Composite fermions and broken symmetries in graphene

ARTICLE Received 21 Feb 2014 | Accepted 13 Nov 2014 | Published 6 Jan 2015 DOI: 10.1038/ncomms6838 Composite fermions and broken symmetries in graphene F. Amet1, A.J. Bestwick2, J.R. Williams2, L. Balicas3, K. Watanabe4, T. Taniguchi4 & D. Goldhaber-Gordon1,2 The electronic properties of graphene are described by a Dirac Hamiltonian with a four-fold symmetry of spin and valley. This symmetry may yield novel fractional quantum Hall (FQH) states at high magnetic field depending on the relative strength of symmetry-breaking interactions. However, observing such states in transport remains challenging in graphene, as they are easily destroyed by disorder. In this work, we observe in the first two Landau levels the two-flux composite-fermion sequences of FQH states between each integer filling factor. In particular, the odd-numerator fractions appear between filling factors 1 and 2, suggesting a broken-valley symmetry, consistent with our observation of a gap at charge neutrality and zero field. Contrary to our expectations, the evolution of gaps in a parallel magnetic field suggests that states in the first Landau level are not spin-polarized even up to very large out-of-plane fields. 1 Department of Applied Physics, Stanford University, Stanford, California 94305, USA. 2 Department of Physics, Stanford University, Stanford, California 94305, USA. 3 National High Magnetic Field Laboratory, Tallahassee, Florida 32310-3706, USA. 4 Advanced Materials Laboratory, National Institute for Materials Science, 1-1 Namiki, Tsukuba 305-0044, Japan. Correspondence and requests for materials should be addressed to F.A. (email: ) or to D.G. (email: ). NATURE COMMUNICATIONS | 6:5838 | DOI: 10.1038/ncomms6838 | www.nature.com/naturecommunications & 2015 Macmillan Publishers Limited. All rights reserved. 1 ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms6838 I n a large magnetic field B, the band structure of twodimensional electrons becomes a discrete set of highly degenerate Landau levels (LLs)1–4. With kinetic energy quenched, the electron interactions determine the ground state of a partially filled LL. This yields new incompressible phases known as fractional quantum Hall (FQH) states5,6, where the longitudinal resistance rxx vanishes exponentially at low temperatures and the transverse resistance rxy is quantized as h/ne2, where e is the electron charge, h is the Planck’s constant and n ¼ nh/eB, the number of filled LLs for an electron density n. In GaAs quantum wells, the most robust FQH states are observed when n ¼ p/(2kp±1) ¼ 1/3, 2/3, 2/5, 3/5...(k and p integers). This may be understood within the composite fermion (CF) theory, where each electron is imagined to bind an even number 2k of magnetic flux quanta f0 ¼ h/e (refs 7–10). CFs experience an effective residual magnetic field B* ¼ B  2knf0, and incompressible phases occur when the number of filled CF LLs p ¼ nf0/B* is an integer. In graphene, the ground state in each of these FQH phases is characterized by the two internal degrees of freedom of the Hamiltonian, the spin and the valley isospin, the latter originating from the hexagonal crystal structure of graphene. These grant LLs an approximate SU(4) symmetry1–4 broken at high magnetic fields11 by Zeeman splitting EZ¼ gmBB and valley symmetry breaking on order ða=lB ÞEc  ae2 ElB2 (refs 12–14), where a is the graphene lattice constant, lB is the the magnetic length, Ec is the the typical energy of Coulomb interactions at a length scale lB and E is the the dielectric constant. While the nature of the brokensymmetry phases at integer filling factors has been under intense scrutiny11–17, little is known about the impact of symmetrybreaking interactions on FQH states18–26. In this work, we present magneto-transport measurements on high-quality graphene devices at magnetic fields up to 45 T and temperatures down to 30 mK. Between each integer filling factor and up to n ¼ 6, we observe sequences of FQH states with unprecedented detail in transport. These follow the two-flux CF sequence dn ¼ p/(2p±1), with pr5 integer. Several devices are gapped at zero field, as a result of their substrate-induced broken sublattice symmetry. These host FQH states at odd filling factors 5/3 and 7/5, which were absent in devices with a preserved valley symmetry. In addition, we report the first study of the inplane field dependence of rxx at fixed out-of-plane field in the FQH regime of graphene. We observe a pronounced weakening of FQH states in the first LL as the in-plane field is increased, suggesting that they are not yet spin-polarized up to very large field. Results Characterization at low magnetic field. Observation of FQH states requires that disorder-induced Fermi level fluctuations dEF be smaller than FQH energy gaps. To minimize dEF, we place monolayer graphene on an atomically flat flake of hexagonal boron nitride27,28, which is typically 15–25 nm thick in the seven heterostructures we studied. The carrier density is tuned with a voltage VBG applied to a graphite back gate (Fig. 1a), which also acts as a screening layer, recently shown to make the potential landscape in graphene devices cleaner29–31. Potential fluctuations should be suppressed on length scales larger than the distance to the back gate, while electron interactions pffiffiffiffiffiffiffiffiffi remain on the scale of B½T, less than the distance to the magnetic length lB ¼ 26 nm the back gate for the relevant magnetic field range. The devices included in this study have field-effect mobilities ranging from 4  105 to 106 cm2 V  1 s  1, as extracted from a linear fit to the conductivity s(n) at low density (typically no2  1011 cm  2). This high quality is seen from the longitudinal resistivity rxx, plotted in logarithmic scale as a function of the carrier density for Device A, at temperature T ¼ 4 K and B ¼ 0 T (Fig. 1b). rxx drops to 20 O at n ¼ 5  1012 cm  2, although in this regime the mean free path is limited by boundary scattering32 and the sheet resistivity is not well defined. The quantum mobility is also estimated from the onset of the n ¼ 2 gaps in rxx(VBG, B) (ref. 33), which for example occurs at 25 mT for Device B (Fig. 1b, inset), indicating a mobility of at least 4  105 cm2 V  1 s  1. Five of our seven devices are strongly insulating at the charge neutrality point, with the peak resistivities rNP exceeding 100 kO at 4 K, well above the theoretical limit of ph/4e2B20 kO expected for gapless pristine graphene34–36 (the last two devices are comparable in quality but with rNP o20 kO). This phenomenon 13 1110 8 3 3 3 3 iac –6 –4 –3 –2 –1 1 2 3 4 6 10 h-BN graphite SiO2 –7 7 –8 8 –10 10 g (S) 106 106 105 105 104 104 B (T) 107 5 VBG(V) 0.05 0.1 0.15 0.2 103 T –1 (K) 60 B (mT) Rxx (Ω) 107 102 10 –5 –2.5 0 n (×1012 cm–2) 2.5 5 0 –3 80 –2 100 50 25 0 0 –1 0 VBG(V) 1 1 Rxx (kΩ) 2 3 4 2 5 3 Figure 1 | Characterization at low field. (a) Schematic of the device, consisting of a graphene/h-BN/grap (...truncated)