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1 DOI: 1.138/NMAT368 Imging currents in HgTe quntum wells in the quntum spin Hll regime Ktj C. Nowck, 1, Eric M. Spnton, 3, Mtthis Benninger, 3, Mrkus König, 3, John R. Kirtley, 1 Been Klisky, 1, 4 C. Ames, 5 Philipp Leuner, 5 Christoph Brüne, 5 Hrtmut Buhmnn, 5 Lurens W. Molenkmp, 5 Dvid Goldher-Gordon, 3, 1,, 3 nd Kthryn A. Moler 1 Deprtment of Applied Physics, Stnford University, Stnford, Cliforni 9435, USA. Stnford Institute for Mterils nd Energy Sciences, SLAC Ntionl Accelertor Lortory, Menlo Prk, Cliforni 945, USA. 3 Deprtment of Physics, Stnford University, Stnford, Cliforni 9435, USA. 4 Deprtment of Physics, Nno-mgnetism Reserch Center, Institute of Nnotechnology nd Advnced Mterils, Br-Iln University, Rmt-Gn 59, Isrel. 5 Physiklisches Institut (EP3), Universität Würzurg, Am Hulnd, D-9774, Würzurg, Germny. CONTENTS Ti 5nm / Au 5nm I. Supplementry methods 1 A. Device detils 1 B. SQUID mesurements C. Effect of finite RC time II. Current inversion III. Fitting procedures to extrct percentge of currents flowing long the top nd ottom edge nd through the ulk 4 A. Fitting of the current profiles shown in Fig. 4 B. Fitting of the flux profiles shown in Fig. 3 4 IV. Dependence on current mplitude 5 Ti 5nm / Au 5nm Al O 3 4nm Hg.3 Cd.7 Te 5nm HgTe d QW Hg.3 Cd.7 Te 1nm CdTe Buffer CdTe (1) Sustrte Al O 3 4nm Hg.3 Cd.7 Te 5nm Hg.3 Cd.7 Te: I 9nm Hg.3 Cd.7 Te 1nm HgTe d QW Hg.3 Cd.7 Te 1nm Hg.3 Cd.7 Te: I 9nm Hg.3 Cd.7 Te 1nm CdTe Buffer CdZnTe (1) Sustrte V. Gte dielectric induced disorder 6 VI. Imge series on Hllr H1 nd H 6 I. SUPPLEMENTARY METHODS A. Device detils We imged three Hll rs, denoted y H1, H nd H3. H1 nd H were fricted from quntum well structures with HgTe lyer thicker thn the criticl thickness, wheres H3 ws fricted from quntum well structure with HgTe lyer thinner thn the criticl thickness. The quntum well structures re shown in Fig. S1. The Hll rs were ptterned using opticl lithogrphy. Ar ion-milling is used to define mess. In/Au in the cse of H1 nd Au/Ge in the cse of H nd H3 is evported to crete ohmic contcts to the quntum wells. The Hll rs re covered with 4 nm thick Al O 3 gte insultor nd Ti (5 nm)-au (5 nm) gte electrode which ws lso ptterned using opticl lithogrphy. See Ref. 5 for more detils on the friction process. For imging we only contct the two top voltge proes of the Hll rs (see Fig. 1). Therefore we only mesure FIG. S1., Schemtic showing the lyers of the quntum well structures from which Hll rs H nd H3 were fricted. The quntum well thickness, d QW, is 8.5 nm for H nd 5 nm for H3., Schemtic showing the lyers of the quntum well structure from which Hll r H1 ws fricted. The quntum well thickness, d QW, is 6.6 nm. Quntum well thicknesses re determined y X-ry reflectivity. the two-terminl resistnce,. In tle I we summrize dditionl trnsport chrcteriztion of devices H1 nd H performed in different mesurement set-up t temperture of pproximtely 4 K. We hve not performed four-terminl mesurements of H3. Bsed on the densities extrcted t top gte voltges = V nd = 1 V mesured on H1 nd H nd the fct tht the resistnce pek in the imging nd chrcteriztion mesurement runs re t similr nd hve similr width, we conclude tht we tune the Fermi level of H1 from the conduction nd through the ulk energy gp into the vlence nd using the top gte during imging. NATURE MATERIALS Mcmilln Pulishers Limited. All rights reserved.

2 DOI: 1.138/NMAT368 Device H1 H TABLE I. Summry of the trnsport chrcteriztion of Hll rs H1 nd H. Mximum four terminl Density resistnce n-type cm t = V; p-type cm 9 kω t V t = 1V TG =.4 V n-type cm t = V; p-type cm 8 kω t V t = 1V TG =.31 V B. SQUID mesurements Imging ws done in two different scnning SQUID systems. H nd H3 were imged in 4 K system, while H1 ws imged in our dilution refrigertor system, however only with the 1K pot ut not the circultion running, such tht the temperture ws pproximtely 3 K. The SQUIDs re operted in flux locked loop, such tht the SQUID response is liner in the flux through the pickup loop with well-chrcterized slope 4. All presented imges nd line cuts were tken y pplying n AC current to the Hll r nd recording the SQUID signl using lock-in mplifier. The AC current ws pplied y connecting the voltge output of the internl oscilltor of the lock-in vi MΩ or 4 MΩ is resistor to one of the two contcted voltge proes of the Hll r, while the other ws grounded. All other contcts were floting. The frequency of the AC current rnged from 77 Hz to 757 Hz. C. Effect of finite RC time While there ws no qulittive dependence on the frequency of the pplied AC current, the ctul current mplitude pplied to the device vried with the frequency due to finite RC time in the system. The finite RC time minly origintes from the comintion of cpcitnce in the connection (stry cpcitnce nd cpcitnce present in filters) nd the device resistnce. The finite RC time results in n ttenution of the mplitude of the current flowing in the device nd phseshift in the current excittion. All presented imges re corrected for the phse shift in the lock-in signl (innd out-of-phse signl of the lock-in re recorded). We chrcterized the RC response y mesuring the voltge cross the device s function of frequency nd voltge pplied to the top gte (which chnges the device resistnce) nd y determining the phse shift we detect in the flux imges. This provides n estimte of the current mplitude t the device. For imges tken t positive top gte voltges t which is low, the RC time is shortest nd hence hs the lest effect. Here, the estimte of the current is the most ccurte nd cn serve s reference for imges tken t higher : from knowing the mount of current tht flows in one flux imge we cn determine the mount of current flowing in ny other imge tken t the sme height y using the fct tht flux profile through one of the Hll r contcts severl tens of microns ove the Hll r (see e.g. Fig. S5,d) is only wekly dependent on how the current distriutes in the Hll r. The fctor used to scle these flux profiles such tht they coincide from imge to imge is equl to the reltive chnge in current flowing in the different imges. The estimte of the current we otin in this wy is used in the min text, prt from Fig. 3 where we rescle with the nominlly pplied current, ecuse we hve not imged the full Hll r t ech temperture. The RC times were more significnt for mesurements of H1 performed in our dilution refrigertor due to lrge cpcitnce in the filters through which the wires to the contcts were pssed. In the min text we normlize the mgnetic (flux) nd current imges nd profiles y dividing y the pplied rms current mplitude, which we determine s just descried. Therefore the units for ll mgnetic imges nd profiles re mφ /na nd 1/µm for current imges nd profiles. This mkes comprison of imges nd profiles tken t different current mplitudes more convenient. II. CURRENT INVERSION In this section we descrie in more detil how we otin the current distriutions J x nd J y from the flux imges. We ssume tht the current density in the Hll r is two-dimensionl nd qusi-sttic ( J = ). Twodimensionl current flow implies, tht the current only flows in thin sheet of thickness d (here ssumed to lie in the x-y plne t z = ), such tht J z s well s the z-dependence of J x nd J y within the sheet cn e neglected. Both re justified ssumptions given the frequencies t which the mesurements re done nd the thickness of the quntum wells. The scnning SQUID is sensitive to the z-component of the mgnetic field generted y the two-dimensionl current density t the scn height z d ove the sheet, which is given y the Biot-Svrt lw : B z (x, y, z ) = µ 4π (x,y )(x x ) J y (x,y )(y y ) ((x x ) +(y y ) + z )3/ dx dy NATURE MATERIALS 13 Mcmilln Pulishers Limited. All rights reserved.

3 DOI: 1.138/NMAT368 SUPPLEMENTARY INFORMATION where J x nd J y re in units A/m. For two-dimensionl nd qusi-sttic J x nd J y eq. 1 cn e inverted. Here we perform the inversion in Fourier spce s descried in Ref. 6 with only one modifiction s descried in the following. We mesure the mount of flux B z genertes in the SQUID pickup loop t ech position. The detected flux φ(x, y) is given y convolution of B z (x,y,z ) with the geometry or more precisely the point spred function (PSF) of the SQUID pickup loop, P SQUID (x, y): to low-pss filter in Fourier spce given y: { 1 + cos (πk) if K<1 H(k x,k y )= otherwise kx with K = (kx mx ) + k y (k mx y ) () φ(x, y) = B z (x,y )P SQUID (x x, y y)dx dy y (μm) x (μm) 1.5 k y (μm -1 ) (1) We determine P SQUID (x, y) from imges of isolted vork x (μm -1 ) FIG. S., Point spred function of the SQUID pick-up loop in rel spce. Scler corresponds to 5 µm., Point spred function of the SQUID pick-up loop in Fourier spce. Red line shows the cut-off for the two-dimensionl Hnning filter given in eq. 3. White line is contour line showing the first zero crossing of the Fourier-trnsformed PSF. Scler corresponds to 1 µm 1. tices in ulk superconductor tken with nominlly identicl SQUID (see Fig. S). Since the penetrtion depth of the superconductor is much smller then the size of the pick-up loop, the mgnetic field generted y vortex cn e pproximted y the field generted y point-like mgnetic monopole one penetrtion depth elow the surfce of the superconductor. As prt of the inversion we divide in Fourier spce y the Fourier trnsform of the point spred function P SQUID (in Ref. 6 the pickup loop is ssumed to e perfect circle). The signl t high sptil frequencies is dominted y noise in the dt, since the imge itself is smoothed out due to the finite size of the SQUID pick-up loop. Also due to the finite size of the pickup loop, PSQUID crosses zero t finite k x,k y. Dividing y P SQUID then mplifies the noise dominted contriutions t k x,k y vlues tht re close to or lrger thn the k x,k y vlues of the first zero crossing of P SQUID. Becuse of this we low-pss filter the Fourier trnsforms of J x nd J y efore performing the inverse Fourier trnsform. We use Hnning filter 1 where kx mx nd ky mx re the cut-offs long x nd y. After low-pss filtering we otin J x (x, y), J y (x, y) y inverse Fourier trnsform. In prctice we use the Fst Fourier trnsform function provided y Mtl. The deconvolution is ccompnied y systemtic errors which we will discuss in the following. First, the Fst Fourier trnsform introduces ringing rtifcts into the rel spce imge, s cn e seen e.g. in dt sets in which noise spike occurred during scnning (see imges in tle II-IV). Second, the finite size of the imge cuses oundry effect. The Fst Fourier trnsform implicitly ssumes periodic oundry conditions. Therefore especilly the top oundry of our imges, where lrge signls from the Hll r contcts re present, cuses strong ringing. To reduce this effect we pd out ech imge to lrger size (typiclly doule the size) nd linerly extrpolte the flux signl to zero. The finite size of the imge introduces nother error: the inversion method gives y construction current density tht fulfills J =. However, in the originl imge the current is not conserved t the top oundry, t which the current flows in nd out of the imges vi the contcts. While J = is locl condition, it implies glolly tht JdS = for ny contour S within the imge. S To fulfill this condition the current inversion produces finite current density outside the Hll r. We sutrct n offset from ech J x nd J y imge to reduce this effect. This is justified, since constnt offset corresponds to the Fourier component t k x = k y = which is not well defined for finite size imge. In ddition to rtifcts from the Fst Fourier trnsform nd the finite imge size sources of systemtic errors re uncertinty in the scn height (solute vlue s well s smll grdient throughout the imge), errors in the point spred function (which y itself is otined y deconvolution of vortex imge) nd smll ngle in etween the plne of the pick-up loop nd the plne of the Hllr. A quntittive indiction for the qulity of the current inversion is tht integrtion of the current profiles shown in Fig. c,d long y yields the correct mount of totl current through the Hllr (estimted s descried in section I) within ±5 %. Note tht ll current profiles shown here nd in the min text re rescled y the estimted pplied current nd re therefore in units 1/µm. NATURE MATERIALS Mcmilln Pulishers Limited. All rights reserved.

4 DOI: 1.138/NMAT368 III. FITTING PROCEDURES TO EXTRACT PERCENTAGE OF CURRENTS FLOWING ALONG THE TOP AND BOTTOM EDGE AND THROUGH THE BULK In Fig. nd Fig. 3 we fit current nd flux profiles with edge nd ulk contriutions to determine which percentge of the current flows long the edges nd through the ulk. Here we descrie the fitting procedures in more detils. A. Fitting of the current profiles shown in Fig. Current density (μm) J x top J x ottom.4 ulk. J x 5 1 Position long y (μm) Current density (μm) V.3.57V..6V.1.65V fit dt.46v.4v.37v.5v 1V.1V Position long y (μm) FIG. S3., The normlized current profiles, top, ottom, ulk, corresponding to the cses tht 1 % of the current flows long the top (lue) nd ottom (green) edge nd through the ulk (red). These re used for fitting the line cuts of the current density long x, J x shown in Fig. c,d., Shown re exmples of the current profiles from Fig. c nd d (lue) together with the fit (red). Profiles re offset for clrity. The top gte voltge corresponding to ech line cut is indicted. The current profiles, J x, shown in Fig. c,d re modeled y sum of three contriutions, top, ottom nd ulk : J x = F top top + F ottom ottom + F ulk ulk. Ech contriution is normlized such tht they correspond to 1 % of the current flowing through the top edge, the ottom edge nd through the ulk respectively. top, ottom nd ulk re shown in Fig. S3 nd re determined from the current profiles shown in Fig. c,d: ulk is the verge of the profiles t = 1. V nd =.1V, wheres top nd ottom re extrcted from the verge of the two profiles t =.49 V nd =.48 V. The fitting results vry minimlly (< %), if we do not verge two profiles or choose line cut t slightly different. We normlize top, ottom nd ulk such tht Hllr J x(y)dy i = 1 for i = top, ottom, ulk. Prior to fitting we lso normlize ech current profile in the sme wy. We hve fitted the current profiles oth with three independent prmeters F top,f ottom nd F ulk s well s with two independent prmeter y imposing F top + F ottom + F ulk = 1. The results for F top,f ottom nd F ulk re the sme within less then 5 % whether or not we impose this condition. Fig. e nd f re sed on fits in which we impose F top + F ottom + F ulk = 1. By construction the fit returns very low mplitudes F ulk t top gte voltges t which the trnsport is dominted y edge conduction. We estimte the mount of current flowing through the ulk t those top gte voltges (e.g. =.49 V) to e less thn 5% sed on integrting the current density in etween the two current peks t the edges. The ringing (section II) imposes limits on how smll edge contriutions cn e relily detected. We therefore limit the xis in Fig. g which shows the effective resistnces of the edges nd the ulk. The qulittive results of this nlysis re roust ginst the detils of the fitting procedure nd its systemtic errors. B. Fitting of the flux profiles shown in Fig. 3 For the temperture dependence shown in Fig. 3 we hve not imged the full Hll r t ech temperture. We therefore cnnot pply the current inversion. Insted we directly fit the flux profiles, φ, with sum of three flux profiles, φ top,φ ottom,φ ulk (see Fig. S4 ). These profiles correspond to 1 % of the current flowing through the top edge, the ottom edge nd the ulk. The profile corresponding to 1 % current flowing through the ulk is otined from scn t top gte voltge fr off the resistnce pek. The profiles corresponding to 1 % of the current flowing through either only the top or only the ottom edge re otined from simultions tking the Hll r geometry including the contcts into ccount. More specificlly, we simulte the current densities y solving Poisson equtions with potentil difference etween the contcts. We ssume thin conducting strip long either the top or the ottom pth of the top gted prt of the Hll r for the cse of edge conduction. The prt of the Hllr tht is not top gted is ssumed to e homogeneouslsy conducting sheet. Using the Biot- Svrt lw, we clculte the z-component of the mgnetic field t the scn height from the current density. Finlly we convolute the mgnetic field with the point spred function of the SQUID (see Fig. S). We fit ech flux profile in Fig. 3 with sum φ = F top φ top + F ottom φ ottom + F ulk φ ulk. To otin the percentge of the current in flowing through the ulk nd long the edges, F top,f ottom,f ulk, we normlize the mplitudes fter the fit: F i = F i /( F top + F ottom + F ulk ) for i = top, ottom, ulk. 4 NATURE MATERIALS 13 Mcmilln Pulishers Limited. All rights reserved.

5 Flux (mφ /na) Flux (mφ /na) Flux (. u.) Flux (Φ ) Flux (mφ /na) Flux (mφ /na) DOI: 1.138/NMAT368 SUPPLEMENTARY INFORMATION top ottom ulk Position long y (μm) K 17.5K 15K 1.5K 1 1K x K 4.5K fit dt 4 Position long y (μm) FIG. S4., Simulted (top nd ottom) nd mesured (ulk) flux profiles corresponding to 1 % of the current flowing long the top (lue) nd ottom (green) edge nd through the ulk (red). These re used for fitting flux profiles shown in Fig. 3. The simultions were performed with the sme mount of current s ws pplied in the mesurement of the ulk profile., Flux profiles from Fig. 3 (lue) together with the fitted profile (red). Trces re offset y Φ for clrity. The pplied current ws nominlly 15 na (not corrected for RC time). The tempertures re noted in the figure. IV. DEPENDENCE ON CURRENT AMPLITUDE The flux imges nd profiles presented in the min text re tken t different current mplitudes which re higher thn typiclly pplied in trnsport mesurement. Specificlly, the flux imge in Fig. 1d ws tken with n rms mplitude of pproximtely 5 na nd the imge shown in Fig. 1e t pproximtely 14 na. Imges shown in section VI nd the corresponding profiles shown in Fig. were tken with current mplitudes rnging from 14 na nd 5 na, prt from the imge t =.1V which ws cquired t n mplitude of 1 µa. The lowest pplied mplitude in tht imge series corresponds to the highest due to the finite RC time in the system (see section I), since ll imges prt from the one t =.1V were tken t 77 Hz nd the sme output mplitude from the lock-in mplifier. 1 x na 1 95 na 14 na 153 na 183 na 33 na Position long y (μm) x Position long x (μm) c d x na 11 na na 5 na 9 na Position long y (μm) 4 x Position long x (μm) FIG. S5., Flux profiles long y through the center of the Hll r H1 for different pplied rms current., Flux profiles long x crossing the upper left contct tken t the sme current s profiles in. At tht position the flux profile is only wekly dependent on how the current flows in the Hll r. c, d, Sme s, showing dt set tken t lower currents. All dt is tken t =.51 V. Profiles re n verge over 5 to 1 profiles in width of 1.5 µm to3µm. At high mplitudes the I-V chrcteristic of the Hll r nd lso the imges ehve non-linerly. In Fig. S5 we show two series of line cuts tken t fixed top gte voltge nd position s function of the rms current mplitude. The line cuts long y re through the center of the Hllr nd show tht t lrger current the shpe of the profile chnges slightly. At lower currents the profiles scle linerly, in the sense tht twice the mplitude results in twice the mount of flux. The profiles re normlized with the pplied current nd hence fll on top of ech other. The profiles long the x-direction re tken severl tens of microns ove the Hllr cutting through the left contct. The fct tht the line cuts fll on top of ech other shows tht we correctly estimte the pplied current for ech line cut. The rnge of currents over which the device ehves linerly depends on the pplied top gte voltge. At top gte voltges fr off the resistnce pek, the device is liner up to currents s lrge s 1 µa. This is consistent with the following simple picture. Applying high current mplitude hs similr effect s gting the device, since the voltge drop long the device chnges the potentil difference etween the Hll r nd the top gte long the device. Therefore the device ehves non-liner t the point t which the voltge drop over the device ecomes comprle to chnge needed in top gte voltge to significntly chnge the device resistnce. An upper ound for the voltge drop long NATURE MATERIALS Mcmilln Pulishers Limited. All rights reserved.

6 DOI: 1.138/NMAT368 (kω) c 8 4 d c Top gte voltge mφ /na x1-4 - d μm FIG. S6., Two terminl resistnce of H1 (sme device s in Fig. 1 of the min text) ut mesured in different therml cycle. -d Mgnetic imges t top gte voltges s indicted in. White dshed lines indicte the outline of the Hll r mes. of the top gte voltge. Here we show line cuts of the current density long x t the sme nd dditionl vlues of the top gte voltge (Fig. S7), s well s line cuts corresponding to different position long the Hllr (Fig. S7). Tle II nd III show prt of the imge series from which the profiles shown in Fig. re extrcted. As discussed in the min text, t vlues of the top gte voltge corresponding to the flnks of the resistnce pek the mount of current flowing long the edge chnges long the Hll r nd some inhomogeneity is present (see imges in tle II, III s well s the difference etween Fig. S7 nd ). Interestingly, smll region t the top nd close to the middle of the Hll r is visile, which stys conductive in rnge of top gte voltges on the p-side t which the rest of the ulk is lredy insulting, ut stys insulting over rnge of top gte voltges on the n-side t which the rest of the ulk is lredy conducting (see gry dshed ox in tle II, III). In tle IV we show n imge series tken on Hll r H (tle IV). the device is given y I pplied, where is the two-terminl resistnce icnluding contct resistnce nd I pplied the mplitude of the current. For ll imges shown in Fig. 1 nd this intervl is smller then 3 mv. From this rgument nd mesurements s shown in Fig. S5 we find tht our conclusions re not ffected y the non-liner ehvior occurring t high pplied current mplitudes. V. GATE DIELECTRIC INDUCED DISORDER Position long y (μm) J x (μm) -1.1 Fig. S6 shows the two-terminl resistnce nd mgnetic imges of Hll r H1 tken in different therml cycle s descried in the min text. We elieve tht the lrge-scle inhomogeneity is cused y inhomogeneous chrge in the gte dielectric cused y n unintentionl electricl shock to the top gte. VI. IMAGE SERIES ON HALLBAR H1 AND H In Fig. we only show line cuts of the mgnetic imges nd imges of the current densities t few vlues -.5 Top gte voltge FIG. S7., Current density long x, J x, t fixed position long the Hll r s function of the top gte voltge. nd correspond to two different positions long the Hll r s indicted in tle II. Verticl lck rs re 1 µm sclers. 6 NATURE MATERIALS 13 Mcmilln Pulishers Limited. All rights reserved.

7 DOI: 1.138/NMAT368 SUPPLEMENTARY INFORMATION TABLE II This tle shows prt of the imge series from which the profiles shown in Fig. re extrcted. Gry rrows ove the mgnetic imge in the first row indicte the positions t which line cuts shown in Fig. S7 re extrcted. Gry dshed oxes in some of the imges of the current density long x highlight the region in the Hll r discussed in the supplementry text. Mgnetic imge 4 4 mφ /na x1-4 Current density long x.1.1 1/μm Current density long y.1.1 1/μm μm V TG µm V TG µm µm µm V TG µm µm V TG µm NATURE MATERIALS Mcmilln Pulishers Limited. All rights reserved.

8 DOI: 1.138/NMAT368 TABLE III Continution of the imge series on H from which the profiles in Fig. re extrcted V TG µm µm µm V TG µm V TG µm µm V TG µm µm µm 8 NATURE MATERIALS 13 Mcmilln Pulishers Limited. All rights reserved.

9 DOI: 1.138/NMAT368 SUPPLEMENTARY INFORMATION TABLE IV This tle shows series of imges tken on Hll r H. Mesurements re done t temperture of pproximtely 4K nd nominl rms current mplitude of 5 na. Distortions in the imges re n rtifct cused y non-linerities in the fine motion long x in this specific cooldown mφ /na x1-4 1/μm 1/μm V TG µm V TG µm µm V TG µm µm V TG µm µm µm NATURE MATERIALS Mcmilln Pulishers Limited. All rights reserved.