Carrier Density and Thickness Dependent Proximity Effect in Doped Topological Insulator -- Metallic Ferromagnet Bilayers
CCarrier Density and Thickness Dependent Proximity Effect in Doped TopologicalInsulator - Metallic Ferromagnet Bilayers
Yaron Jarach, Gad Koren, Netanel Lindner , Amit Kanigel
Physics Department, Technion-Israel Institute of Technology, Haifa 32000, Israel.
We use magneto-conductivity to study magnetic proximity effect on surface states of doped topo-logical insulators. Our bilayers consist of a layer of Fe Se , which is a metallic ferrimagnet anda layer of Bi . Sb . Te which is a highly hole-doped topological insulator. Using transport mea-surements and a modified Hikami-Larkin-Nagaoka model, we show that the ferromagnet shortenssignificantly the effective coherence length of the surface states, suggesting that a gap is opened atthe Dirac point. We show that the magnetically induced gap persists on surface states which areseparated from the magnet by a topological insulator layer as thick as 170nm. Furthermore, thesize of the gap is found to be proportional to the magnetization that we extract from the anomalousHall effect. Our results give information on the ties between carrier density, induced magnetizationand magnetically induced gap in topological insulator/ferromagnet bilayers. This information isimportant both to theoretical understanding of magnetic interactions in topological insulators andto the practical fabrication of such bilayers, which are the basis of various suggested technologieswhich depends on these interactions, such as spintronic devices, far infra-red detectors etc. I. INTRODUCTION
Strong topological insulators (TIs) are band insulatorswith time reversal symmetry (TRS) protected gaplesssurface states . The dispersion of the surface statesexhibits an odd number of massless Dirac fermions and astrong spin-momentum locking . Magnetic fields nor-mal to the surface will open a gap at the Dirac points .Magnetically gapped surface states host a variety of fas-cinating phenomena, including the quantum anomalousHall effect (QAHE) .One way to break TRS is by interfacing a TI with a fer-romagnetic (FM) material . The surface states inter-act with the magnetic moments and acquire a magneticproximity effect induced energy gap . One major ad-vantage of this method is the ability to pattern the FMlayer and create spatially varying magnetic interactions .Magnetic proximity effect induced energy gaps on sur-face states in TI/FM bilayers may result for two differentmechanisms : 1) induction of spin polarization in theTI. 2) exchange mechanisms such as RKKY that cou-ple the surface states and the magnetic material leav-ing the TI non-magnetic . Polarized neutron reflectionand magnetic second harmonic generation measurementsshow that only 1-2 nanometers in the TI are magnetizedby the magnetic interface . However, magnetic ex-change coupling in FM/Metal structures can penetratetens of nanometers into the metal .While magnetic proximity effect is known to gap thesurface states on the interface that is in direct contactwith the FM , an important question is whether mag-netic exchange coupling can also gap surface states thatare not in direct contact with the FM. This remains sofar an open question . In this work, we studiedthis question for FM/doped-TI bilayers, for which one ofthe surface states is in direct contact with the FM, whilethe opposing (exposed) surface is separated from the FMby the TI bulk. A direct way to probe the surface states dispersion isvia angle-resolved photoemission spectroscopy (ARPES)or scanning tunneling microscopy (STM). However, theseare surface methods that require high quality surfaceswhich are difficult to achieve in TI/FM bilayers. Simplermethods to probe the surface states energy gap involveelectrical transport measurements. The anomalous Halleffect in the TI can indicate TRS breaking and gappedsurface states . Furthermore, the surface states gapstrongly affects the magneto-conductivity . The gapsize can be calculated using a modified Hikami-Larkin-Nagaoka (m-HLN) equation . We discuss this Eq.insection II.Naturally, to explore the surface states gap using trans-port measurements insulating TIs and FM are prefer-able. Thus, previous transport studies on TI/FM bilay-ers used low carrier density (cid:0) n (cid:46) (cid:2) cm − (cid:3)(cid:1) ultra-thin( t (cid:46) nm ) TIs and insulating FM .However, low carrier density samples might lack the ex-change mechanisms needed to open a gap in the exposedsurface states, since exchange mechanisms are highly sen-sitive to carrier density . We therefore useddoped TIs in this study.In addition, proximity to a FM metal can be con-siderably different from a proximity to a FM insulator.This is because the metal can modify the TI’s bandstructure, chemical potential and the magnetic couplingmechanism . Therefore, in this study we focusedon TI/FM metal bilayers. This is also of practical interestsince there are only a few TI-compatible ferromagneticinsulators and some possible TI/FM applications requirea FM metal .Here, we study the electrical transport properties ofhighly doped TI/FM-metal bilayers with varying TIthicknesses and carrier densities. By fitting the bilayersmagneto-conductivity with the modified Hikami-Larkin-Nagaoka (m-HLN) for surface states magneto-conductivity we extract the magnetic proximity effect in- a r X i v : . [ c ond - m a t . m e s - h a ll ] J a n duced energy gap. Our results indicate a long-range, car-rier density dependent proximity effect which gaps bothsurface states, the one in direct contact with the FMbut also the surface states on the exposed surface of theTI. We also verify the validity of m-HLN equation inour system despite the large amount of bulk conductingchannels. II. RESULTS AND DISCUSSION
Figure 1 shows the structure of our bilayers. The TIlayer is a highly p-type doped Bi . Sb . T e film withthickness ranging from 17 nm to 170 nm . This compoundhas an hexagonal lattice structure and a ∼ . eV bandgap . We note that for TIs thinner than 5 nm the sur-face states may be gapped due to a hybridization betweenthem . As we show, the thickness variation results alsoin carrier density variation, enabling us to study boththickness and carrier density effects.The FM layer is a F e Se film, which is a hexagonalmetallic ferrimagnet with T c ∼ K and a canted easymagnetization axis . We grew the bilayers on Al O substrates using pulsed laser deposition (PLD) at 280 C and a pressure of ∼ ∗ − [ T orr ]. X-ray diffractionshows that the FM and the TI grow along the c-axis,with grain size of about 45nm (see appendix E). TheFM layer thickness was 12nm in all of the bilayers. Alltransport measurements were done at T= 1 . K using thestandard 4-contact Van-der-Pauw (VDP) technique. Surface states properties:
The
F e Se film out-of-plane magnetization component opens an energy gap (∆)at the Dirac node of the surface states dispersion. Inthis case, the surface states magneto-conductivity (MC)obeys a modified Hikami-Larkin-Nagaoka model . Webegin our analysis using the small-gap limit of the model(for details see appendix A):∆ σ ( B ) = − e πh α (cid:34) ψ (cid:32)
12 + l B l φ (cid:33) − ln (cid:32) l B l φ (cid:33)(cid:35) (1) l − φ ≡ l − c + l − e f ( (cid:101) ∆ ) (2)Where ∆ σ ( B ) ≡ σ ( B ) − σ (0) is the MC of one sur-face and ψ is the digamma function . The param-eter l B = (cid:114) (cid:126) eB is the magnetic length. The lengthscale l φ is the effective coherence length of the surfacestates in the presence of an induced gap. As indicatedby Eq.2, l φ depends on l c , which is the surface statescoherence length; l e , which is the elastic mean free path;and (cid:101) ∆ ≡ ∆ /(cid:15) F is the normalized magnetically inducedenergy gap. Furthermore, in Eq.1 α and f are functionsof (cid:101) ∆, whose form is given in appendix A .We can therefore study magnetic interactions in ourbilayers by fitting the MC signal with Eq.1. In Figure 2we show a comparison between the MC data for a bilayerand for a bare TI film for several different TI thicknesses. (a)(b) FIG. 1: (a) The bilayer scheme (b) A Time Of FlightSecondary Ion Mass Spectroscopy (TOF-SIMS) 3Dimage of Bi (green) Fe (red) atoms relative intensities in170nm Bi . Sb . T e on 12 nm F e Se on Al O . Xand Y axes are spatial coordinates and Z is etchingtime. The FM etching rate is ˜5 times lower than theTI rate so the layer thicknesses are not scaled. Thebottom transparent layer is the Al O substrate. Theimage shows that the bilayer consists of two distinctphases with an interface volume much smaller than thesample volume.We fit the data with the m-HLN form for the MC givenby Eq1. Our next steps will be to show that the bilayerMC data is dominated by the exposed surface states andnot by the bulk transport channels, and discuss the fit-ting parameters of Eq.1. Afterwards we will return toour main topic which is the extracted energy gap ( (cid:101) ∆)properties.Eq.1 gives the conductivity of a system with a single 2Dmassive Dirac Fermion dispersion. The fitting betweenEq.1 and our data demonstrates the surface states ro-bustness even in proximity to a FM-metal interface .It also indicates that the MC of both the F e Se and theTI bulk are negligible.To verify that the F e Se MC is negligible we hadmeasured a bare
F e Se film and found MC values below1% of the total MC of the BLs. For the TI bulk, theleading order obeys Kohler’s equation : σ bulkxx ( B ) ∼ σ bulkxx (0) (cid:0) − µ xx B (cid:1) , with µ xx the mobility. By fittingthis equation to the high-field MC data and extrapolatingFIG. 2: Magneto conductivity (MC),∆ σ xx = σ xx ( B ) − σ xx (0), of bare TI films (diamonds)and TI/FM-metal bilayers (filled circles) with a m-HLNmodel fit. The TI thicknesses are 17nm (red), 73nm(blue) and 170nm (black). (Inset) Conductivity vsmagnetic field of a bilayer with 170nm TI, with aKohler quadratic fit in the high fields. The fit gives anupper limit on the TI bulk MC term, showing it isnegligible at low fields.down to zero field, we found that the bulk MC is anorder of magnitude smaller than the total MC of thebilayers. The inset in Fig. 2 shows this fit for a bilayerwith 170nm thick TI (thickest in this study). From thisfit we obtained a mobility of about 90 cm /(v · s). Thismobility value is within 10% from the value obtained byHall measurements, which verifies our MC extrapolatingprocedure.Previous studies have shown that although in the bi-layers the TI layer has two surfaces, the MC data canbe fit using Eq.1, which corresponds to a single surface.One explanation is that the short coherence length atthe surface in direct contact with the FM makes its MCsignal negligible . Another possible explanation isthat the two surfaces are coupled even for TI thicknessof ∼ . This means that either the extracted en-ergy gap (cid:101) ∆ is a property only of the exposed surfacestates, or both surfaces are gapped since they are cou-pled. Therefore, we can conclude that the reduced MCof the bilayers is a property of the exposed surface states.This is a crucial point in our analysis. In appendix Fwe give another detailed justification for this conclusion.One important consequence is that we cannot explain thereduced bilayer MC as a result of magnetic scattering atthe surface, because there are no magnetic particles atthe exposed surface. Moreover, it allows us to use theMC data to study the thickness dependence of the mag-netically induced energy gap, since the thickness shouldaffect mainly the exposed surface states, or the couplingof these states to the magnetic interface.In the m-HLN model the function α in Eq.1 depends (a)(b) FIG. 3: (a) The bilayers’ effective coherence length l φ vs TI layer thickness (cid:0) t T I (cid:1) , extracted by fitting the lowfield MC data (Fig.2) using Eq.1. (b) The bilayers’normalized magnetically induced energy gap (cid:101) ∆ = ∆ /(cid:15) F ,obtained by fitting the low field MC data assuming thebilayers’ coherence length l c is the same as in our bareTI films (blue), or fourth of this value (red). See maintext and the for a discussion of this result. Error barsare from the fitting and thickness uncertainty. Note, theinflection point around 73nm.solely on (cid:101) ∆ , but previous studies have shown thatthis is true only up to an experimental prefactor .The experimental prefactor masks the dependence of α on (cid:101) ∆, so we can regard α as an independent parameter.Consequently, there are two independent fit parametersin Eq.1: α and the effective coherence length l φ . Belowwe explain how we estimated the different contributionsof the gap, the mean free path and coherence length tothe effective coherence length l φ (Eq.2).Figure 2 shows a strong effect of the FM layer on theTI surface states transport properties, as expected if thesurface states are gapped . The gap reduces the sur-face states coherence length (Eq.2) and consequently re-duces their MC (Eq.1). Figure 3a shows the l φ valuesfor bilayers with various TI thicknesses and a constant12nm FM. The bilayers’ l φ are of order 100nm, muchshorter than in bare TI films where we found l c of order500nm . We therefore argue that the exposed sur-face states acquire a gap by a magnetic proximity effect,and that this gap dictates the bilayer effective coherencelength, l φ (see Eq.2). Below we will further justify theseconclusions by showing self-consistency with other partsof our data. In particular, we will show that the gap isproportional to the magnetization we extract from theanomalous Hall data. We will also explain why we claimthat the reduced effective coherence length is due to mag-netic effects and not due to structural differences betweenthe bare TIs and the TI layers in the bilayers.To extract the normalized energy gap (cid:101) ∆ quantitativelywe assume that l c in the bilayers is the same as l c in thebare TIs. Under this assumption, we use the full m-HLNequation to fit the MC data and extract (cid:101) ∆ (see appendixA for details). Figure 3b shows (cid:101) ∆ of the bilayers as afunction of TI thickness. The values we find correspondto energy gaps between ∼ . (cid:15) F and ∼ . (cid:15) F . We alsocompare between the values we got for l c equals that ofthe bare TIs (blue), and the values we got for fourth ofthis value (red). As figure 3b shows, our assumption isrobust and the results remain almost the same, as longas l c is at least 20% than l c in the bare TI. Our resultsshow that the exposed surface states are gapped evenin the thickest bilayer with a 170nm TI layer, suggest-ing a coupling between the magnetic interface and theexposed surface states. As we will show below, this cou-pling highly depends on the TI bulk carrier density andthickness.Figure 3b shows that (cid:101) ∆ has an inflection pointas a function of thickness between 50nm and 100nm.We will show that this corresponds to a minimum inthe bulk magnetization and carrier density. To studythis point, and since the magnetic coupling and (cid:101) ∆could depend on the TI carrier density and magnetization , we discuss now these properties. Thickness dependent bulk carrier density:
Figure 4 showsthe carrier density of the TI layer (cid:0) n bulk (cid:1) as a function ofits thickness. Note that in the notation (cid:0) n bulk (cid:1) we mean n of the TI bulk. Importantly, n bulk is minimal wherethe gap has its inflection point.We extracted n bulk from Hall measurements using theDrude model for conductors connected in parallel. Weonly needed to consider the FM layer and TI bulk trans-port channels because the high resistance of the surfacestates makes their contribution negligible . The Drudemodel gives: R totH ≡ V totxy BI ∼ (cid:18) R totxx R F Mxx (cid:19) R F MH + (cid:18) R totxx R bulkxx (cid:19) R bulkH (3)Where R totH and R totxx are the measured Hall coefficient FIG. 4: The bilayer TI layer carrier density vs TI layerthickness calculated using Eq.3. The FM thickness is12nm.and linear resistance of the bilayer. We use the approx-imation R xx = ( nteµ xx ) − with t the thickness and µ xx the mobility since in our samples R xx (cid:29) R xy . Notethat Eq.3 holds only for the linear Hall terms withoutthe anomalous Hall terms which are a function of samplemagnetization (see below). We could extract the linearterms from the high magnetic field data where the mag-netization saturates.To extract the TI bulk properties from Eq.3 we need R F Mxx and R F MH . We could not measure them directlyfrom a bare FM because the TI layer growth changesthe FM properties. We thus grew a bilayer with only5nm thick TI layer, where we assumed that the FM layerdominates the transport properties. This assumption isjustified by the resistance of a bare 5nm thick TI filmbeing ten times larger than that of a 12nm thick FMfilm. Therefore, we used the measured properties of theBL with 5nm TI to represent the properties of the FMlayer in our analysis. For example, we took R totH ( t =5 nm ) ≈ R F MH .The carrier density non-monotonic behavior (Fig.4)may result from a combination of different reasons. First,lattice mismatch creates defects and strain that decreaseswith film thickness. Second, the metal-semiconductor in-terface leads to a band bending due to the Shottky barriereffect . Third, Se diffusion from the F e Se layer canreduce hole concentration since Se is an electron donor in BST compounds . Different thickness dependence ofthe various contributions to the carrier density variationcan lead to a non-monotonic carrier density thickness de-pendence. However, to quantify these effects will requirea detailed chemical and structural study, which is beyondthe scope of this paper.
Bulk magnetization:
We turn now to the magnetiza-tion. In a FM, the Hall resistivity includes an anomalousHall effect (AHE) term that is proportional to perpen-dicular magnetization : ρ xy ≡ V xy I xx t = R totH tB z + α tot M totz (4)With B z and M totz being the out of plane componentof magnetic field and bilayer magnetization, respectively. (a)(b) FIG. 5: (a) The bilayer (total) Hall resistivitysymmetric part, which is the anomalous Hall effect, forbare
F e Se (black, right y axis) and bilayers withdifferent TI thickness (left y axis): 17nm (red), 73nm(green), 102nm (blue). Note the opposite sign of the F e Se and bilayer signals. The displayed data is forpositive to negative magnetic field direction in thehysteresis curve. (b) The bilayer remnant magnetizationvs thickness normalized by the minimal value to cancelunknown constants. As explained in main text, thesignal arises mainly from the magnetization that theFM induces in the TI layer.According to the theory of Luttinger and Berger, α tot = C ∗ ρ βxx with β ∼ C a constant. For simplicity wetake β = 2.To differentiate between the ordinary and anomalousterms we take the symmetric part with respect to themagnetic field: ρ symxy ≡ ( ρ xy ( B ) + ρ xy ( − B )) / C ∗ (cid:0) ρ xx M z (cid:1) sym . Figure 5a shows ρ symxy for a bare 12nmthick F e Se film (black, right axis) and three bilayers(colored, left axis).Intriguingly, the bilayers have a negative AHE termwhile for a bare Fe Se the sign is positive. The bilayercurves FWHM is wider, namely their coercive fields arehigher than that of the bare Fe Se .Previous works have shown negative AHE terms inTI/FM-insulator bilayers, where the AHE arises fromthe TI layer . They have also shown that the FMmagnetization penetrates the TI layer with a flipped di-rection, explaining the negative AHE . Bulk mediatedmagnetic exchange coupling is also diamagnetic in Bi compounds . We therefore conclude that the TI layerhas the dominant contribution to the AHE of the bi-layer. The AHE in the TI is indicative of TRS breakingand gapped surface states , as we argued by analyzingthe effective coherence length (Fig.3a).We are interested in the zero field surface states energygap so we will focus now on the remnant magnetization,defined as the magnetization at zero external magneticfield. Using Eq.3 and Eq.4 we can write (appendix B): M rem ≡ M z (0) ∼ C − (cid:32) R totxy (0) t T I ( R totxx (0)) (cid:33) (5)Figure 5b shows M rem vs TI thickness normalized byits minimal value. A non-zero value of M rem leads to agap in the surface states. Notably, the remnant magne-tization M rem and carrier density n bulk (Fig.4) share aminimum just where the normalized magnetic gap has itsinflection (Fig.3). Analyzing the gap using bulk properties:
So far we onlydiscussed the normalized gap, (cid:101) ∆ ≡ ∆ (cid:15) F , which we obtainfrom the m-HLN fit. To estimate the actual gap valueswe first need to estimate (cid:15) F . Using the parabolic bandapproximation, we can write: (cid:15) F = (cid:126) m ∗ (cid:0) π n T I (cid:1) / (6)Here, m ∗ is the effective electron mass and n T I is thecarrier density of the TI layer (Fig.4).Figure 6a shows the magnetic gap ∆ as a function ofthe TI layer thickness. We normalized the values by theminimal value to cancel the mass prefactor. Figure 6bshows a clear correlation between the bulk magnetizationand the energy gap. We note that this correlation veri-fies our assumption that the clear difference between themagneto-conductivity of the bilayers and bare TI films(Fig.2) is due to a gap opening in the surface states.The gap must be opened since the sample magnetizationbreaks time reversal symmetry .Previous work found that the bulk states effective elec-tron mass in Bi − x Sb x T e compounds is about one elec-tron mass . Inserting m ∗ = m e in Eq.6 we get (cid:15) F val-ues of ∼ − meV and corresponding magnetic gapvalues of ∼ . − meV , depending on TI thickness.These gap values are similar to gaps reported in otherTI/FM systems . However, the parabolic band ap-proximation does not fully describe the band structure of Bi − x Sb x T e , thus the real gap values may differ fromour estimation.The results indicate that there is a gap even for thethickest TI layer of 170nm. A natural length scale forexchange interactions is l int = (cid:126) v F πk B T (7)Using the parabolic band approximation with m ∗ = m e and our carrier densities we get length scales of ∼ − (a)(b) FIG. 6: (a) The magnetic gap ∆ vs t T I normalized byits minimal value, showing a similar thicknessdependence as the magnetization and carrier density(Fig. 4 and Fig. 5b). We calculated ∆ assumingparabolic band approximation. (b) Magneticallyinduced energy gap (∆) vs remnant bulk magnetization( M rem ). We normalized the values by the minimalvalue to cancel unknown constants. Importantly, thefigure shows that the gap monotonically depends on thebulk magnetization.300 nm . For example, l int is about 100 nm in our 170nmsample. These length scales are of the same order asthe samples thickness which can explain how we foundproximity effect is such thick samples.The minimum in the energy gap that we find for a spe-cific thickness may seem surprising. One would expectthe interactions between the exposed surface states andthe magnetic interface to decay monotonically with theTI thickness. However, the interaction length scale l int depends on v F which is a function of the carrier density.This can result in a non-monotonic thickness dependenceof the exchange interaction, because the carrier densitydependence on the thickness is non monotonic in our bi-layers (Fig.4).In our bilayers we grew the TI layers on the FM layer,while we grew the bare TIs directly on the Al O sub-strate. Therefore one might assume that the reducedsurface states effective coherence length (figures 2 and3) is due to stronger strains and smaller grain size, andnot due to magnetic proximity effect as we suggest here.Indeed, using X-ray data we show in appendix E that there is slightly higher strain in the TI layers of the bi-layers compared to the bare TIs. However, we show thatthis is a tensile in-plane strain, which is known to en-hance surface properties and therefore cannot reduce thecoherence length in the bilayers . We also show in ap-pendix E that the TI grain size is about 45 nm both inthe bare TIs and the bilayers, confirming that grain sizedifference is not responsible for our results. Moreover,in appendix D we explain why we could neglect possibleeffects of Fe doping of the TI layer from the underliningFM layer in our analysis.Previous publications have shown that ferromagnetismpenetrates only ∼ nm into the TI . This mightseem in contradiction with our results. However, in thesepublications the films have very low carrier concentra-tion of n bulk (cid:46) , and the measurements were doneat a slightly higher temperature where equation 7 in-deed predicts a small magnetization penetration lengthof l int (cid:46) nm . More importantly, these publicationspresent induction of ferromagnetism into the TI, whileRKKY-like interactions are antiferromagnetic. It mightbe that only a small portion of our TI layers is ferromag-netically magnetized and gives the AHE signal. However,the exposed TI surface can still interact with this ferro-magnetic part via bulk states mediated by long rangedRKKY-like antiferromagnetic interactions . Such in-teractions break the time reversal symmetry of the ex-posed surface states and open a gap. III. SUMMARY
In this study we measured the electrical transportproperties of doped-TI/FM-metal bilayers, made of 17-170nm Bi . Sb . T e on 12nm F e Se . By fitting thelow field MC data to the m-HLN model we demonstratedits validity in doped-TI/FM-metal interfaces. As a fit pa-rameter we extracted the exposed surface states effectivecoherence length l φ , which is a geometric average of thecoherence length and a term arising from a magneticallyinduced gap.We compared the MC and l φ between our bilayersand bare TIs and showed that l φ in the bilayers is muchshorter. We therefore argued that the gap term dom-inates l φ in the bilayers. Assuming that the ordinarycoherence length l c of the surface states in the bilayersis not changed by more than a factor of five comparedto its value in the bare TIs, we extracted the normal-ized gap (cid:101) ∆ = ∆ /(cid:15) F . We find that the normalized energygap of the exposed surface states (cid:101) ∆ remains finite evenfor a 170nm thick TI layer. This indicates a couplingbetween the magnetic interface and the exposed surfacestates which is mediated by the bulk states. We con-clude that high bulk carrier density of n bulk (cid:38) cm − is necessary for the coupling in such thick layers.To justify our analysis, we first show a strong correla-tion between the carrier density the magnetization and (cid:101) ∆. Second, we show that the bilayer anomalous Hall sig-nal is dominated by the magnetic interactions inside theTI layer. Such magnetic interactions break TRS in theTI and will open a gap in the surface states. Lastly, us-ing a simple model for the electronic dispersion we showthat for thin bilayers (where data by other groups exists)the gap values we find are comparable to values found inother TI/FM systems .More studies with different FM materials and TI thick-nesses and carrier densities, as well as direct energy bandmeasurements, are needed to fully understand the phys-ical mechanism behind the carrier density and thicknessdependent magnetic proximity effect we found in thisstudy. IV. METHODS
We grew our bilayers on polished Al O (0001) sub-strates of a 1 cm area and 0 . cm thickness. The averagesubstrate RMS roughness was < . nm . Before each bi-layer growth we used a mask during the deposition topattern the bilayers into a VDP compatible geometry.After annealing the substrate at 330 C for an hour, wehad used a homemade pulsed laser deposition (PLD) de-vice to grow the bilayer. The base pressure of the depo-sition chamber was ˜ 1.5 × C for both F e Se and B . Sb . T e layers. In the PLD process we used a 355 nm Nd:YAGlaser with a ∼ Hz repetition rate for the F e Se and ∼ Hz for the B . Sb . T e . The pulse energy flux was0 . Jcm and 0 . Jcm respectively. The F e Se target wasa pressed pellet of powders of F e and Se with a ratio of3 : 10. The B . Sb . T e target was a pressed pelletmade of grounded B . Sb . T e single crystals.We characterized the bilayers orientation and phaseusing a Bruker D8 eco X-ray diffractometer, and foundthe characteristic c-axis peaks of the
F e Se and B . Sb . T e phases. We measured thickness and rough-ness using a Veeco Wyko NT1100 profilometer. The char-acteristic bilayer RMS roughness was < nm . To verifythe bilayers stoichiometry and homogeneity we used aQuanta SEM-EDS. We did not find any detectable chem-ical inhomogeneity up to the device resolution. To fur-ther analyze the thickness dependent chemical composi-tion and the quality of the interface we used TOF-SIMS.We discussed the results in the main text.For electrical transport measurements we used an Ox-ford Teslatron and homemade electronics. We madethe contacts from silver paint in a 4-contacts Van derPauw configuration (VDP). The bilayer’s area was ∼ ∗ mm , and the thicknesses were 5 − nm . Thecontacts area was less than 5% of the sample size andreciprocity theorem was valid within < . mA at 1.5K with temperature fluctuations of∆ T ≤ mK . V. DATA AVAILABILITY
The data that support our findings are available fromthe corresponding author upon reasonable request.
VI. ACKNOWLEDGMENTS
We thank B. Shapiro and E. Akkerman for useful dis-cussions. This work was supported by the Israel ScienceFoundation grant no. 320/17. Y. Jarach was supportedby the Israel Ministry of Energy under the program ofBSc to PhD students scholarships in the field of energy.
VII. CONTRIBUTIONS
Y.J. grew samples, performed experiments and anal-ysed data, G.K. helped with the sample growth, Y.J.,N.L. and A.K. wrote the paper.
VIII. COMPETING INTERESTS
The authors declare that there are no competing inter-ests.
Appendix A: The Modified HLN formula for SurfaceStates
For the massive Dirac cone dispersion, the mag-netoconductivity can be fitted by a modified HLNformula with an experimental global pre-factor : σ ( B ) − σ (0) = A (cid:88) i =1 , α i e πh (cid:34) ψ (cid:32)
12 + l B l φi (cid:33) − ln (cid:32) l B l φi (cid:33)(cid:35) (A1)Where A is the experimental prefactor, l B is the mag-netic length, l B = (cid:113) (cid:126) e | B | ∼ √ | B | (cid:104) nm √ T (cid:105) and l − φi = l c l i l c + l i , l c is the surface states’ phase coherence length,and ψ = ∂ x ln (cid:0) Γ ( x ) (cid:1) (the digamma function) . Forcos ( θ ) = ∆ gap E Fermi ≡ (cid:101) ∆2 , l e the mean free path and( a, b ) = (cid:0) cos (cid:0) θ (cid:1) , sin (cid:0) θ (cid:1)(cid:1) , the l i and α i parameters aregiven by : l = l e (cid:0) a + b (cid:1) (cid:104) a b (cid:0) a − b (cid:1) (cid:105) − (A2) α = − a b (cid:2)(cid:0) a + b − a b (cid:1) (cid:0) a + b (cid:1)(cid:3) − (A3) l = l e a (cid:0) a + b − a b (cid:1) (cid:104) b (cid:0) a − b (cid:1) (cid:105) − (A4) α = (cid:0) a + b (cid:1) (cid:0) a − b (cid:1) (cid:104) (cid:0) a + b − a b (cid:1) (cid:105) − (A5)For pure weak antilocalization, namely without magneticinteractions , ( α , α ) = ( − . ,
0) . By fitting themagnetoconductivity data with this formula the normal-ized magnetically induced energy gap, (cid:101) ∆, can be ex-tracted. This formula assumes l e (cid:28) l c and l e (cid:28) l B ,which limits its validity only for B (cid:46) T in topolog-ical insulators where l e ∼ − nm . It also as-sumes that the phase coherence length includes all theinformation about disorder and scattering from magneticimpurities .The full model have four fit parameters- A, (cid:101) ∆, l e and l c . In the low magnetically induced energy gap limit α is smaller than α and the MC is mainly dominatedby the α and l φ terms. In this case the experimentalprefactor almost masks the α dependence on (cid:101) ∆ since theprefactor also depends on sample properties. In addition, l φ depends on parameters (cid:101) ∆, l e and l c . Hence, in the lowgap limit there are two dominant effective fit parameters,A α and l φ , and one cannot get a trustworthy MC fitusing all four parameters. To simplify notations we define Aα ≡ α when we discuss this limit in the article (Eq.1).In this article we expect energy gaps in the low gapregime, so we begin with analysis of this limit (see thediscussion of Eq.1 and its results in the main text). Thisis self-consistent with our results. However, to disentan-gle the gap term (cid:101) ∆ from the mean free path l e we needthe full model (Eq.A1). To do this despite the low gaplimit fitting problems just discussed, we first show thatour results really indicate that such a gap indeed exists.Next, we show that the bilayer fitted l φ values are aboutfive times smaller than the real coherence length l c in bareTIs. This allows us to assume that in the bilayer, the realcoherence length contribution to l φ is small. Therefore,in the bilayers we did not use l c as a fit parameter butsubstitute its values with the values obtained for bare TIfilms. This process is valid unless the bilayer l c is aboutfive times smaller than in the bare TI films. After con-cluding that a gap exists, and substituting l c , we onlyused the full model to separate the contributions of l e and (cid:101) ∆ to l φ and l φ . Appendix B: Calculation of the remnantmagnetization
In a system where only the ordinary Hall effect takesplace the transverse resistance is R xy = R H B z , with R H ≡ / ( nte ); B z the out-of-plane magnetic field; n the 3D carrier density; t the thickness; and e the elec-tron charge . As explained in the main text (see Eq.3in the main text), we can get the total transverse (Hall)resistance of our bilayers by using the Drude model forparallel connection of the FM layer and the TI bulk: R totxy = (cid:18) R totxx R F Mxx (cid:19) R F Mxy + (cid:18) R totxx R bulkxx (cid:19) R bulkxy (B1)where R totxy and R totxx are the values we got from mea-surements of the transverse and parallel resistance of a bilayer, respectively. The TI surface states are negligiblehere due to their high parallel resistance R xx .In a general system with the anomalous Hall effect, thetransverse resistance takes the form : R xy = R H B + αM z t (B2) M z is the out-of-plane magnetization component. In themain text we discuss α and explain that according toanomalous Hall effect theories we can approximate α ∼ Cρ xx with C a constant . Since ρ xx = R xx t , we canwrite α ∼ C ( R xx t ) .In the main text we conclude that the TI layer has thelargest contribution to the AHE of the bilayers. There-fore, we neglect the anomalous term of the FM layer.Now, in Eq.B1 we can replace R F Mxy with R H B z (ordi-nary Hall term), and R bulkxy with Eq.B2. We thereforeget: R totxy = (cid:18) R totxx R F Mxx (cid:19) R F MH B z + (cid:18) R totxx R bulkxx (cid:19) (cid:18) R H B z + ( αM z ) bulk t T I (cid:19) (B3)We are interested in the remnant magnetization, namelythe magnetization at zero external magnetic field. Sub-stituting B = 0 we get: R totxy (0) = (cid:18) R totxx (0) R bulkxx (0) (cid:19) (cid:18) α bulk (0) M bulkz (0) tT I (cid:19) (B4)Since the magnetization term comes from the TI, then α ∼ C (cid:0) R bulkxx t T I (cid:1) so finally we get: R totxy (0) ∼ C (cid:0) R totxx (cid:1) t T I M bulkz (B5)Now we can extract the remnant magnetization: M rem ≡ M z (0) ∼ C − (cid:32) R totxy (0) t T I ( R totxx (0)) (cid:33) (B6) Appendix C: A simple model for the remnantmagnetization
To get some insights into the non-monotonic thicknessdependence of the bilayers remnant magnetization (andas a result, of the energy gap), we use a simple exponen-tial decay model: M ( t ) = M exp (cid:18) − tl int (cid:19) (C1)Here M ( t ) is the sample magnetization and t is the dis-tance from the FM layer. M is the magnetization atthe FM-TI interface. We assume that the FM dictates M such that it is sample independent. The magneticinteraction length l int is: l int = (cid:126) v F πk B T (C2) (a)(b) FIG. 7: (a) The magnetic interaction length l int vs theTI layer thickness in our bilayers. See the text for l int definition. (b) The averaged magnetization M av of ourmodel as defined in the main text vs the TI layerthickness.where v F is the Fermi velocity and T is the temperature.We can calculate l int under the parabolic band approx-imation using Eq.C2 and the TI bulk carrier densities(Fig.4 in the main text). Fig. 7a shows l int of the bilay-ers vs the TI layer thickness. Up to our thickest TI layer(170nm thick) the magnetic interaction length remainsof the same order as the TI thickness. This explains howthe gap is finite even in such thick samples.As Eq.C1 presents, the magnetization induced into theTI decays with the distance from the FM-TI interface.The remnant magnetization we got from the anomalousHall effect measurements (Fig.5b in the main text) isthough an averaged value over the TI thickness. We de- fine the averaged magnetization in our model as: M av ( t T I ) = (cid:82) t TI M ( z ) dz (cid:82) t dz = M l int t T I (cid:18) − exp (cid:18) − t T I l int (cid:19)(cid:19) (C3)Fig. 7b shows M av (cid:0) t T I (cid:1) for our bilayers againstthe TI layer thickness in units of M . We get that M av (cid:0) t T I = 170 nm (cid:1) > M av (cid:0) t T I = 73 nm (cid:1) . As can beseen from Eq.C3 and Eq.C2, this inequality and the non-monotonic behavior of the magnetization (Fig.5b in themain text and Fig.7b here) result from the dependence of l int on the BL thickness, which in turn results from thecarrier density dependence on the BL thickness (Fig.4 inthe main text). Appendix D: Role of Iron doping in the TI layer
Tof-SIMS measurements put an upper limit of about1% on Fe dopant concentration in the TI layers. Thisis far below the ∼
6% Fe doping needed to open a gapin the surface states dispersion . In addition, our out-of-plane coercive fields are almost an order of magnitudelarger than in Fe-doped TIs . Last, Fe dopant in BSTScompounds are known as electron donors . Since oursamples are p-type, Fe doping reduces the charge den-sity. If the Fe dopants are dominant in the bilayer mag-netic properties, the magnetization should be maximalwhere the charge density is minimal (maximal Fe den-sity). This is opposite to our results where the chargedensity, magnetization, and magnetization induced gapshare the same minimum. We therefore conclude that Fedopants are not dominant in the bilayers magnetization.We claim that the dominant mechanism is interactionsbetween the TI bulk, the upper TI surface, and the FMlayer, which the TI conductive bulk states mediate. Appendix E: Role of strain and grain size
In figure 8a we show X-ray diffraction data of a bare102nm TI film (black), and a bilayer with 12nm FM and102nm TI (red). Clearly, the films are C-axis oriented.The TI peaks intensities are similar, and the small differ-ence is due to a different sample alignment and samplearea.Figure 8b shows a zoom on in the TI [0,0,2] peak.The bilayer peak is shifted by a ∆2 θ = 0 . o with re-spect to the bare TI peak. This means a tensile in-plainstrain in the bilayer which shortens the TI C-axis latticevector . It is known from the literature that a tensilein-plane strain enhances the surface states properties ,and therefore this strain cannot be the cause for the re-duced surface states coherence length we found in thisarticle.The bilayer peak is also broader, which can result froma smaller TI grain size. Using the Scherrer equation (a)(b) FIG. 8: (a) X-ray diffraction data in a θ − θ method ofa 102nm thick bare TI film (black) and a bilayer with a12nm thick FM layer and a 102nm thick TI layer(red).The y axis is the counts per second root. We subtractedthe background from both samples. The X-raywavelength is 1.54[A]. (b) A zoom in on the TI [0,0,2]peak.with a shape factor K of 0.9 we get a grain size of 42 nm for the TI in the bilayer, and 46 nm in the bare TI. Thisclearly shows that grain size is also not the source forreduced coherence length in the bilayers.In addition, in the bare TI the surface states coher-ence length is about 500nm, which we extracted fromits magneto conductivity as explained in the main arti-cle. This is much higher than the grain size, which isreasonable since the non-gapped surface states are pro-tected from back-scattering by the grain boundaries . Inthe bilayers, the surfaces states are gapped and are notprotected from back-scattering. This may explain whyin the bilayer the coherence length is about 85 nm (seemain article), similar to the grain size. Note that ourcalculated grain size is a lower boundary on real grainsize, which may be somewhat larger since the Scherrerequation does not take into account peak broadening by randomly oriented strain . Appendix F: The dominance of the exposed surfacein our magneto-conductivity data
In the main text we concluded that our measured lowfield magneto-conductivity (MC) is either a property ofthe exposed surface only, or that both surfaces are cou-pled and behaves as one effective transport channel suchthat any magnetic effect in one surface affects both. Wejustify this by the ability to fit the MC of bare TIs andbilayers using a one-surface modified HLN model , aswas reported also by many previous publications .From this we concluded that the effective coherencelength is reduced in both surfaces by the magnetic layer,so there is a gap in both. Therefore, magnetic scatter-ing on the interface cannot explain our effect, becausethere are no magnetic particles at the exposed surface(see appendix D why Fe dopant are not important in oursystem).Here we wish to give another justification for our con-clusion that our measured reduced MC in the bilayers is aproperty of the exposed surface. In the end, we will onlyuse the assumption that in bare TIs, the contributionof the exposed surface states at least equals that of theinterface surface. This is a reasonable assumption sincethe interface suffers more from strain, defects, impuritiesetc.Let us first assume the opposite assumption- the sur-face states at the exposed surface of the TI in the bilayersremains intact, and the change occurs only at the inter-face surface. We are also assuming now that the surfacestates at the two surfaces are not coupled and behaves astwo separate transport channels. We can therefore treatthe two surfaces as two transport channels connected inparallel. In this case the total conductance is the sumof the conductance of each surface, and so is the totalmagneto conductivity (MC):∆ σ tot ( B ) = σ tot ( B ) − σ tot (0) = σ ( B )+ σ ( B ) − ( σ (0) + σ σ ( B )+∆ σ ( B )(F1)The exposed surface is probably of a better quality thanthe interface surface, since the interface suffers more fromimpurities, strain, defects etc. Therefore, in a bare TIfilm we can assume that the contribution of the interfacesurface to the MC is less or equal to that of the exposedsurface.In our bilayers the total MC is about fifth of its valuein the bare TI films. Remember we are assuming that themagnetic layer reduces the MC only at the TI interfacesurface in the bilayers. Since the total MC is the sumof the MC of both surfaces, and the contribution of theinterface surface is less or equal to that of the exposedsurface, than the total MC in the bilayers cannot be lessthan half of the MC in bare TI. This is so unless either1the MC of the interface surface changes sign or one ofour assumptions is false.If the MC of the interface surface changes sign, itmeans that the surface states magnetically induced en-ergy gap is greater than the Fermi energy . For exam-ple, in our bilayer with 17nm-thick TI layer this means agap larger than 300 meV (see the Fermi energy analysisin the main text). This value is more than three timeslarger than any previously reported or predicted mag-netically induced gap at TI-FM interfaces, so we assumethis scenario as highly improbable. Therefore, we mustconclude that one of our assumptions is false.We used three assumptions: 1) In the bare TIs thecontribution of the exposed surface states at least equalsthat of the interface surface. 2) The two surfaces arenot coupled. 3) The FM layer affects only the interfacesurface. As we already explained, we see the first as-sumption as highly probable due the inferior quality ofthe interface surface .If the second assumption is false, and the two surfacesbehaves as a one coupled transport channel, then we can-not differ between the surface states at the interface andthe exposed surfaces. Therefore, the magnetically in-duced gap which our MC data reflects is a property of all the surface states in the bilayers, including the exposedsurface states. But this is exactly our claim- the surfacestates at the exposed surface of our bilayers are gapped.Therefore, we have to conclude that the third assump-tion is false, and the FM does affect the exposed sur-face and reduces its MC, namely reduces the effectivecoherence length of its surface states (see main text inthe discussion of figure 2). As we thoroughly explain inthe main text, this can be done either by reducing thecoherence length or the surface states, or by opening amagnetically induced gap.In the main text we explained that to assume thatthere is no gap at the exposed surface, we have to as-sume that the coherence length of its surface states isless than fifth of its value in the bare TIs. In appendixE we showed that the average grain size of the TI in thebilayers is almost the same as in the bare TIs (with 10%difference), and the additional strain leads only to a shiftof 0 . o in the [0,0,2] peak of the TI. We therefore claimthat such drastic reduction of the coherence length lessprobable than the opening of a magnetically induced en-ergy gap at the exposed surface. Such a gap can occurdue to magnetic coupling to the FM layer, and not due tomagnetic scattering since there are no magnetic particlesat the exposed surface. B. A. 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