Quantum parity in Bernal-stacked trilayer graphene

Petr Stepanova,1, Yafis Barlasb,1, Shi Chea, Kevin Myhroc, Greyson Voigtc, Ziqi Pic, Kenji Watanabed, Takashi Taniguchid, Dmitry Smirnove, Fan Zhangf, Roger K. Lakeg, Allan H. MacDonaldh,2, and Chun Ning Laua,2

aDepartment of Physics and Astronomy, The Ohio State University, Columbus, OH 43210; bDepartment of Physics, Yeshiva University, New York, NY 10033; cDepartment of Physics and Astronomy, University of California, Riverside, CA 92521; dNational Institute for Materials Science, Ibaraki 305-0044, Japan; eNational High Laboratory, Tallahassee, FL 32310; fDepartment of Physics, University of Texas at Dallas, Richardson, TX 75080; gDepartment of Electrical Engineering, University of California, Riverside, CA 92521; and hDepartment of Physics, University of Texas at Austin, Austin, TX 78712-1192

Contributed by Allan MacDonald, March 27, 2019 (sent for review December 12, 2018; reviewed by Andre Geim and Francisco Guinea)

The has recently been generalized from mirror symmetry is preserved, Landau levels (LLs) belonging to transport of conserved charges to include transport of other different mirror symmetry representations do not couple. approximately conserved-state variables, including and val- The complex interplay between the spin, valley, parity, and ley, via spin- or valley-polarized boundary states with different electronic interactions in ABA graphene suggests the possibility chiralities. Here, we report a class of quantum Hall effect in of SPT phases at the carrier neutrality point (CNP) that have Bernal- or ABA-stacked trilayer graphene (TLG), the quantum not been identified in previous studies (24–29). In this paper, parity Hall (QPH) effect, in which boundary channels are distin- we demonstrate SPT phases in ABA TLG in which mirror sym- guished by even or odd parity under the system’s mirror reflec- metry preserves counterpropagating edge modes. We observe a tion symmetry. At the charge neutrality point, the longitudinal sequence of transitions between different SPT phases that are 2 conductance σxx is first quantized to 4e /h at a small perpendic- driven by magnetic field-dependent interactions between elec- ular magnetic field B⊥, establishing the presence of four edge trons close to the Fermi level and the Dirac sea. When mirror 2 channels. As B⊥ increases, σxx first decreases to 2e /h, indi- symmetry is broken by an out-of-plane displacement field, the cating spin-polarized counterpropagating edge states, and then, two-probe longitudinal conductance decreases dramatically to to approximately zero. These behaviors arise from level cross- form a layer-polarized . ings between even- and odd-parity bulk Landau levels driven Our experiments were performed on dual-gated TLG devices by exchange interactions with the underlying Fermi sea, which encapsulated between two hexagonal boron nitride (hBN) sheets favor an ordinary insulator ground state in the strong B⊥ limit (Fig. 1B) etched into Hall-bar geometries and with edge contacts and a spin-polarized state at intermediate fields. The transitions (30, 31). The device quality was enhanced by the presence of between spin-polarized and -unpolarized states can be tuned by a local graphite gate underneath the TLG channel, which pro- varying Zeeman energy. Our findings demonstrate a topologi- vides for additional screening of charged impurities and also, cal phase that is protected by a gate-controllable symmetry and enables independent control of the Fermi levels of the channel sensitive to Coulomb interactions. and the leads (32). The device is very conductive in the absence of

2D materials | quantum Hall effect | topological insulators | Significance symmetry-protected phases | trilayer graphene The quantum Hall effect, which is characterized by an insulat- n the conventional Hall effect, a charge current combines with ing bulk and chiral edge states, has recently been generalized Ia perpendicular magnetic field B⊥ to yield a steady state with from transport of charges to transport of other degrees of a transverse chemical potential gradient. A quantum version of freedom, leading to quantum spin Hall and quantum valley the Hall effect (QHE), in which the chemical potential gradient Hall effects. Here, we report a class of quantum Hall effect, the is replaced by a chemical potential difference between oppo- quantum parity Hall (QPH) effect, in which boundary channels site sample edges, can occur when the 2D bulk is insulating. are distinguished by even or odd parity under the system’s Recently, the Hall effect and the QHE have been generalized mirror reflection symmetry. The QPH is realized in charge neu- from transport of conserved charge to transport of other approx- tral Bernal- or ABA-stacked trilayer graphene, which displays imately conserved-state variables, including spin (1, 2) and valley a striking phase diagram with four different phases, arising (3, 4); their quantum versions (5–8) are then characterized by from an interplay between topologically protected crystalline spin- or valley-polarized boundary states with different chirali- symmetry and Coulomb interactions. Our results motivate ties. For material systems that host these topologically nontrivial a search for classes of topological states in van der Waals phenomena, discrete symmetries play an important role. For materials. example, time-reversal symmetry in the is essential to protect 1D counterpropagating edge modes from Author contributions: P.S., Y.B., S.C., D.S., F.Z., R.K.L., A.H.M., and C.N.L. designed research; P.S., Y.B., S.C., K.M., G.V., Z.P., D.S., F.Z., R.K.L., A.H.M., and C.N.L. performed backscattering. research; K.W. and T.T. contributed new reagents/analytic tools; P.S., Y.B., F.Z., R.K.L., Multiband Dirac systems, such as Bernal- or ABA-stacked tri- A.H.M., and C.N.L. analyzed data; and P.S., Y.B., F.Z., R.K.L., A.H.M., and C.N.L. wrote layer graphene (TLG), afford richer and more exotic symmetry- the paper.y protected topological (SPT) phases. For instance, in addition Reviewers: A.G., University of Manchester; and F.G., IMDEA Nanociencia.y to approximate spin and valley symmetries, TLG has an addi- Conflict of interest statement: K.W. and T.T. are coauthors with A.G. on a 2017 article.y tional discrete mirror symmetry (Fig. 1A) that allows bands to be Published under the PNAS license.y classified by their parity, the eigenvalue of the operator for reflec- 1 P.S. and Y.B. contributed equally to this work.y tion in the plane of the middle layer. TLG has two low-energy 2 To whom correspondence may be addressed. Email: [email protected] or odd-parity π-bands described by a massless Dirac Hamiltonian [email protected] with a monolayer graphene (MLG)-like spectrum and four low- This article contains supporting information online at www.pnas.org/lookup/suppl/doi:10. energy even-parity bands that exhibit a massive Dirac bilayer 1073/pnas.1820835116/-/DCSupplemental.y graphene (BLG)-like spectrum (9–23) (SI Appendix). As long as Published online May 3, 2019.

10286–10290 | PNAS | May 21, 2019 | vol. 116 | no. 21 www.pnas.org/cgi/doi/10.1073/pnas.1820835116 Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 tpnve al. et Stepanov ietrace Line (D) I–IV. labeled are phases electronic different The mK. 1. Fig. h oitrcigeeto rudsaefor state ground electron noninteracting the As B 1E, branch. BLG-like Fig. above the four- in well in the branch shown LLs lowest MLG-like places degenerate the displacement eightfold in the this LLs lowest field, degenerate magnetic fold unstacked a and In stacked between atoms. differences interlayer energy remote from and arises even- hopping which the of points, between Dirac feature displacement odd-parity energetic and important the particularly is graphene A ABA respectively. lines, and dashed lines, red with by associated represented LLs are odd- branches lines, (M) blue which MLG-like by in parity represented 1E, are Fig. LLs 30, in (B) 27, displayed BLG-like is (24, even-parity spectrum work LL previous fan calculated with The Landau consistent 33). experimental are the they in and crossings diagram, LL cal- to fitting spectra by continuum extracted culated were a parameters using hopping remote calculate TLG various first of we structure states, these band the of the at nature phases the SPT understand to To CNP. points and values conductivity longitudinal with IV phase insulating an into large sufficiently of application measured in the range with wide a state over resistive sists a III, phase σ to transitions device of (σ I phase zero of near observed fields, are small very phases three first at The as E labeled Strikingly, plot. are the 1C. which on identified, Fig. be I–IV in can phases shown different is four least diagram the phase resulting explore The to us device’s allows which phases. film, topological the with ABA associated symmetries the of (E symmetry field displacement density nal carrier charge the (V control independently top varying By of Appendix). mobility quantum estimated and conductivity CNP longitudinal with fields, external indicate respectively. lines operation, () reflection Dashed mirror field. under even B and the of function a as A B xx ⊥ ⊥ hssrkn hs iga dnie ifrn hssb their by phases different identifies diagram phase striking This the measure we CNP, the at phases topological explore To B V hs ihqatzdconductance quantized with I phase : mirror ' plane ∼ xx ⊥

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PHYSICS These shifts, which are related to the established Dirac velocity enhancement in graphene monolayers in the absence of a field, have the same sign as the carrier and therefore, lower the MLG- like LL energies while raising the BLG-like LL energies (detailed calculations are in SI Appendix). For intermediate B⊥, the (M, h, ↑) LL is occupied in preference to the orbital N = 0 (B, e, ↓) LL (Fig. 3A). Phase II has ν = −1 for odd parity because of an empty spin ↓ LL and ν = +1 for even parity because of an occupied spin ↑ LL; therefore, it has counterpropagating edge states with opposite spin and opposite parity, explaining its accu- rate conductance quantization. We classify phase II as a quantum parity Hall ferromagnet (QPHF). The quantized conductance of both phases I and II is sup- pressed by application of a large |E⊥|. Because |E⊥| breaks mirror symmetry, edge conduction is no longer protected by crys- tal symmetry when |E⊥| 6= 0. Because the |E⊥| 6= 0 suppression occurs to phase II, we conclude that spin-rotational invariance, which would otherwise protect ballistic conduction at finite |E⊥|, must be broken by spin-orbit coupling in our samples. Even with spin-orbit coupling, mirror symmetry protects the |E⊥| = 0 spin-polarized counterpropagating states. For particles with spin, Mˆ 2 = −1, where Mˆ is the mirror symmetry operator (SI Appendix), with eigenvalues ±ı for the even-parity and ∓ı for the odd-parity up-spin (down-spin) eigenstates. Therefore, in the QPHF phase, the projected mirror symmetry operator M¯ sat- isfies M¯ 2 = −1, providing an effective Kramer degeneracy for the QPHF phase. When |E⊥| 6= 0, the spin-polarized edge states intermix due to spin-orbit coupling, which was originally forbid- Fig. 3. Phases II and III of TLG. (A and B) Schematics of edge-state config- den due to the mirror symmetry. Since classification on the basis urations for phases II and III, respectively. (C) σxx (E⊥, B⊥) at different tilt of mirror symmetry is not relevant at |E⊥| 6= 0, we identify phase ◦ ◦ angles. (D and E) σxx (E⊥, B⊥) at tilt angles θ = 30 and 50 , respectively. Color scale: black, low conductance; red, high conductance. (F) Critical field B⊥,c for transition between phases II and III vs. total magnetic field Bt .(G) High field dependence σxx (B⊥) at θ = 0 for B⊥ up to 31 T.

that it is protected by mirror symmetry. These two properties identify phase I as the spin-unpolarized noninteracting QPH effect ground state expected on the basis of single-particle physics. The quantized σxx is associated with two even-parity and two odd-parity chiral edge states that propagate from source to drain along opposite edges (Fig. 2D). These approximately spin-degenerate counterpropagating edge states are protected against backscattering by an underlying crystalline symmetry, since they correspond to different representations of the mirror reflection symmetry of the TLG crystal lattice that is preserved at E⊥ = 0. (The slight deviation of σxx from the quantized value may be accounted for by the presence of disorder that breaks mirror symmetry.) In the presence of a finite displace- ment field, backscattering between counterpropagating states is allowed, the conductance is quickly reduced to a small value, and the ground state is a partially layer-polarized ordinary insulator. 2 As B⊥ increases from 1 to 5 T, σxx drops to ∼ 2e /h, signaling a reduction in the number of chiral channels from four in phase I to two in phase II (Fig. 1D). This reduction in σxx is not expected in a noninteracting electron theory and can explained only by tak- ing Coulomb interactions into consideration. Two aspects play an essential role: (i) intra-LL interactions along with Zeeman energy can stabilize strongly spin-polarized monolayer-like and bilayer-like N = 0 LLs, and (ii) exchange interactions between the N = 0 LLs and Dirac sea induce a magnetic field-dependent change in the energy separation between LLs with different par- ity and in the BLG-like case, different orbital character. The Fig. 4. Phase IV and overall phase diagram. (A) σxx (E⊥, B⊥) at the tilt angle latter effect, which also plays a role in phase I by selecting N = 0 ◦ θ = 0 for 4 ≤ B⊥ ≥ 8 T. The dotted line is plotted using E⊥(mV/nm) = p BLG-like LLs for occupation over N = 1 LLs, can be captured 135 (B⊥ − 4). (B) Line trace σxx (E⊥) at B⊥ = 8 T. (C) Schematics of the by adding self-energy corrections (33–35) to the electron and phase diagram of charge neutral TLG as a function of E⊥, B⊥, and Bk. PLP, hole LL energies that are larger for N = 1 in the BLG-like case. partially layer polarized.

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WK, and Tse Theory ZH, effect: Qiao Hall spin 6. quantum The (2008) al. et M, Konig 5. .GrahvR,e l 21)Dtcigtplgclcret ngahn superlattices. graphene in currents topological Detecting (2014) al. et RV, Gorbachev 4. .MkK,MGl L akJ cunP 21)Tevle alefc nMS transistors. in MoS2 in effect electrons Hall valley of The (2014) PL orientation McEuen J, spin Park KL, McGill Current-induced KF, Mak (1971) 3. VI Perel current. MI, with spins Dyakonov electron orienting 2. of Possibility (1971) VI Perel MI, Dyakonov 1. Eq. tblt odtosfrthe for conditions Stability larger even For ttsi hrlysakdfwlyrgahn systems. graphene few-layer stacked chirally in states study. ab-initio An B adsorption: metal-atom Rev via graphene in states Hall lous/valley graphene. bilayer in transition phase 256801. topological and state lator Jpn Soc Science Science . Lett JETP Phys B 3 84:195444. 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(2011) al. semimetal- et and SH, effect Jhang Hall 13. trilayer Quantum ABC-stacked (2012) JP and Eisenstein ABA- D, of Nandi characterization EA, Raman Henriksen (2011) 12. al. et C, trilayer in Cong transport quantum and 11. gap band Stacking-dependent (2011) al. et W, Bao 10. utvt safnto fdslcmn edi o monotonic. not is to field sharply displacement of function As a as ductivity II. phase as linearly almost 3G, Fig. in of dependence nonmonotonic 91F1--46adUiest fTxsa alsrsac enhancement research Dallas Grant at Office Texas funds. Research of Army University by and supported W911NF-18-1-0416 was F.Z. Welch F1473. Division and DOE DE-FG03-02ER45958 Grant by Grant the Foundation part Engineering for in and Society supported Sciences was Materials Japan Research Austin of from in Scientific Layers” Work for Science. Atomic of of Grant-in-Aid Promotion “Science a Areas Culture, and Innovative Education, Technology on of Ele- and Ministry the by the Science the by by Sports, supported supported and conducted was is Initiative Florida, crystals Strategy nitride which per- of mental boron was Lab, State hexagonal of work the Field Growth this Research-0654118, DOE. Magnetic of Materials High Part of National SC0012670. NSF/Division the Research Award Frontier at BES Energy DOE formed an by is which funded and Systems, Spin Center by Electronic enabled Nanoscale are Energy in experiment Heat Basic and works theory (DOE) theoretical between The Energy collaboration 46940-DE-SC0010597. and of ER Grant Department Division (BES) by Sciences supported are experiments ACKNOWLEDGMENTS. lock-in standard using top refrigerators Cr/Au He3 and in Hall- techniques. contacts measured multiterminal metallic are Cr/Au devices the to The define coupled gates. then to is used TLG geometry. are bar by plasma etching gas ion hexafluoride reactive and sulfur lithography beam Electron stacks. bulk hBN/TLG/hBN from exfoliated mechanically Si/SiO are onto crystals hBN few-layer and TLG ABA-stacked Methods and crystals Materials 2D in symmetries group space the heterostructures. phases and and open SPT point gate-tunable observations by of plethora protected our a discovering Additionally, topo- of possibility (38). interacting play- rich symmetry-protected phases a exotic logical provide realize multilayer, Dirac to ABA-stacked Multiband ground parities. as different such and with systems, interactions, bands exchange of energy, symme- self-energies crystalline Zeeman and localization, spin between tries, interplay intricate an from and conductivity. zero Hall with valley the states of separate values that nonzero boundary-state walls to domain due along expected is transport transition in of the tuned values at nonzero peak is by ductivity field state spin-polarized intermediate the spin-polarized of favor the and strong state spin-unpolarized field the between transition phase first-order nonmonotonic The = nanometer) per equation can the regimes by two phenomenologically these described separates be that peak conductivity local the .AakvV,CarbryT(02 lcrclytnbecag n pntransitions spin and charge tunable Electrically (2012) T Chakraborty VM, Apalkov 9. ial,w n ht tlarge at that, find we Finally, h utpepae bevdi h ihpaedarmarise diagram phase rich the in observed phases multiple The h lcrnctasoti rlyrgraphene. trilayer in transport electronic the graphene. trilayer in gaps particle rlyrgraphene. trilayer graphene. trilayer graphene. ABA-stacked dual-gated of 011004. behavior lic graphene. graphene. trilayer graphene. in fermions Dirac interacting of 86:035401. levels Landau in E ⊥ increases, hsRvB Rev Phys Nano ACS Phys Nat ∼ σ 0.1e xx ece iiu t1. n hn increase then, and T 12.5 at minimum a reaches 2 hsRvB Rev Phys 2 7:948–952. usrts eueadytase ehiu oassemble to technique dry-transfer a use We substrates. 5:8760–8768. /h 84:161408. σ B 135 PNAS xx σ etakMxmKaioo o icsin.The discussions. for Kharitonov Maxim thank We ⊥ .Tepsto n( E in position The 4B). (Fig. xx p rtrssto rises first nrae o3 stedvc reenters device the as T 31 to increases (E 79:125443. | B ⊥ ⊥ a 1 2019 21, May a Commun Nat ) (Tesla) eednedmntae htthe that demonstrates dependence σ xx B tvr large very at ⊥ C Nano ACS − ∼ h eairo h con- the of behavior the , 5:5656. 4 1.5e | dte iei i.4A). Fig. in line (dotted o.116 vol. 2 4:7087–7092. /h n hn drops then, and B | ⊥ E ⊥ o 21 no. , hsRvX Rev Phys ⊥ splotted As . B E (millivolts pc of space ) ⊥ hsRvB Rev Phys con- A . | 10289 2:

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