Evidence of orbital ferromagnetism in twisted bilayer graphene aligned to hexagonal boron nitride
Aaron L. Sharpe, Eli J. Fox, Arthur W. Barnard, Joe Finney, Kenji Watanabe, Takashi Taniguchi, Marc A. Kastner, David Goldhaber-Gordon
EEvidence of orbital ferromagnetism in twistedbilayer graphene aligned to hexagonal boronnitride
Aaron L. Sharpe, ∗ , † , ‡ , @ Eli J. Fox, ¶ , ‡ Arthur W. Barnard, ¶ , ‡ , § Joe Finney, ¶ , ‡ KenjiWatanabe, (cid:107)
Takashi Taniguchi, ⊥ M. A. Kastner, ¶ , ‡ , and DavidGoldhaber-Gordon ∗ , ¶ , ‡ † Department of Applied Physics, Stanford University, 348 Via Pueblo Mall, Stanford, CA94305, USA ‡ Stanford Institute for Materials and Energy Science, SLAC National AcceleratorLaboratory, 2575 Sand Hill Road, Menlo Park, California 94025, USA ¶ Department of Physics, Stanford University, 382 Via Pueblo Mall, Stanford, CA 94305,USA § Department of Physics and Department of Materials Science and Engineering, Universityof Washington, 302 Roberts Hall, Seattle, Washington 98195, USA (cid:107)
Research Center for Functional Materials, National Institute for Materials Science, 1-1Namiki, Tsukuba 305-0044, Japan ⊥ International Center for Materials Nanoarchitectonics, National Institute for MaterialsScience, 1-1 Namiki, Tsukuba 305-0044, Japan
Department of Physics, Massachusetts Institute of Technology, 77 Massachusetts Avenue,Cambridge, MA 02139, USA @ Present address: Quantum and Electronic Materials Department, Sandia NationalLaboratories, Livermore CA 94550, USA
E-mail: [email protected]; [email protected] a r X i v : . [ c ond - m a t . m e s - h a ll ] F e b bstract We have previously reported ferromagnetism evinced by a large hysteretic anoma-lous Hall effect in twisted bilayer graphene (tBLG). Subsequent measurements of aquantized Hall resistance and small longitudinal resistance confirmed that this mag-netic state is a Chern insulator. Here we report that, when tilting the sample in anexternal magnetic field, the ferromagnetism is highly anisotropic. Because spin-orbitcoupling is negligible in graphene such anisotropy is unlikely to come from spin, butrather favors theories in which the ferromagnetism is orbital. We know of no other casein which ferromagnetism has a purely orbital origin. For an applied in-plane field largerthan , the out-of-plane magnetization is destroyed, suggesting a transition to a newphase. The possibility of flat bands in Van der Waals heterostructures, beginning with magic-angle twisted bilayer graphene (tBLG), has drawn much experimental and theoreticalattention. In such weakly dispersing bands the kinetic energy is reduced, allowing theelectron-electron interactions to favor correlated states. Evidence of correlated behavior hasbeen observed, for example, as the appearance of resistive states at fractional filling of bandsin tBLG with an interlayer twist of ≈ . ◦ , ABC-trilayer graphene/hBN moiré, twistedbi-bilayer, monolayer-bilayer graphene, and twisted
WSe . One recently observedconsequence of these correlations is ferromagnetism in tBLG in a narrow range of carrier den-sities around / filling of the flat conduction band, where full filling corresponds to fourelectrons per moiré unit cell accounting for spin and valley degeneracies. The magnetismwas initially revealed by a hysteretic anomalous Hall effect as large as . . Initial evi-dence of chiral edge states from nonlocal transport indicated that the magnetic state couldbe an incipient Chern insulator. Subsequent measurements of similarly-configured tBLGrevealed precise quantization of the Hall resistivity coincident with longitudinal resistivityas small as , conclusively demonstrating it is possible to achieve a Chern insulator withChern number C = 1 at 3/4 filling. At optimal doping, this Chern insulator has a coer-cive field of tens of millitesla and survives up to . Evidence of Chern insulators has2lso been predicted and observed in ABC-trilayer graphene/hBN moiré superlattice andmonolayer-bilayer graphene heterostructures.
A Chern insulator requires nontrivial band topology and a gap between bands of differentChern numbers. tBLG samples are generally encapsulated in hexagonal boron nitride (hBN)flakes to protect from disorder and serve as dielectrics for electrostatic gating. A gap at theDirac point that would favor forming a Chern insulator could be opened by aligning the tBLGcrystal axis to that of one of the cladding hBN layers, breaking C , the in-plane two-foldrotation symmetry, that could otherwise protect the Dirac crossings of bands (in con-junction with time reversal symmetry T ). Though self-consistent Hartree-Fock calculationsshow that in-plane two-fold rotation symmetry may be broken spontaneously, both tBLGsamples that exhibit ferromagnetism appear to have hBN aligned to the tBLG, whereassuch alignment has typically been intentionally avoided when fabricating other samples.The precise nature of the ferromagnetic ground state is an open question. In ferromag-netic materials, exchange interactions break time reversal symmetry, favoring long-rangeorder of electron spins. Though the motion of electrons can generate an orbital magneticdipole moment, we are unaware of any magnetic material where the magnetism is dominatedby the orbital magnetic moment independent of spin. However, for tBLG this is preciselythe prediction of Refs. 20,22–25,28–31. Recent measurements using a superconducting quan-tum interference device found the magnetization to be approximately 2-4 Bohr magnetonsper moiré unit cell. Those measurements were performed near the / state, which corre-sponds to a single conduction-band hole per moiré unit cell. The measured magnetizationsignificantly exceeds the expected 1 Bohr magneton per moiré unit cell for a single spin,suggesting that the magnetism has a strong orbital component. Here, we demonstrate bytransport measurements in magnetic fields at angles ranging from normal to the plane ofthe sample to completely in the plane of the sample that the magnetism in tBLG is highlyanisotropic. Given that spin-orbit coupling in graphene is weak, this observation impliesthat the magnetism is dominated by the orbital magnetic moment (expected to be highly3nisotropic) rather than the isotropic spin.We used the “tear-and-stack” dry transfer method and standard lithography tech-niques to fabricate a tBLG Hall bar device which was previously characterized in Ref. 1.The device was fabricated with both a silicon back gate and a Ti/Au top gate, allowingfor independent control of the charge carrier density n and perpendicular displacement field D (see Methods). We measured the longitudinal resistance R xx and Hall resistance R yx using standard lock-in techniques with a 5-nA root mean square (RMS) AC bias current.The angle of the top graphene sheet relative to the top cladding hBN was measured byoptical microscopy to be . ◦ ± . ◦ clockwise (see the Supplemental Information). Featuresin electron transport apparently due to this alignment correspond to a hBN twist angle of . ◦ ± . ◦ relative to the nearer graphene sheet. A rough measure of the twist angle be-tween the sheets of graphene of . ◦ ± . ◦ can be obtained from the superlattice density n s which corresponds to four electrons (or holes) per superlattice unit cell. n s is determinedfrom the distance in gate voltage between the charge neutrality point (CNP) and the peak inresistance corresponding to n s , multiplied by dn/dV tg extracted for top gate voltages V tg nearthe CNP. Typically, a more accurate measure of the twist angle can be obtained by fittingquantum oscillations. However, for this sample these features are not very sharp and yielda twist angle between the graphene sheets of . ◦ ± . ◦ . This twist angle correspondsto ± n s = 3 . × cm − . This measure of the twist angle between the two graphenesheets is consistent with such a measurement performed on an atomic force microscopy im-age, which shows that the bottom graphene layer is rotated clockwise relative to the topgraphene layer (see the Supplemental Image). This sample exhibits a magnetic state near / filling. Though the low-field ground state with strong and hysteretic anomalous Halleffect could in principle be metallic, we have identified it as an incipient Chern insulator. By mounting the sample on a two-axis piezoelectric rotating stage equipped with resis-tive positional readout, we can control the orientation of the device relative to the appliedmagnetic field. One axis of the stage controls the tilt angle θ of the sample plane relative to4he field (see Fig. 1a inset). The second axis allows for rotation of the sample stage aboutits normal, which controls the orientation of the in-plane component of the field relative tothe sample (see the Supplemental Information for a more complete discussion). All mea-surements were performed with the same orientation of the in-plane field component unlessotherwise noted. To calibrate the tilt angle θ , we tuned the device to a regime where noanomalous Hall signal is present ( n = 0 . n s ) and used the Hall resistance R yx as a measureof the out-of-plane component of the field. We paid particular attention to precisely deter-mining when the field was parallel to the sample plane (see the Supplemental Information).We define the tilt angle θ such that ◦ corresponds to a fully out-of-plane field while ◦ corresponds to a fully in-plane field as seen by the device. The in-plane component andout-of-plane component of the field are then B (cid:107) = B cos( θ ) and B ⊥ = B sin( θ ) , respectively,where B is the magnitude of the applied field.We have measured hysteresis loops of R yx for different tilt angles of the device, with thedevice tuned to be ferromagnetic ( n/n s = 0 . and D/(cid:15) = − .
30 V / nm ). The coercivefield, identified by the field at which the largest step in the Hall signal occurs, increases as thesample is rotated so that the field is in the plane of the sample (Fig. 1a). When the hysteresisloops at the various angles are plotted as a function of the out-of-plane component of themagnetic field, B ⊥ , the largest step in R yx consistently occurs at ± for measuredtilt angles down to about ◦ (Fig. 1b), indicating that the magnetization is indeed highlyanisotropic and likely dominated by the orbital magnetic moment.The increased magnitude of the in-plane field as we lower the tilt angle does not stronglyaffect the hysteresis loops down to a loop performed at a tilt angle of . ◦ ± . ◦ (plottedvs. B ⊥ in Fig. 1b and vs. B in Fig. 2a): at this angle, a transition in R yx is still seenwhen out of plane field roughly matches the coercive field measured at larger tilt angles(marked with dashed vertical lines). At this tilt angle, the in-plane field reaches a maximumof B (cid:107) = 0 .
69 T . However, as the sample is tilted closer to perfectly in-plane ( . ◦ ± . ◦ in Fig. 2b and beyond), we no longer see a dominant transition in R yx at the same value of5ut-of-plane field, and the measured magnitude of hysteresis in Hall resistance is significantlyreduced. As the tilt angle is further reduced such that the out-of-plane field just reaches thecoercive field (Fig. 2c) or does not reach the coercive field (Fig. 2d), any semblance of thehysteresis loops seen at larger tilt angles is lost.To explore whether the effects seen when the behavior in nearly in-plane field resultsfrom a small residual out-of-plane field, we compare hysteresis loops performed at smallangles of similar magnitude but opposite sign (Fig. 2d). The two R yx vs. B curves are verysimilar despite magnetic field angle deviating from in-plane in opposite directions, so theout-of-plane components are opposite for the two curves (see Fig. S3d in the SupplementalInformation for corresponding R xx curves). It is unlikely that this behavior results from asignificant B ⊥ : at , the maximum B applied in these loops, a tilt angle of more than
700 mdeg would be needed for B ⊥ to exceed the out of plane coercive field of
119 mT . Suchmisalignment is well outside our experimental error for the two traces in Fig. 2D (the traceswere performed at +0 . ◦ ± . ◦ and − . ◦ ± . ◦ ) so that, unlike in the other tracesin Fig. 2, the out-of-plane coercive field is not reached (see the Supplemental Informationfor characterization of the rotating probe). Furthermore, were significant misalignment thesource of the residual Hall signal, we would expect R yx to be antisymmetric in field and thedevice would likely recover some portion of its initial magnetization upon cycling the field,as is the case for larger tilt angles. For . ◦ and . ◦ the in-plane field has a significanteffect: the shape of the hysteresis loop is quite different and the difference in R yx betweenthe upward and downward sweeping traces remains substantial but approximately half thatseen in a perpendicular field. The two traces in Fig. 2d happened to be acquired at differenttemperatures – in perpendicular field we have found that the coercive field is ∼ lessfor the higher temperature while the size of the Hall signal is substantially unchanged. Acomparison performed at a constant temperature with the sample at a different in-planeangle yields similar results (Fig. S7 of the Supplemental Information).Up to now, hysteresis loops were acquired sequentially without explicitly repolarizing6he sample between traces. To study the response of the orbitally polarized state to an in-plane field, we now start with the sample initially magnetized by an out-of-plane magneticfield. This training field is returned to zero, and the sample is then rotated to as close toin-plane as possible in zero magnetic field (for this measurement, the resultant tilt angle is − ±
21 mdeg ). Once rotated, a magnetic field is applied at this very small angle to thesample (red trace in Fig. 3). As the in-plane field is increased, | R yx | initially rises, thenbegins to decrease at B (cid:107) = 2 . (red trace in Fig. 3). As B (cid:107) is increased through , | R yx | rapidly falls, and we observe no hysteresis or steps in the Hall resistance for fields of largermagnitude, indicating an apparent transition from the Chern insulating state to a differentstate: above , both R xx and R yx show repeatable oscillations of order in size thatappear to depend only on the magnitude of B (cid:107) (see the Supplemental Information). Theaccessible range of field is insufficient to say whether or not these oscillations are periodic,and if so whether the oscillations are periodic in B (cid:107) or /B (cid:107) . The high field state may or maynot be polarized in spin and/or orbit. If it does have an orbital polarization, it is no longerset by the out-of-plane field component. The high-in-plane-field state also does not show astrong anomalous Hall signal. Were the Hall signal arising primarily from coupling to themagnetization, one would expect R yx ( − M ) = − R yx ( M ) , where M is the magnetization ofthe sample. If high field were indeed fixing the orbital polarization, its direction and thus theHall component of the signal should reverse with field direction. Instead R yx measured in thisregime depends almost entirely on | B | (Supplemental Information), so either this high-in-plane-field state lacks orbital polarization or the large in-plane field changes the topologicalcharacter of bands so there is no net Chern number. Below , the orbital polarization isbeing modified by in-plane field somehow, not by the small out of plane component. Upondecreasing B (cid:107) , R yx shows evidence of magnetism but never recovers its initial value whichnominally corresponds to a maximally polarized state (blue traces in Fig. 3). This behaviorbelow may result from a repeatable pattern of out-of-plane orbital domains that is setby an in-plane field. 7hough this device does not exhibit quantized Hall resistance or zero longitudinal resis-tance, as previously noted it does appear to exhibit incipient Chern insulating behavior. Thesimplest model for a Chern insulator at / filling would have complete spin- and valley- po-larization. Other possible Chern insulator states have been considered in our previouspublication. As spin-orbit coupling is extremely weak in graphene, there should not be significantanisotropy in the direction spins prefer. In fact, there might be no relation between the di-rection of spin polarization and valley polarization. Starting from the out-of-plane orbitallypolarized state with no external magnetic field, and then applying in-plane field, the initialrise in | R yx | (red trace in Fig. 3, up to 2T), may indicate that the spin is oriented by theexternal field, widening the gap to charge-carrying spin excitations which were causing de-parture from quantized transport. As B (cid:107) is increased beyond , we observe that | R yx | rapidly falls and is no longer hysteretic, perhaps indicating a field induced transition fromthe Chern insulator to some other state. If the low-field Chern insulator is valley-polarizedbut spin-unpolarized, polarizing spin by applying a large in-plane field could suppress theHall signal by mixing with higher-order bands or by favoring spin instead of valley polar-ization. Another possible mechanism for the observed transition is that, because of thefinite thickness of tBLG, an in-plane field directly couples to the orbital moments and asufficiently large in-plane field could then drive the sample into a valley unpolarized state.Regardless of the dominant mechanism by which in-plane field couples to the device, it seemsthat either the bands are losing their topological character by the mixing in of higher bands, or the in-plane field is shifting population among a fixed set of flat bands. Thus far, in tBLGaligned with hBN no evidence of magnetism has been observed at n/n s = 1 / , whichshould nominally be similar to n/n s = 3 / . Therefore there may be a competing state whichdoes not have a net Chern number and would not exhibit a large Hall signal. Calculationsshow that both a gapless C T -symmetric nematic state and a gapped C T -symmetric stripestate are nearby in energy to the spin- and valley-polarized Chern insulator with C = 1 . We have observed that the magnetization of tBLG is sensitive primarily to the out-of-plane component of the field, requiring a threshold coercive field to flip the magnetization.A single value of B ⊥ required to switch the measured anomalous Hall resistance is consistentwith uniaxial magnetization. It is unlikely that this uniaxial behavior is related to theelectron spin because of the extremely low spin-orbit coupling in graphene. Rather, it islikely that the / state is an orbital ferromagnet. The confinement of circulating electroncurrents to the plane would provide the high degree of anisotropy observed.9 eferences (1) Sharpe, A. L.; Fox, E. J.; Barnard, A. W.; Finney, J.; Watanabe, K.; Taniguchi, T.;Kastner, M. A.; Goldhaber-Gordon, D. Emergent ferromagnetism near three-quartersfilling in twisted bilayer graphene. Science , , 605–608(2) Serlin, M.; Tschirhart, C. L.; Polshyn, H.; Zhang, Y.; Zhu, J.; Watanabe, K.;Taniguchi, T.; Balents, L.; Young, A. F. 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Data for: Evidence of orbital ferromag-netism in twisted bilayer graphene aligned to hexagonal Boron Nitride, Version 1.0,Stanford Digital Repository (2020); https://doi.org/10.25740/dq349wz5558(44) Wang, L.; Meric, I.; Huang, P. Y.; Gao, Q.; Gao, Y.; Tran, H.; Taniguchi, T.; Watan-abe, K.; Campos, L. M.; Muller, D. A.; Guo, J.; Kim, P.; Hone, J.; Shepard, K. L.;Dean, C. R. One-Dimensional Electrical Contact to a Two-Dimensional Material. Sci-ence , , 614–617 15 igures (a) (b) B θ BB Figure 1:
Angular dependence of magnetic hysteresis loops.
Magnetic field depen-dence of the Hall resistance R yx with n/n s = 0 . and D/(cid:15) = − .
30 V / nm at
29 mK as afunction of the angle of the device relative to the field direction; ◦ corresponds to field in theplane of the sample. The hysteresis loops are plotted as a function of (a) the applied field B and (b) the component of the field perpendicular to the plane of the sample B ⊥ . The solidand dashed lines correspond to sweeping the magnetic field B up and down, respectively.Inset: schematic diagram displaying the components of the magnetic field B at the sample(shown in purple) for a given tilt angle θ . 16 a)(b) (c)(d) Figure 2:
Hysteresis loops for small tilt angles.
Angular dependence of R yx vs B with n/n s = 0 . and D/(cid:15) = −
30 V / nm for angles of the field relative to the plane of thesample: (a) . ◦ ± . ◦ , (b) . ◦ ± . ◦ , (c) . ◦ ± . ◦ , (d) +0 . ◦ ± . ◦ , and − . ◦ ± . ◦ . Vertical dashed black lines indicate where the out-of-plane componentof the field equals the coercive field ±
119 mT . The out-of-plane component of the field israised beyond the coercive field in panels (a),(b), just reaches the coercive field in (c), anddoes not reach it in (d). All traces were taken at
27 mK except for the trace with tilt angle +0 . ◦ ± . ◦ , which was taken at .
35 K .17igure 3:
Erasing the initial magnetic state.
In-plane hysteresis loops of R yx with n/n s = 0 . and D/(cid:15) = −
30 V / nm at
26 mK . The sample is initially polarized withan out-of-plane field. The sample is then rotated to − ±
21 mdeg in zero magnetic field.The in-plane magnetic field B (cid:107) is then increased from zero (red trace) before completing ahysteresis loop (blue solid and dashed traces).18 cknowledgement We acknowledge fruitful discussions with Greg Fuchs, Michael Zaletel, Allan MacDonald, T.Senthil, Ashvin Vishwanath, Eslam Khalaf, Oskar Vafek, Steve Kivelson, Yoni Schattner,Andrea Young, Matt Yankowitz, Feng Wang, and Guorui Chen. Some of these began atthe Aspen Center for Physics, which is supported by National Science Foundation grantPHY-1607611. Yuan Cao and Pablo Jarillo-Herrero generously taught us about their fabprocess and their insights into tBLG. Hava Schwartz and Sungyeon Yang helped with devicefabrication, and they and Anthony Chen performed preliminary measurements as part of aproject-based lab class at Stanford.
Funding:
Device fabrication, measurements, and analysis were supported by the U.S.Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engi-neering Division, under Contract DE-AC02-76SF00515. Infrastructure and cryostat supportwere funded in part by the Gordon and Betty Moore Foundation through Grant GBMF3429.Part of this work was performed at the Stanford Nano Shared Facilities (SNSF), supported bythe National Science Foundation under award ECCS-1542152. A. S. acknowledges supportfrom an ARCS Foundation Fellowship, a Ford Foundation Predoctoral Fellowship, and a Na-tional Science Foundation Graduate Research Fellowship. E. F. acknowledges support froman ARCS Foundation Fellowship. K.W. and T.T. acknowledge support from the Elemen-tal Strategy Initiative conducted by the MEXT, Japan, Grant Number JPMXP0112101001,JSPS KAKENHI Grant Number JP20H00354 and the CREST (JPMJCR15F3), JST.
Au-thor contributions:
A.S. and J.F. fabricated devices. K.W. and T.T. provided the hBNcrystals used for fabrication. A.S. and E.F. performed transport measurements. A.S., E.F.,A.B., and J.F. analyzed the data. M.K. and D.G.-G. supervised the experiments and analy-sis. The manuscript was prepared by A.S. and E.F. with input from all authors.
Competinginterests:
The authors have filed a patent disclosure on using the low-power switching ofthe magnetic state for low-temperature memory applications. M.K. was a member of theScience Advisory Board of the Gordon and Betty Moore Foundation until December 201919nd remains a chair of the DOE Basic Energy Science Advisory Committee. Both the MooreFoundation and Basic Energy Sciences provided funding for this work.
Data and materialsavailability:
The data from this study are available at the Stanford Digital Repository. Supporting Information Available
Methods
Our device was previously characterized in a paper by the authors. The device consistsof twisted bilayer graphene (tBLG) encapsulated in two hexagonal boron nitride (hBN)cladding layers, each ∼
50 nm thick. The heterostructure was assembled using a “tear-and-stack” technique.
A poly(bisphenol A carbonate) film/gel (Gel-Pak DGL-17-X8) stampon a glass slide heated to ◦ C was used to pick up the top hBN flake. To stack two layersof graphene with a well defined twist angle, we used the Van der Waals attraction betweenhBN and monolayer graphene to tear off and pick up a portion of monolayer graphene froma larger flake. The remaining portion of monolayer graphene was then controllably rotatedand picked up. The completed stack was transferred onto a 5x5 mm chip of - nm -thickSiO atop degenerately doped Si substrate. The doped Si is used as a back gate.The completed heterostructure was then fabricated into a measurable device using stan-dard nanopatterning techniques. Patterned Ti/Au was deposited to serve as a top gate, andwas then used as a hard mask for a CHF /O (50/5 sccm) etch to define a Hall bar geometry.During this etch, regions of the heterostructure were protected by resist extending outwardfrom the top gate near each of the leads of the Hall bar to provide space for making Cr/Auedge contacts without risk of shorting to the top gate. The sample temperature was keptbelow ◦ C throughout all processing in an effort to prevent relaxation of the twist angleof the tBLG.The Au top gate and Si back gate can be used to tune both the carrier density in thetBLG and the displacement field applied to the device. The gates can be modeled as parallel20late capacitors such that the density under the top gated region is given by n = C BG ( V BG − V BG ) + C TG ( V TG − V TG ) , where BG (TG) indicates the back (top) gate, C is the capacitance per unit area determinedfrom low-density Hall slope measurements, and V BG ( V TG ) is the charge neutrality point ofthe back (top) gated region at zero displacement field. We define the applied displacementfield as D = ( D BG − D TG ) / , where the displacement field within a given dielectric D i = (cid:15) i ( V i − V i ) /d i , (cid:15) i is the relativedielectric constant, d i is the thickness of each dielectric, and i = { BG , TG } . The relativedielectric constant of hBN is assumed to be (cid:15) TG = 3 . As described previously, we do not seeany clear features to ascribe to a true zero in displacement field, so we assume that that thedisplacement field D ≈ when both gates are tuned to . This assumption is reasonablegiven that the expected displacement field due to differences in the work functions betweenthe top and back gate is small ( − .
01 V / nm ). Any nonzero displacement field when the gatevoltages are zero should then simply yield a constant offset to our reported values.We mounted the sample in a Kyocera custom 32 contact ceramic leadless chip carrier(drawing PB-44567-Mod with no nickel sticking layer under gold, to reduce magnetic effects).The device was measured in a dilution refrigerator capable of reaching a base temperature of
30 mK . The measurement lines are equipped with electronic filtering at the mixing chamberstage to obtain a low electron temperature in the device and reduce high-frequency noise.There are two stages of filtering. The wires are passed through a cured mixture of epoxyand bronze powder to filter GHz frequencies, then low-pass RC filters mounted on sapphireplates filter MHz frequencies. The sample was mounted in an attocube atto3DR two-axispiezoelectric rotating stage equipped with resistive positional readout.Stanford Research Systems SR830 lock-in amplifiers with NF Corporation LI-75A voltage21reamplfiers were used to perform four-terminal resistance measurements. A biasresistor was used to apply an AC bias current of
RMS at a frequency of . .Keithley 2400 SourceMeters were used to apply voltages to the gates. All standard Hallconfiguration measurements were performed using the same voltage probes. One voltagecontact behaved inconsistently and was not used in any of the reported measurements. Ascribing relative twist angles
Analysis of an optical microscopy image of the completed heterostructure (Fig. S1 of Ref.1) yields that the top graphene layer is rotated clockwise from the hBN by . ◦ ± . ◦ .Similar analysis of an atomic force microscopy image (Fig. S1b) shows that the bottomgraphene layer is rotated clockwise by . ◦ ± . ◦ relative to the top graphene layer. Edgescorresponding to specific layers of the tBLG are identified by comparing the atomic forcemicroscopy image to an optical microscopy image of the graphene flake before it was torn(Fig. S1a). Rotator calibration
Summary:
To control the orientation of the device relative to the field from the solenoid, the samplechip carrier was mounted in an attocube atto3DR two-axis piezoelectric rotator. 28 of the 32contacts on the Kyocera chip carrier were available for measurement given space constraintsin the probe. The atto3DR combines two rotators: one to control the tilt angle θ of thesample relative to the field to tune the relative magnitudes of the out-of-plane and in-planecomponents of the field (see Fig. 1a inset of the main text) and one to rotate about the normalof the stage to control the direction of the in-plane component of the field relative to thesample. Rotation is eucentric: the sample location in the solenoid is fixed as its orientationis changed. The rotator is equipped with a resistive readout for each axis for determining22 a) (b) Figure S1:
Graphene flake prior to and after tearing. (a) Optical image of the grapheneflake used in the complete device prior to tearing. The dashed black line indicates theapproximate axis the flake was torn about. Blue lines indicate edges of the portion of theflake that will become the top layer of the tBLG. The green line indicates an edge on theportion of the flake that will become the bottom layer of the tBLG. The second graphene flaketo the right of the annotated flake was not used to fabricate a device. The grey lines abovethe graphene flakes are tape residue. (b) Atomic force microscopy image of the completedheterostructure prior to device fabrication. Specific edges are indicated in the same manneras they were in (a). The green line has been manually rotated by . ◦ clockwise relative tothe rightmost blue line. The device was fabricated in a region without any bubbles.23he angular position of the stage. We calibrated this readout using the Hall resistance R yx of the device in a regime where the (ordinary) Hall effect is a measure of the out-of-planefield. All the data we present are accompanied by estimates of the angular position and errorbounds based on this calibration. A precise calibration is particularly important when thefield is nearly in-plane to avoid a significant unintended out-of-plane component when themagnitude of the total field is large. For all our nominally in-plane field measurements, theout-of-plane component remained smaller than the out-of-plane coercivity. Details:
Since samples may not sit perfectly flat in the ceramic chip carriers we use to mount them,and the chip carriers may not always sit in exactly the same position in the pogo pin socketof the rotator stage, with every cooldown it is necessary to calibrate the angular position ofa sample as a function of the resistive position readout for accurate positioning. To calibratethe tilt angle θ for this device, we tuned the device to a carrier density n = 0 . n s whereno anomalous Hall effect was present and a linear ordinary Hall effect allowed us to extractthe out-of-plane component of the field. The resistive position readout was then calibratedby rotating the sample in a fixed field while measuring the Hall resistance and the resistivereadout. The measured Hall resistance may have a small offset at zero applied field due tomixing in of the longitudinal resistivity. Therefore, to ensure an accurate identification ofthe angle corresponding to an in-plane field, we rotated the sample in both a +6 T field anda − field. The in-plane-field position was then determined as the angle corresponding tothe value of the resistive readout where the measured R yx from the two angular sweeps wereequal.Uncertainty in the true angular position arises from several sources. Noise in the mea-surement of the resistive position readout can be reduced, but not completely removed, bymeasuring with a lock-in amplifier. This noise, along with uncertainty in the relationshipbetween the Hall signal used for calibration and the true out-of-plane field component, con-24ributes to error in the calibration. The resistive readout is also hysteretic with rotationbecause of backlash in the piezoelectric rotators. This backlash can be controlled for byconsistently rotating to a final position from the same direction or generating a separatecalibration for each direction of rotation. Additionally, the sample will have some smalltilt relative to the rotator stage. Thus, except when the normal to the rotator stage isaligned with the solenoid axis, rotation of the in-plane angle will change the tilt angle ofthe sample relative to the field. One solution to this problem is simply to calibrate θ foreach in-plane angle used. Finally, a large magnetic field can apply a torque to the sampleor rotator. We have observed a resulting field-induced rotation in a separate Bernal bilayergraphene sample under nearly in-plane field: compared to the angle calibrated in a field of magnitude, θ can change by up to
15 mdeg at zero field, and
40 mdeg at
14 T . Thisrotation is measurable using a Hall signal, but is not reflected in the rotator resistive read-out. The torque-induced rotation is equal rather than opposite for opposite magnetic fielddirections (i.e. opposite solenoid current). We account for all of these sources of error in ourangular position uncertainty estimates, except for field-induced rotation. This omission isjustified because our calibration is performed at 6 T, roughly the maximum field at whichwe observe sharp transitions in the Hall signal in nearly in-plane field; the provided errorestimate therefore accurately characterizes the maximum possible out-of-plane field duringthese transitions.Based on the precision and accuracy of the angular position calibration near θ = 0 (in-plane field), we are confident that for all measurements in nominally in-plane field, theout-of-plane field component remained substantially smaller than the out-of-plane coercivityof the tBLG device, which was approximately
119 mT for the density and displacement fieldused. To achieve a
119 mT out-of-plane component in a field requires | θ | ≥ . ◦ ,whereas of our nominally in-plane sweeps the largest deviation from zero angle is 200 mdeg;the magnetic transitions observed in the Hall signal between and in Figs. 2(d) and3 of the main text therefore do not appear to be driven simply by the out-of-plane field25omponent flipping an orbital magnetization of a domain. Even at
14 T , the out-of-planecomponent remains below
119 mT for | θ | < mdeg, and each of the measurements shownin Figs. 2(d) and 3 of the main text and Figs. S3D, S4, S5, S6, S7, and S8 of the supplementinformation are well within this range, even accounting for possible field-induced rotation ofthe rotator. Longitudinal resistance data
In this section, we provide the corresponding longitudinal resistance data for the figures of themain text. Fig. S2 shows the longitudinal resistance R xx corresponding to the Hall resistancedata of Fig. 1 of the main text. As was seen previously with this device, R xx displays visiblehysteresis, presumably due to mixing in of the Hall signal from inhomogeneity in the deviceor its domain structure. When plotted as a function of the perpendicular field component(Fig. S2b), we see that the longitudinal resistance depends mostly on the perpendicular fieldcomponent with some small variations from angle to angle, perhaps due to an effect of thein-plane magnetic field.Fig. S3 shows the longitudinal resistance R xx corresponding to the Hall resistance dataof Fig. 2 of the main text. As the angle of the sample is tuned closer and closer to a purelyin-plane magnetic field, the variations in the longitudinal resistance become smaller. Asthe field becomes almost perfectly in-plane (Fig. S3d), the magnitude of hysteretic R yx isdiminished compared to that seen at larger out-of-plane tilt angles. We see jumps in thelongitudinal resistance that likely correspond to the flipping of magnetic domains.Finally, Fig. S4 shows the (a) longitudinal and (b) Hall response to a purely in-planefield of a state initially magnetized out-of-plane. Panel (b) is reproduced from Fig. 3 of themain text. As was discussed in the main text, applying an in-plane magnetic field erases theinitial magnetic state and drives a phase transition to a different state above 5 T in-planefield. Below this critical field, the magnetic state is recovered but the full polarization of thesystem is not. This leads to a reduction in the magnitude of the Hall resistance.26 a) (b) Figure S2:
Angular dependence of longitudinal resistance in magnetic hysteresisloops.
Magnetic field dependence of the longitudinal resistance R xx corresponding to thedata shown in Fig. 1 of the main text, with n/n s = 0 . and D/(cid:15) = − .
30 V / nm at
29 mK as a function of the angle of the device relative to the field direction; ◦ correspondsto field in the plane of the sample. The hysteresis loops are plotted as a function of (a) theapplied field B and (b) the component of the field perpendicular to the plane of the sample B ⊥ . The solid and dashed lines correspond to sweeping the magnetic field B up and down,respectively. 27 a) (c)(b) (d) Figure S3:
Longitudinal resistance hysteresis loops for small tilt angles.
Angulardependence of the longitudinal resistance R xx vs B corresponding to the data shown inFig. 2 of the main text with n/n s = 0 . and D/(cid:15) = −
30 V / nm for angles of the fieldrelative to the plane of the sample: (a) . ◦ ± . ◦ , (b) . ◦ ± . ◦ , (c) . ◦ ± . ◦ ,(d) +0 . ◦ ± . ◦ , and − . ◦ ± . ◦ . Vertical dashed black lines indicate where theout-of-plane component of the field equals the coercive field ±
119 mT . The out-of-planecomponent of the field is raised beyond the coercive field in panels (a),(b), just reaches thecoercive field in (c), and does not reach it in (d).All traces were taken at
27 mK except forthe trace with tilt angle +0 . ◦ ± . ◦ , which was taken at .
35 K .28 a) (b)
Figure S4:
Longitudinal resistance under an in-plane field from an initially magne-tized state.
In-plane hysteresis loops of the (a) longitudinal resistance R xx correspondingto the data shown in Fig. 3 of the main text with n/n s = 0 . and D/(cid:15) = −
30 V / nm at
26 mK . The Hall resistance data R yx of Fig. 3 of the main text are replicated in panel (b).The sample is initially polarized with an out-of-plane field. The sample is then rotated to − ± in zero magnetic field. The field B (cid:107) is then increased from zero (red trace)before completing a hysteresis loop (blue solid and dashed traces).29 ntisymmetric components and dependence on magnitude of in-plane field We can antisymmetrize the hysteresis loop shown in Fig. 3 of the main text by consider-ing R asym yx ( B ) = ( R yx ( B, sweeping upwards) − R yx ( − B, sweeping downwards)) / (Fig. S5a).The final upward sweep is completed only up to , so we have restricted the symmetriza-tion to [ − , . We see that for small in-plane fields, the system is strongly hysteretic.Although a typical ferromagnetic hysteretic loop is not observed as a function of in-planefield, we do see that there is a large discrepancy between the up and down sweeps. Abovean in-plane field of ∼ , the up and down sweeps differ only slightly. Additionally, themagnitude of the antisymmetric component of the Hall signal is quite small, (cid:46) . , overthis high field range, in contrast to the values at lower in-plane fields, or any out-of-planefields. A comparison with a loop performed at a tilt angle of . ◦ ± . ◦ such that thefield is nearly out-of-plane (Fig. S5b) shows that this symmetrization process preserves thehysteresis loop (which is shown schematically in Fig.Fig. S5c).When the data from Fig. S4 are plotted against the magnitude of the field (shown inFig. S6 for fields above 6 T), we again see that those parts of the data outside of [ − , are remarkably similar. For large in-plane fields, the longitudinal and Hall resistances arequalitatively similar, and each shows an offset of order . between data for the two fieldpolarities. Expanding on what we said in the main text, the similarity of the longitudinaland Hall resistance and the fact that both are mostly symmetric in field suggest that thesample is no longer orbitally polarized and that the apparent Hall signal results from mixingin of the longitudinal signal. Effect of out-of-plane field at small tilt angles
In this section we present additional data similar to those in Fig. 2d to elucidate the effect ofthe sign of a small out-of-plane field in the background of a large in-plane field at small tilt30 a) (b)(c)
Figure S5:
Antisymmetric contribution of the Hall resistance under an in-planefield. (a) Antisymmetric component of the Hall resistance R asym yx corresponding to the Halldata shown in Fig. 3 of the main text. ± R asym yx is plotted as a solid (dashed) line. For and above (indicated by the large tick on the horizontal axis), we report R asym yx for the initialsweep up and the sweep down. Otherwise we report R asym yx for the sweep down and thesweep up (which was only completed up to .) (b) Antisymmetric component of the Hallresistance R asym yx for the nearly out-of-plane hysteresis loop performed at . ◦ ± . ◦ , shownin Fig. 1 of the main text. (c) Schematic diagram of the symmetrization process for an idealhysteresis loop that is offset from zero in the vertical direction. (a) (b) Figure S6:
Dependence on the magnitude of the in-plane field.
In-plane hysteresisloops of corresponding longitudinal resistance R xx to the data shown in Fig. 3 of the maintext with n/n s = 0 . and D/(cid:15) = −
30 V / nm at
26 mK . The sample is initially polarizedwith an out-of-plane field. The sample is then rotated to − ±
21 mdeg in zero magneticfield. The field B (cid:107) is then increased from zero (red trace) before completing a hysteresis loop(blue traces). Though the absolute resistance is offset between the two panels, as seen in thevertical axis labels, the size of the resistance range is the same in both panels.31ngles (Fig. S7). Note that for the presented data, the sample has been rotated in the planeby ◦ relative to the traces performed in Fig. 2d so that the in-plane field is in a differentdirection relative to the sample’s crystal axis and direction of current flow. Between thetwo traces shown in Fig. S7, there are numerous jumps in R yx , some which occur at similarvalues of B and some which are slightly shifted between the two traces. There is an orderof magnitude difference in the magnitude of the out-of-plane field between the two traces,suggesting that the features in common between the two traces of Fig. S7 are primarily aresponse to the in-plane field.Figure S7: Hysteresis loops at small angles.
Magnetic field hysteresis loops where R yx is measured at two small angles of very different magnitude and likely opposite sign: +172 ±
29 mdeg in red and − ±
22 mdeg in blue. Both traces were taken at
28 mK with n/n s = 0 . and D/(cid:15) = −
30 V / nm . The sample has been rotated in the plane by an angleof ◦ relative to the measurements performed in Fig. 2 of the main text. Effect of in-plane field on longitudinal resistance
For completeness, we have examined the effect of an in-plane field at other carrier densities.The longitudinal resistance R xx does not depend significantly on B (cid:107) except in the range of32ensities n/n s = 0 . to . , where R xx increases with increasing B (cid:107) (Fig. S8). This is theopposite dependence compared to the behavior in a superconducting device presented in Ref.7.Figure S8: In-plane field dependence of longitudinal resistance.
Longitudinal resis-tance R xx as a function of carrier density n for several different in-plane magnetic fields at afixed displacement field of D/(cid:15) = − .
30 V / nm and . for a tilt angle of − ±
23 mdeg23 mdeg