field. The B (T) first scenario describes disordered spinless one dimensional quantum wire25. For strong enough disorder elec- 0) tron paths frequently deviate into the bulk region en- (1NL,0 closing AB flux before returning back to the edge, la- R B)/ (b) beled in figure 6 by number 2. The conventional WAL (NL approach can be used. The average with respect to the R differentsizeoftheloopsleadstothepositivelinearmag- 0,9 netoresistance. Second scenario relies to the localization of the helical edge sates due to the collective action of the random magnetic flux through the loops, naturally -1 0 1 formed by rough edges of the sample (labeled by num- B (T) ber 1 in the Figure 6)19. In accordance with this scenario immediately after magnetic field is switched on, FIG. 4: (Color online) Color online) (a) The relative local resistance RL(B)/RL(0)(RL = RI=1,4;V=2,3) as a function edge states become localized. The magnetic field pene- of the magnetic field for the two values of the gate volt- tratesthroughtherandomhelicaledgestatesloops(Fig- age Vg −VCNP(V): 0.35 (black), 0.53 (blue, red), and two ure 6 ), which acts as magnetic flux impurity and in- temperatures(deviceB).(b)Therelativenonlocal resistance troduces the backscattering between edge states on each RNL(B)/RNL(0) (RNL = RI=2,6;V=3,5) as a function of the boundary. Similar to the first scenario the average with magneticfieldforthetwovaluesofthegatevoltageVg−VCNP respect to the different magnetic fluxes should be per- : 0.35(black),0.5(red)(deviceA).DashedlinesareB-linear formed. Themodel19predictspositivemagnetoresistance approximations. and B2 dependence of the inverse localization length γ in small magnetic field, which becomes more linear with increasing magnetic field. Our experimentally observed linear positive magnetoresistance near CNP are in good 40000 2 3 qualitative agreement with both models. Detailed com- parisonwithmodel25 hasbeenperformedinourprevious hm) m publication24. Discrimination among two scenarios re- O R (1,4;2,3 400 1 4 qbuoitrhesmfoudrtehlsercoenxspiedreirmbeanltliasltiwcotrrka.nsItpoisrtwaotrtzhernoomtinaggntehtaict field, while our samples demonstrate diffusive transport. Despite the fact that while the both models give a sat- 20000 isfactory description of the linear positive magnetoresis- 6 5 tance near CNP, the explanation of negative MR and AB-likeoscillationsinFig.3-4awayfromCNPrequiresa 20 m further elaboration of the models. -1.0 -0.5 0.0 0.5 1.0 Absenceoftheresistancequantizationinsampleswith B (T) dimensions above a few microns11,12,24 is another un- FIG.5: (Coloronline)Coloronline)ThelocalresistanceRfor solved problem. One of the possible explanation is the device C as a function of the magnetic field for the different fluctuations of the local insulating gap width induced values of the gate voltages. (a) Vg−VCNP (V) (from top to by smooth inhomogeneities,which canbe representedas bottom): 0.1, 0.3, 0.5, 0.7. metallic puddles or dots. The well-localized metallic re- gionsalongtheedgehavebeenfoundusingscanninggate microscopy31 and in microwavesexperiments32. Accord- Fig.5, when the Fermi energy passes the electronic side ingtothemodel21chargecarrierpuddlescoupledtoaco- of the resistance peak away from CNP. Note, however, herentconductorresultsinincoherentinelasticprocesses that the evolution of the oscillations with gate occurs and modifies the ballistic transport. Therefore metallic 5 while the edge roughnessaswellas the bulk disorderare important and play a dominant role in the region near CNP in magnetic field, away from CNP the role of the 1 B metallic puddles in MR becomes more pronounced and essential. WL effects inballistic cavitiesindeedhas been sample boundary studied both theoretically35 and experimentally36. The WL peak and periodic oscillations are observed which 2 were attributed to the Aharonov-Bohm effect through a periodic orbit within the cavities. Therefore, it is 3 naturally to explain the negative magnetoresistance and helical edge states AB-like oscillations in figures 3,4 by existence of metallic ballistic puddles near the edge. It is worth noting that the Fermi energy of electrons in puddles lies in the bulk trajectories parabolicpartoftheenergyspectrum(figure1)because quantum dot of the low density. Therefore, one would expect that the spin orbit coupling is weak, and magnetoresistance is negative in agreement with our observations. It is FIG. 6: (Color online) Schematics of helical edge state prop- easy to estimate the characteristic sizes of these loops agation along a disordered edge in 2DTI in a perpendicu- using the period of the oscillations in fig.3-4, which is lar magnetic field. Electrons tunnel in and out quantum dot δB = 0.3−0.4 T. Then the characteristic area of the (puddle)createdbyinhomogeneouschargedistribution. Elec- tron helical trajectories form loops resulted from the edge loops is S = δB/Φ0(Φ0 = h/e) = 10−10cm2 and respec- roughness (1), near bulk disorder (2) and metallic puddles tivelysizeisabout100nm. Indeedthisvalueagreeswith (3). Bulk electron trajectories form loops inside of the pud- estimationsofthepuddlesizeanddensity33andscanning dle. gatemicroscopy31. Note that for explanationofthe negative signof the MR the role of the bulk electrons in the puddles is emphasized, however the bulk state and edge puddles canleadto spindephasing,since anelectronen- state may co-exist (figure 6). The interplay between the tering the puddle is thermalized by dissipation and later topological insulators helical states and bulk electrons onfed back into the system. Therefore ballistic coherent requires further theoretical study. When the Fermi level transportisexpectedonlyintheregionbetweenthepud- moves to valence band, it is expected that the puddles dles, and total 4-terminal resistance exceeds the quan- shouldbeoccupiedbytheholes. Notehowever,thatboth tized value. Self-averaging resistance of the sample with dephasing and spin relaxation times for holes are found edge state dominated contribution to transport is given to be much smaller than for electrons in similar condi- by21: tions and WAL effect should be considerably smaller in agreement with our observations. 3 h 1 T R∼ n λ L (1) When the Fermi levellies in the conductive band, and e2g2 p (cid:18)δ(cid:19) transport becomes dominant by massive fermions with Dirac spectrum (figure 1), the weak antilocalization be- where np is the density of the puddles, λ = ~v/Eg ≈ haviour has been observed34. 18nm is the electron penetration depth into the puddles In conclusion, we observed interplay between posi- (v ≈5.5×107cm/sis the electronvelocity,E ≃20meV g tive and negative magnetoresistance, when Fermi level is the forbidden gap), g is the dimensionless conduc- shifts with respect to the charge neutrality point, but tance within the dot (puddle), δ is the mean level spac- still lies inside of the gap, and transport occurs via edge ing within the dot, L is the distance between probes states. We consider three contributions to the magne- (length of the edge states). It has been found that toresistance: edge state penetration to the bulk, edge the equation 1 gives a satisfactory explanation of the statescatteringbymagneticflux formedbyroughedges, high resistance value, obtained in experiments33. Com- and WL of the bulk electrons in the puddles formed by bining all parameters we calculate ρ = R/(hL) = 0 e2 inhomogeneous charge distributions. The negative mag- 8 × 103 T 3(e2/h)/cm, which is comparable with ex- netoresistance and AB-like oscillations are attributed to δ periment(cid:0)al(cid:1)value ρ = 15×103(e2/h)/cm. The density weak localization of the bulk electrons in the metallic 0 of the puddles increases, and the size of the puddles is puddlesformedbyfluctuationofthelocalinsulatinggap. growing, when the Fermi level lies nearer to the con- We thank O.K.Raichev for helpful discussions. A fi- ductive band. 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