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Huang L, Zhong J, Sheng W, Zhou A. Scaling of energy gaps in phosphorene nanoflakes. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 34:085502. [PMID: 34814118 DOI: 10.1088/1361-648x/ac3c68] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/26/2021] [Accepted: 11/23/2021] [Indexed: 06/13/2023]
Abstract
Electronic structure of phosphorene nanoflakes which consist of hundreds of phosphorus atoms are studied in the framework of unrestricted Hartree-Fock approach. On the base of Pariser-Parr-Pople model for electron-electron interactions, a simplified Bethe-Salpeter formalism is established for the calculation of excitation states of the system. Taking into account the electron-hole interaction in various dielectric environments, the optical gap of a triangular phosphorene nanoflake is shown to increase as the screening effect becomes stronger while its graphene counterpart exhibits just the opposite dependence. After confirming an exponential dependence of the optical gap on the effective dielectric constant, the quasiparticle and optical gaps are also found to obey an exponential scaling rule against the total number of atoms in the nanoflakes, respectively. By extrapolating the dependence on the size of the system, one is able to estimate the exciton binding energy of a monolayer phosphorene sheet on a SiO2substrate to be 0.894 eV. The result is found to agree well with the previous experimental result of 0.9 eV.
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Affiliation(s)
- Linan Huang
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, People's Republic of China
| | - Jun Zhong
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, People's Republic of China
| | - Weidong Sheng
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, People's Republic of China
| | - Aiping Zhou
- Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing 211167, People's Republic of China
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2
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Zheng Z, Ma Q, Bi Z, de la Barrera S, Liu MH, Mao N, Zhang Y, Kiper N, Watanabe K, Taniguchi T, Kong J, Tisdale WA, Ashoori R, Gedik N, Fu L, Xu SY, Jarillo-Herrero P. Unconventional ferroelectricity in moiré heterostructures. Nature 2020; 588:71-76. [DOI: 10.1038/s41586-020-2970-9] [Citation(s) in RCA: 77] [Impact Index Per Article: 19.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/06/2020] [Accepted: 10/07/2020] [Indexed: 11/09/2022]
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3
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Stepanov P, Das I, Lu X, Fahimniya A, Watanabe K, Taniguchi T, Koppens FHL, Lischner J, Levitov L, Efetov DK. Untying the insulating and superconducting orders in magic-angle graphene. Nature 2020; 583:375-378. [PMID: 32632215 DOI: 10.1038/s41586-020-2459-6] [Citation(s) in RCA: 105] [Impact Index Per Article: 26.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2019] [Accepted: 04/17/2020] [Indexed: 11/09/2022]
Abstract
The coexistence of superconducting and correlated insulating states in magic-angle twisted bilayer graphene1-11 prompts fascinating questions about their relationship. Independent control of the microscopic mechanisms that govern these phases could help uncover their individual roles and shed light on their intricate interplay. Here we report on direct tuning of electronic interactions in this system by changing the separation distance between the graphene and a metallic screening layer12,13. We observe quenching of correlated insulators in devices with screening layer separations that are smaller than the typical Wannier orbital size of 15 nanometres and with twist angles that deviate slightly from the magic angle of 1.10 ± 0.05 degrees. Upon extinction of the insulating orders, the vacated phase space is taken over by superconducting domes that feature critical temperatures comparable to those in devices with strong insulators. In addition, we find that insulators at half-filling can reappear in small out-of-plane magnetic fields of 0.4 tesla, giving rise to quantized Hall states with a Chern number of 2. Our study suggests re-examination of the often-assumed 'parent-and-child' relation between the insulating and superconducting phases in moiré graphene, and suggests a way of directly probing the microscopic mechanisms of superconductivity in strongly correlated systems.
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Affiliation(s)
- Petr Stepanov
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Barcelona, Spain
| | - Ipsita Das
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Barcelona, Spain
| | - Xiaobo Lu
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Barcelona, Spain
| | - Ali Fahimniya
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA
| | - Kenji Watanabe
- National Institute for Materials Science, Tsukuba, Japan
| | | | - Frank H L Koppens
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Barcelona, Spain
| | - Johannes Lischner
- Departments of Materials and Physics and the Thomas Young Centre for Theory and Simulation of Materials, Imperial College London, London, UK
| | - Leonid Levitov
- Department of Physics, Massachusetts Institute of Technology, Cambridge, MA, USA
| | - Dmitri K Efetov
- ICFO-Institut de Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Barcelona, Spain.
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4
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Karimi S, Rezania H. Effect of magnetic field on specific heat and magnetic susceptibility of biased bilayer graphene: A full band approach. Chem Phys 2019. [DOI: 10.1016/j.chemphys.2019.110417] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/25/2022]
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5
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Nam Y, Ki DK, Soler-Delgado D, Morpurgo AF. A family of finite-temperature electronic phase transitions in graphene multilayers. Science 2018; 362:324-328. [PMID: 30337406 DOI: 10.1126/science.aar6855] [Citation(s) in RCA: 27] [Impact Index Per Article: 4.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/19/2017] [Accepted: 08/28/2018] [Indexed: 12/24/2022]
Abstract
Suspended Bernal-stacked graphene multilayers up to an unexpectedly large thickness exhibit a broken-symmetry ground state whose origin remains to be understood. We show that a finite-temperature second-order phase transition occurs in multilayers whose critical temperature (T c) increases from 12 kelvins (K) in bilayers to 100 K in heptalayers. A comparison of the data with a phenomenological model inspired by a mean-field approach suggests that the transition is associated with the appearance of a self-consistent valley- and spin-dependent staggered potential that changes sign from one layer to the next, appearing at T c and increasing upon cooling. The systematic evolution with thickness of several measured quantities imposes constraints on any microscopic theory aiming to analyze the nature of electronic correlations in this system.
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Affiliation(s)
- Youngwoo Nam
- Department of Quantum Matter Physics (DQMP) and Group of Applied Physics (GAP), University of Geneva, 24 Quai Ernest-Ansermet, CH1211 Genéve 4, Switzerland.,Department of Physics, Gyeongsang National University, Jinju-daero 501, Jinju-si, South Korea
| | - Dong-Keun Ki
- Department of Quantum Matter Physics (DQMP) and Group of Applied Physics (GAP), University of Geneva, 24 Quai Ernest-Ansermet, CH1211 Genéve 4, Switzerland.,Department of Physics, The University of Hong Kong, Hong Kong, China
| | - David Soler-Delgado
- Department of Quantum Matter Physics (DQMP) and Group of Applied Physics (GAP), University of Geneva, 24 Quai Ernest-Ansermet, CH1211 Genéve 4, Switzerland
| | - Alberto F Morpurgo
- Department of Quantum Matter Physics (DQMP) and Group of Applied Physics (GAP), University of Geneva, 24 Quai Ernest-Ansermet, CH1211 Genéve 4, Switzerland.
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Sboychakov AO, Rozhkov AV, Rakhmanov AL, Nori F. Externally Controlled Magnetism and Band Gap in Twisted Bilayer Graphene. PHYSICAL REVIEW LETTERS 2018; 120:266402. [PMID: 30004746 DOI: 10.1103/physrevlett.120.266402] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/10/2017] [Indexed: 06/08/2023]
Abstract
We theoretically study the effects of electron-electron interaction in twisted bilayer graphene in a transverse dc electric field. When the twist angle is not very small, the electronic spectrum of the bilayer consists of four Dirac cones inherited from each graphene layer. An applied bias voltage leads to the appearance of two holelike and two electronlike Fermi surface sheets with perfect nesting among electron and hole components. Such a band structure is unstable with respect to the exciton band-gap opening due to the screened Coulomb interaction. The exciton order parameter is accompanied by spin-density-wave order. The gap depends on the twist angle and can be varied by a bias voltage. This result correlates well with recent transport measurements [J.-B. Liu et al., Sci. Rep. 5, 15285 (2015)SRCEC32045-232210.1038/srep15285]. Our proposal allows the coexistence of (i) an externally controlled semiconducting gap and (ii) a nontrivial multicomponent magnetic order. This is interesting for both fundamental research and applications.
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Affiliation(s)
- A O Sboychakov
- Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
- Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, 125412 Russia
| | - A V Rozhkov
- Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
- Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, 125412 Russia
- Moscow Institute for Physics and Technology (State University), Dolgoprudnyi, 141700 Russia
| | - A L Rakhmanov
- Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
- Institute for Theoretical and Applied Electrodynamics, Russian Academy of Sciences, Moscow, 125412 Russia
- Moscow Institute for Physics and Technology (State University), Dolgoprudnyi, 141700 Russia
- Dukhov Research Institute of Automatics, Moscow, 127055 Russia
| | - Franco Nori
- Theoretical Quantum Physics Laboratory, RIKEN Cluster for Pioneering Research, Wako-shi, Saitama 351-0198, Japan
- Department of Physics, University of Michigan, Ann Arbor, Michigan 48109-1040, USA
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Abstract
In a conventional semiconductor, when the dielectric screening effect is suppressed, the exciton binding energy increases and the corresponding excitonic transition would exhibit a redshift in the spectrum. In this work, I study the optical properties of hexagonal graphene nanodots by using a configuration interaction approach and reveal that the edge of the absorption spectrum shows an abnormal blueshift as the environmental dielectric constant ϵr decreases. The two dominant many-body effects in the nanodot: the quasiparticle and excitonic effects are both found to scale almost linearly with ϵr-1. The former is shown to have a larger proportionality constant and thus accounts for the blueshift of the absorption edge. In contrast to the long-range Coulomb interaction, the on-site Coulomb energy is found to have a negative impact on the bright excitonic states. In the presence of a strong dielectric screening effect, a strong short-range Coulomb interaction is revealed to be responsible for the disintegration of the bright exciton.
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Affiliation(s)
- Weidong Sheng
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, People's Republic of China and Collaborative Innovation Center of Advanced Microstructures, Nanjing 210093, China
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9
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Rudenko AN, Stepanov EA, Lichtenstein AI, Katsnelson MI. Excitonic Instability and Pseudogap Formation in Nodal Line Semimetal ZrSiS. PHYSICAL REVIEW LETTERS 2018; 120:216401. [PMID: 29883184 DOI: 10.1103/physrevlett.120.216401] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/21/2017] [Revised: 03/26/2018] [Indexed: 06/08/2023]
Abstract
Electron correlation effects are studied in ZrSiS using a combination of first-principles and model approaches. We show that basic electronic properties of ZrSiS can be described within a two-dimensional lattice model of two nested square lattices. A high degree of electron-hole symmetry characteristic for ZrSiS is one of the key features of this model. Having determined model parameters from first-principles calculations, we then explicitly take electron-electron interactions into account and show that, at moderately low temperatures, ZrSiS exhibits excitonic instability, leading to the formation of a pseudogap in the electronic spectrum. The results can be understood in terms of Coulomb-interaction-assisted pairing of electrons and holes reminiscent of that of an excitonic insulator. Our finding allows us to provide a physical interpretation of the unusual mass enhancement of charge carriers in ZrSiS recently observed experimentally.
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Affiliation(s)
- A N Rudenko
- School of Physics and Technology, Wuhan University, Wuhan 430072, China
- Institute for Molecules and Materials, Radboud University, Heijendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
- Theoretical Physics and Applied Mathematics Department, Ural Federal University, Mira Street 19, 620002 Ekaterinburg, Russia
| | - E A Stepanov
- Institute for Molecules and Materials, Radboud University, Heijendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
- Theoretical Physics and Applied Mathematics Department, Ural Federal University, Mira Street 19, 620002 Ekaterinburg, Russia
| | - A I Lichtenstein
- Institute for Theoretical Physics, University of Hamburg, Jungiusstrasse 9, D-20355 Hamburg, Germany
- Theoretical Physics and Applied Mathematics Department, Ural Federal University, Mira Street 19, 620002 Ekaterinburg, Russia
| | - M I Katsnelson
- Institute for Molecules and Materials, Radboud University, Heijendaalseweg 135, 6525 AJ Nijmegen, The Netherlands
- Theoretical Physics and Applied Mathematics Department, Ural Federal University, Mira Street 19, 620002 Ekaterinburg, Russia
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10
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Xu X, Liu C, Sun Z, Cao T, Zhang Z, Wang E, Liu Z, Liu K. Interfacial engineering in graphene bandgap. Chem Soc Rev 2018. [PMID: 29513306 DOI: 10.1039/c7cs00836h] [Citation(s) in RCA: 63] [Impact Index Per Article: 10.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/23/2022]
Abstract
Graphene exhibits superior mechanical strength, high thermal conductivity, strong light-matter interactions, and, in particular, exceptional electronic properties. These merits make graphene an outstanding material for numerous potential applications. However, a graphene-based high-performance transistor, which is the most appealing application, has not yet been produced, which is mainly due to the absence of an intrinsic electronic bandgap in this material. Therefore, bandgap opening in graphene is urgently needed, and great efforts have been made regarding this topic over the past decade. In this review article, we summarise recent theoretical and experimental advances in interfacial engineering to achieve bandgap opening. These developments are divided into two categories: chemical engineering and physical engineering. Chemical engineering is usually destructive to the pristine graphene lattice via chemical functionalization, the introduction of defects, doping, chemical bonds with substrates, and quantum confinement; the latter largely maintains the atomic structure of graphene intact and includes the application of an external field, interactions with substrates, physical adsorption, strain, electron many-body effects and spin-orbit coupling. Although these pioneering works have not met all the requirements for electronic applications of graphene at once, they hold great promise in this direction and may eventually lead to future applications of graphene in semiconductor electronics and beyond.
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Affiliation(s)
- Xiaozhi Xu
- State Key Laboratory for Mesoscopic Physics, School of Physics, Peking University, Beijing 100871, China.
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11
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Gonzalez-Arraga LA, Lado JL, Guinea F, San-Jose P. Electrically Controllable Magnetism in Twisted Bilayer Graphene. PHYSICAL REVIEW LETTERS 2017; 119:107201. [PMID: 28949176 DOI: 10.1103/physrevlett.119.107201] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/27/2017] [Indexed: 06/07/2023]
Abstract
Twisted graphene bilayers develop highly localized states around AA-stacked regions for small twist angles. We show that interaction effects may induce either an antiferromagnetic or a ferromagnetic (FM) polarization of said regions, depending on the electrical bias between layers. Remarkably, FM-polarized AA regions under bias develop spiral magnetic ordering, with a relative 120° misalignment between neighboring regions due to a frustrated antiferromagnetic exchange. This remarkable spiral magnetism emerges naturally without the need of spin-orbit coupling, and competes with the more conventional lattice-antiferromagnetic instability, which interestingly develops at smaller bias under weaker interactions than in monolayer graphene, due to Fermi velocity suppression. This rich and electrically controllable magnetism could turn twisted bilayer graphene into an ideal system to study frustrated magnetism in two dimensions.
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Affiliation(s)
| | - J L Lado
- QuantaLab, International Iberian Nanotechnology Laboratory (INL), Avenida Mestre Jose Veiga, 4715-330 Braga, Portugal
| | - Francisco Guinea
- IMDEA Nanociencia, Calle de Faraday, 9, Cantoblanco, 28049 Madrid, Spain
- School of Physics and Astronomy, University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
| | - Pablo San-Jose
- Instituto de Ciencia de Materiales de Madrid (ICMM-CSIC), Cantoblanco, 28049 Madrid, Spain
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12
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Zhang Y, Sheng W, Li Y. Dark excitons and tunable optical gap in graphene nanodots. Phys Chem Chem Phys 2017; 19:23131-23137. [DOI: 10.1039/c7cp04591c] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/21/2022]
Abstract
By using a configuration interaction approach with up to the fifth excitations taken into account, we study the excitonic effect in the optical absorption in graphene nanodots.
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Affiliation(s)
- Yingjie Zhang
- State Key Laboratory of Surface Physics and Department of Physics
- Fudan University
- Shanghai 200433
- China
| | - Weidong Sheng
- State Key Laboratory of Surface Physics and Department of Physics
- Fudan University
- Shanghai 200433
- China
- Collaborative Innovation Center of Advanced Microstructures
| | - Yang Li
- State Key Laboratory of Surface Physics and Department of Physics
- Fudan University
- Shanghai 200433
- China
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13
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Abstract
The binding energy of an exciton in a semiconductor or an insulator is known to scale linearly with εr-2, where εr is its dielectric constant. In graphene however, since the kinetic energy scales linearly with the wave number instead of its square, the exciton binding energy is thus expected to scale with εr-1. In this work we make use of the configuration interaction approach to study the properties of excitons in graphene nanodots embedded in various dielectric environments. With tens of million configurations taken into account in the calculation, we find that the exciton binding energy can be well described by a single scaling rule in which the scaling factor is found to vary with the dimension of the nanodots as well as with the on-site interaction parameter, which agrees well with a recent experiment. The linear relation of the exciton binding energy found with the quasi-particle gap also agrees with the previous work on bulk graphene and other two-dimensional materials.
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Affiliation(s)
- Weidong Sheng
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China. and Collaborative Innovation Center of Advanced Microstructures, Nanjing, 210093, China
| | - Hao Wang
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai 200433, China.
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Cao Y, Luo JY, Fatemi V, Fang S, Sanchez-Yamagishi JD, Watanabe K, Taniguchi T, Kaxiras E, Jarillo-Herrero P. Superlattice-Induced Insulating States and Valley-Protected Orbits in Twisted Bilayer Graphene. PHYSICAL REVIEW LETTERS 2016; 117:116804. [PMID: 27661712 DOI: 10.1103/physrevlett.117.116804] [Citation(s) in RCA: 137] [Impact Index Per Article: 17.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/07/2016] [Indexed: 05/25/2023]
Abstract
Twisted bilayer graphene (TBLG) is one of the simplest van der Waals heterostructures, yet it yields a complex electronic system with intricate interplay between moiré physics and interlayer hybridization effects. We report on electronic transport measurements of high mobility small angle TBLG devices showing clear evidence for insulating states at the superlattice band edges, with thermal activation gaps several times larger than theoretically predicted. Moreover, Shubnikov-de Haas oscillations and tight binding calculations reveal that the band structure consists of two intersecting Fermi contours whose crossing points are effectively unhybridized. We attribute this to exponentially suppressed interlayer hopping amplitudes for momentum transfers larger than the moiré wave vector.
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Affiliation(s)
- Y Cao
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - J Y Luo
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - V Fatemi
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
| | - S Fang
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
| | | | - K Watanabe
- National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan
| | - T Taniguchi
- National Institute for Materials Science, Namiki 1-1, Tsukuba, Ibaraki 305-0044, Japan
| | - E Kaxiras
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
- John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, Massachusetts 02138, USA
| | - P Jarillo-Herrero
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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15
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Pujari S, Lang TC, Murthy G, Kaul RK. Interaction-Induced Dirac Fermions from Quadratic Band Touching in Bilayer Graphene. PHYSICAL REVIEW LETTERS 2016; 117:086404. [PMID: 27588872 DOI: 10.1103/physrevlett.117.086404] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/26/2016] [Indexed: 06/06/2023]
Abstract
We revisit the effect of local interactions on the quadratic band touching (QBT) of the Bernal honeycomb bilayer model using renormalization group (RG) arguments and quantum Monte Carlo (QMC) simulations. We present a RG argument which predicts, contrary to previous studies, that weak interactions do not flow to strong coupling even if the free dispersion has a QBT. Instead, they generate a linear term in the dispersion, which causes the interactions to flow back to weak coupling. Consistent with this RG scenario, in unbiased QMC simulations of the Hubbard model we find compelling evidence that antiferromagnetism turns on at a finite U/t despite the U=0 hopping problem having a QBT. The onset of antiferromagnetism takes place at a continuous transition which is consistent with (2+1)D Gross-Neveu criticality. We conclude that generically in models of bilayer graphene, even if the free dispersion has a QBT, small local interactions generate a Dirac phase with no symmetry breaking and that there is a finite-coupling transition out of this phase to a symmetry-broken state.
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Affiliation(s)
- Sumiran Pujari
- Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506-0055, USA
| | - Thomas C Lang
- Institute for Theoretical Physics, University of Innsbruck, 6020 Innsbruck, Austria
| | - Ganpathy Murthy
- Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506-0055, USA
| | - Ribhu K Kaul
- Department of Physics and Astronomy, University of Kentucky, Lexington, Kentucky 40506-0055, USA
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16
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Yan J, Li C, Zhan D, Liu L, Shen D, Kuo JL, Chen S, Shen Z. Graphene homojunction: closed-edge bilayer graphene by pseudospin interaction. NANOSCALE 2016; 8:9102-9106. [PMID: 26809883 DOI: 10.1039/c5nr08083e] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Depending on the sublattices they are propagated in, low-energy electrons or holes are labeled with pseudospin. By engineering pseudospin interactions, we propose that two critical features of a junction, i.e., band gap opening and spatial charge separation, can be realized in graphene layers with proper stacking. We also demonstrate theoretically that such a graphene diode may play a role in future pseudospin electronics such as for harvesting solar energy.
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Affiliation(s)
- Jiaxu Yan
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore.
| | - Chao Li
- College of Physics, Jilin University, Changchun 130012, People's Republic of China
| | - Da Zhan
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore.
| | - Lei Liu
- Key Laboratory of Excited State Processes, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, People's Republic of China.
| | - Dezhen Shen
- Key Laboratory of Excited State Processes, Changchun Institute of Optics, Fine Mechanics and Physics, Chinese Academy of Sciences, Changchun 130033, People's Republic of China.
| | - Jer-Lai Kuo
- Institute of Atomic and Molecular Sciences, Academia Sinica, Taipei 10617, Taiwan
| | - Shoushun Chen
- School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798, Singapore
| | - Zexiang Shen
- Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore 637371, Singapore.
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17
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Sheng W, Luo K, Zhou A. Communication: Generalization of Koopmans’ theorem to optical transitions in the Hubbard model of graphene nanodots. J Chem Phys 2015; 142:021102. [DOI: 10.1063/1.4905789] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Weidong Sheng
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai, China
- Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China
- Department of Physics, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada
| | - Kaikai Luo
- State Key Laboratory of Surface Physics and Department of Physics, Fudan University, Shanghai, China
| | - Aiping Zhou
- Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing 211167, China
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18
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Li X, Zhang F, Niu Q, MacDonald AH. Spontaneous layer-pseudospin domain walls in bilayer graphene. PHYSICAL REVIEW LETTERS 2014; 113:116803. [PMID: 25259998 DOI: 10.1103/physrevlett.113.116803] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/15/2014] [Indexed: 06/03/2023]
Abstract
Bilayer graphene is susceptible to a family of unusual broken symmetry states with spin and valley dependent layer polarization. We report on a microscopic study of the domain walls in these systems, demonstrating that they have interesting microscopic structure related to the topological character of the ordered states. We use our results to show that the metal-insulator transition temperature in bilayer graphene is reduced from mean-field estimates by thermal excitation of domain walls.
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Affiliation(s)
- Xiao Li
- Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
| | - Fan Zhang
- Department of Physics, University of Texas at Dallas, Richardson, Texas 75080, USA
| | - Qian Niu
- Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA and School of Physics, International Center for Quantum Materials and Collaborative Innovation Center of Quantum Matter, Peking University, Beijing 100871, China
| | - A H MacDonald
- Department of Physics, The University of Texas at Austin, Austin, Texas 78712, USA
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19
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Kou A, Feldman BE, Levin AJ, Halperin BI, Watanabe K, Taniguchi T, Yacoby A. Electron-hole asymmetric integer and fractional quantum Hall effect in bilayer graphene. Science 2014; 345:55-7. [DOI: 10.1126/science.1250270] [Citation(s) in RCA: 84] [Impact Index Per Article: 8.4] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
Affiliation(s)
- A. Kou
- Department of Physics, Harvard University, Cambridge, MA 02138, USA
- Department of Applied Physics, Yale University, New Haven, CT 06520, USA
| | - B. E. Feldman
- Department of Physics, Harvard University, Cambridge, MA 02138, USA
| | - A. J. Levin
- Department of Physics, Harvard University, Cambridge, MA 02138, USA
| | - B. I. Halperin
- Department of Physics, Harvard University, Cambridge, MA 02138, USA
| | - K. Watanabe
- National Institute for Materials Science, Tsukuba, Japan
| | - T. Taniguchi
- National Institute for Materials Science, Tsukuba, Japan
| | - A. Yacoby
- Department of Physics, Harvard University, Cambridge, MA 02138, USA
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20
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Vafek O, Murray JM, Cvetkovic V. Superconductivity on the brink of spin-charge order in a doped honeycomb bilayer. PHYSICAL REVIEW LETTERS 2014; 112:147002. [PMID: 24766005 DOI: 10.1103/physrevlett.112.147002] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/12/2013] [Indexed: 06/03/2023]
Abstract
Using a controlled weak-coupling renormalization group approach, we establish the mechanism of unconventional superconductivity in the vicinity of spin or charge ordered excitonic states for the case of electrons on the Bernal stacked bilayer honeycomb lattice. With one electron per site, this system, physically realized in bilayer graphene, is unstable towards a spontaneous symmetry breaking. Repulsive interactions favor excitonic order, such as a charge nematic and/or a layer antiferromagnet. We find that upon adding charge carriers to the system, the excitonic order is suppressed, and unconventional superconductivity appears in its place, before it is replaced by a Fermi liquid. We focus on firmly establishing this phenomenon using the renormalization group formalism within an idealized model with parabolic touching of conduction and valence bands.
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Affiliation(s)
- Oskar Vafek
- National High Magnetic Field Laboratory and Department of Physics, Florida State University, Tallahasse, Florida 32306, USA
| | - James M Murray
- National High Magnetic Field Laboratory and Department of Physics, Florida State University, Tallahasse, Florida 32306, USA and Department of Physics and Astronomy, Johns Hopkins University, Baltimore, Maryland 21218, USA
| | - Vladimir Cvetkovic
- National High Magnetic Field Laboratory and Department of Physics, Florida State University, Tallahasse, Florida 32306, USA
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21
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McCann E, Koshino M. The electronic properties of bilayer graphene. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2013; 76:056503. [PMID: 23604050 DOI: 10.1088/0034-4885/76/5/056503] [Citation(s) in RCA: 208] [Impact Index Per Article: 18.9] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/21/2023]
Abstract
We review the electronic properties of bilayer graphene, beginning with a description of the tight-binding model of bilayer graphene and the derivation of the effective Hamiltonian describing massive chiral quasiparticles in two parabolic bands at low energies. We take into account five tight-binding parameters of the Slonczewski-Weiss-McClure model of bulk graphite plus intra- and interlayer asymmetry between atomic sites which induce band gaps in the low-energy spectrum. The Hartree model of screening and band-gap opening due to interlayer asymmetry in the presence of external gates is presented. The tight-binding model is used to describe optical and transport properties including the integer quantum Hall effect, and we also discuss orbital magnetism, phonons and the influence of strain on electronic properties. We conclude with an overview of electronic interaction effects.
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Affiliation(s)
- Edward McCann
- Department of Physics, Lancaster University, Lancaster LA1 4YB, UK
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22
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Mkhitaryan VV, Mishchenko EG. Localized states due to expulsion of resonant impurity levels from the continuum in bilayer graphene. PHYSICAL REVIEW LETTERS 2013; 110:086805. [PMID: 23473187 DOI: 10.1103/physrevlett.110.086805] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/13/2012] [Indexed: 06/01/2023]
Abstract
The Anderson impurity problem is considered for a graphene bilayer subject to a gap-opening bias. In-gap localized states are produced even when the impurity level overlaps with the continuum of band electrons. The effect depends strongly on the polarity of the applied bias as long as hybridization with the impurity occurs within a single layer. For an impurity level inside the conduction band a positive bias creates the new localized in-gap state. A negative bias does not produce the same result and leads to a simple broadening of the impurity level. The implications for transport are discussed including a possibility of the gate-controlled Kondo effect.
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Affiliation(s)
- V V Mkhitaryan
- Department of Physics and Astronomy, University of Utah, Salt Lake City, Utah 84112, USA
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23
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Rakhmanov AL, Rozhkov AV, Sboychakov AO, Nori F. Instabilities of the AA-stacked graphene bilayer. PHYSICAL REVIEW LETTERS 2012; 109:206801. [PMID: 23215515 DOI: 10.1103/physrevlett.109.206801] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/21/2011] [Indexed: 06/01/2023]
Abstract
Tight-binding calculations predict that the AA-stacked bilayer graphene has one electron and one hole conducting band, and that the Fermi surfaces of these bands coincide. We demonstrate that as a result of this degeneracy, the bilayer becomes unstable with respect to a set of spontaneous symmetry violations. Which of the symmetries is broken depends on the microscopic details of the system. For strong on-site Coulomb interaction we find that antiferromagnetism is the most stable order parameter. For an on-site repulsion energy typical for graphene systems, the antiferromagnetic gap can exist up to room temperature.
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Affiliation(s)
- A L Rakhmanov
- Advanced Science Institute, RIKEN, Wako-shi, Saitama, 351-0198, Japan
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24
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Chern GW, Batista CD. Spontaneous quantum Hall effect via a thermally induced quadratic Fermi point. PHYSICAL REVIEW LETTERS 2012; 109:156801. [PMID: 23102349 DOI: 10.1103/physrevlett.109.156801] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/30/2012] [Indexed: 06/01/2023]
Abstract
Gapless electronic systems containing topologically nontrivial Fermi points are sources of various topological insulators. Whereas, most of these special band-crossing points are built in the electronic structure of the noninteracting lattice models, we show that a quadratic Fermi point characterized by a nonzero winding number emerges with a collinear triple-Q spin-density-wave state that arises from a perfectly nested but topologically trivial Fermi surface. We obtain a universal low-energy Hamiltonian for the quadratic Fermi point and show that such collinear orderings are unstable against the onset of scalar spin chirality that opens a gap and induces a spontaneous quantum Hall insulator as the temperature tends to zero.
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Affiliation(s)
- Gia-Wei Chern
- Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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25
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Lang TC, Meng ZY, Scherer MM, Uebelacker S, Assaad FF, Muramatsu A, Honerkamp C, Wessel S. Antiferromagnetism in the Hubbard model on the Bernal-stacked honeycomb bilayer. PHYSICAL REVIEW LETTERS 2012; 109:126402. [PMID: 23005964 DOI: 10.1103/physrevlett.109.126402] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/02/2012] [Indexed: 06/01/2023]
Abstract
Using a combination of quantum Monte Carlo simulations, functional renormalization group calculations and mean-field theory, we study the Hubbard model on the Bernal-stacked honeycomb bilayer at half-filling as a model system for bilayer graphene. The free bands consisting of two Fermi points with quadratic dispersions lead to a finite density of states at the Fermi level, which triggers an antiferromagnetic instability that spontaneously breaks sublattice and spin rotational symmetry once local Coulomb repulsions are introduced. Our results reveal an inhomogeneous participation of the spin moments in the ordered ground state, with enhanced moments at the threefold coordinated sites. Furthermore, we find the antiferromagnetic ground state to be robust with respect to enhanced interlayer couplings and extended Coulomb interactions.
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Affiliation(s)
- Thomas C Lang
- Institute for Theoretical Solid State Physics, RWTH Aachen University, Germany.
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26
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Kharitonov M. Canted antiferromagnetic phase of the ν=0 quantum Hall state in bilayer graphene. PHYSICAL REVIEW LETTERS 2012; 109:046803. [PMID: 23006103 DOI: 10.1103/physrevlett.109.046803] [Citation(s) in RCA: 10] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/26/2011] [Indexed: 06/01/2023]
Abstract
Motivated to understand the nature of the strongly insulating ν=0 quantum Hall state in bilayer graphene, we develop the theory of the state in the framework of quantum Hall ferromagnetism. The generic phase diagram, obtained in the presence of the isospin anisotropy, perpendicular electric field, and Zeeman effect, consists of the spin-polarized ferromagnetic (F), canted antiferromagnetic (CAF), and partially (PLP) and fully (FLP) layer-polarized phases. We address the edge transport properties of the phases. Comparing our findings with the recent data on suspended dual-gated devices, we conclude that the insulating ν=0 state realized in bilayer graphene at lower electric field is the CAF phase. We also predict a continuous and a sharp insulator-metal phase transition upon tilting the magnetic field from the insulating CAF and FLP phases, respectively, to the F phase with metallic edge conductance 2e(2)/h, which could be within the reach of available fields and could allow one to identify and distinguish the phases experimentally.
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Affiliation(s)
- Maxim Kharitonov
- Center for Materials Theory, Rutgers University, Piscataway, New Jersey 08854, USA
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27
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Zhang F, MacDonald AH. Distinguishing spontaneous quantum Hall states in bilayer graphene. PHYSICAL REVIEW LETTERS 2012; 108:186804. [PMID: 22681103 DOI: 10.1103/physrevlett.108.186804] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/22/2011] [Indexed: 06/01/2023]
Abstract
Chirally stacked N-layer graphene with N≥2 is susceptible to a variety of distinct broken symmetry states in which each spin-valley flavor spontaneously transfers charge between layers. In mean-field theory, one of the likely candidate ground states for a neutral bilayer is the layer antiferromagnet that has opposite spin polarizations in opposite layers. In this Letter, we analyze how the layer antiferromagnet and other competing states are influenced by Zeeman fields that couple to spin and by interlayer electric fields that couple to layer pseudospin, and comment on the possibility of using Zeeman responses and edge state signatures to identify the character of the bilayer ground state experimentally.
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Affiliation(s)
- Fan Zhang
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA.
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28
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Freitag F, Trbovic J, Weiss M, Schönenberger C. Spontaneously gapped ground state in suspended bilayer graphene. PHYSICAL REVIEW LETTERS 2012; 108:076602. [PMID: 22401232 DOI: 10.1103/physrevlett.108.076602] [Citation(s) in RCA: 14] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/23/2011] [Indexed: 05/31/2023]
Abstract
Bilayer graphene bears an eightfold degeneracy due to spin, valley, and layer symmetry, allowing for a wealth of broken symmetry states induced by magnetic or electric fields, by strain, or even spontaneously by interaction. We study the electrical transport in clean current annealed suspended bilayer graphene. We find two kinds of devices. In bilayers of type B1 the eightfold zero-energy Landau level is partially lifted above a threshold field revealing an insulating ν=0 quantum-Hall state at the charge neutrality point. In bilayers of type B2 the Landau level lifting is full and a gap appears in the differential conductance even at zero magnetic field, suggesting an insulating spontaneously broken symmetry state. Unlike B1, the minimum conductance in B2 is not exponentially suppressed, but remains finite with a value G is < or approximately equall to e(2)/h even in a large magnetic field. We suggest that this phase of B2 is insulating in the bulk and bound by compressible edge states.
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Affiliation(s)
- F Freitag
- Department of Physics, University of Basel, Basel, Switzerland
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29
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Barlas Y, Yang K, MacDonald AH. Quantum Hall effects in graphene-based two-dimensional electron systems. NANOTECHNOLOGY 2012; 23:052001. [PMID: 22238249 DOI: 10.1088/0957-4484/23/5/052001] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/31/2023]
Abstract
In this article we review the quantum Hall physics of graphene-based two-dimensional electron systems, with a special focus on recent experimental and theoretical developments. We explain why graphene and bilayer graphene can be viewed respectively as J D 1 and 2 chiral two-dimensional electron gases (C2DEGs), and why this property frames their quantum Hall physics. The current status of experimental and theoretical work on the role of electron-electron interactions is reviewed at length with an emphasis on unresolved issues in the field, including the role of disorder in current experiments. Special attention is given to the interesting low magnetic field limit, and to the relationship between quantum Hall effects and the spontaneous anomalous Hall effects that might occur in bilayer graphene systems in the absence of a magnetic field.
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Affiliation(s)
- Yafis Barlas
- National High Magnetic Field Laboratory and Department of Physics, Florida State University, FL 32306, USA
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30
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Zhang F, Jung J, MacDonald AH. Spontaneous Quantum Hall States and Novel Luttinger Liquids in Chiral Graphene. ACTA ACUST UNITED AC 2011. [DOI: 10.1088/1742-6596/334/1/012002] [Citation(s) in RCA: 8] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022]
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31
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Nandkishore R, Levitov L. Polar Kerr effect and time reversal symmetry breaking in bilayer graphene. PHYSICAL REVIEW LETTERS 2011; 107:097402. [PMID: 21929269 DOI: 10.1103/physrevlett.107.097402] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/25/2011] [Indexed: 05/31/2023]
Abstract
The unique sensitivity of optical response to different types of symmetry breaking can be used to detect and identify spontaneously ordered many-body states in bilayer graphene. We predict a strong response at optical frequencies, sensitive to electronic phenomena at low energies, which arises because of nonzero interband matrix elements of the electric current operator. In particular, the polar Kerr rotation and reflection anisotropy provide fingerprints of the quantum anomalous Hall state and the nematic state, characterized by spontaneously broken time-reversal symmetry and lattice rotation symmetry, respectively. These optical signatures, which undergo a resonant enhancement in the near-infrared regime, lie well within reach of existing experimental techniques.
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Affiliation(s)
- Rahul Nandkishore
- Department of Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
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32
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Mayorov AS, Elias DC, Mucha-Kruczynski M, Gorbachev RV, Tudorovskiy T, Zhukov A, Morozov SV, Katsnelson MI, Geim AK, Novoselov KS. Interaction-Driven Spectrum Reconstruction in Bilayer Graphene. Science 2011; 333:860-3. [DOI: 10.1126/science.1208683] [Citation(s) in RCA: 236] [Impact Index Per Article: 18.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
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33
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Lv M, Wan S. Orbital diamagnetism of weakly doped bilayer graphene in a magnetic field. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2011; 23:215306. [PMID: 21558597 DOI: 10.1088/0953-8984/23/21/215306] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/30/2023]
Abstract
We investigate the orbital diamagnetism of weakly doped bilayer graphene (BLG) in a spatially smoothly varying magnetic field and obtain the general analytic expression for the orbital susceptibility of BLG, with finite wavenumber and Fermi energy, at zero temperature. We find that the magnetic field screening factor of BLG is dependent on the wavenumber, which results in a more complicated screening behavior compared with that of monolayer graphene (MLG). We also study the induced magnetization and electric current in BLG, under a nonuniform magnetic field, and find that they are qualitatively different from those for MLG and the two-dimensional electron gas (2DEG). However, as for MLG, a magnetic object placed above BLG is repelled by a diamagnetic force from the BLG, which is approximately equivalent to the force produced by its mirror image on the other side of the BLG with a reduced amplitude dependent on the typical length of the systems. BLG shows crossover behaviors in the responses to the external magnetic field, intermediate between those of MLG and 2DEG.
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Affiliation(s)
- Min Lv
- Institute for Theoretical Physics, University of Science and Technology of China, Hefei, People's Republic of China
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34
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Zhang F, Jung J, Fiete GA, Niu Q, MacDonald AH. Spontaneous quantum Hall states in chirally stacked few-layer graphene systems. PHYSICAL REVIEW LETTERS 2011; 106:156801. [PMID: 21568592 DOI: 10.1103/physrevlett.106.156801] [Citation(s) in RCA: 123] [Impact Index Per Article: 9.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/19/2010] [Indexed: 05/30/2023]
Abstract
Chirally stacked N-layer graphene systems with N≥2 exhibit a variety of distinct broken symmetry states in which charge density contributions from different spins and valleys are spontaneously transferred between layers. We explain how these states are distinguished by their charge, spin, and valley Hall conductivities, by their orbital magnetizations, and by their edge state properties. We argue that valley Hall states have [N/2] edge channels per spin valley.
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Affiliation(s)
- Fan Zhang
- Department of Physics, University of Texas at Austin, Austin, Texas 78712, USA.
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35
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Martin J, Feldman BE, Weitz RT, Allen MT, Yacoby A. Local compressibility measurements of correlated states in suspended bilayer graphene. PHYSICAL REVIEW LETTERS 2010; 105:256806. [PMID: 21231612 DOI: 10.1103/physrevlett.105.256806] [Citation(s) in RCA: 38] [Impact Index Per Article: 2.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 09/08/2010] [Indexed: 05/30/2023]
Abstract
Bilayer graphene has attracted considerable interest due to the important role played by many-body effects, particularly at low energies. Here we report local compressibility measurements of a suspended graphene bilayer. We find that the energy gaps at filling factors ν= ± 4 do not vanish at low fields, but instead merge into an incompressible region near the charge neutrality point at zero electric and magnetic field. These results indicate the existence of a zero-field ordered state and are consistent with the formation of either an anomalous quantum Hall state or a nematic phase with broken rotational symmetry. At higher fields, we measure the intrinsic energy gaps of broken-symmetry states at ν=0, ± 1, and ± 2, and find that they scale linearly with magnetic field, yet another manifestation of the strong Coulomb interactions in bilayer graphene.
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Affiliation(s)
- J Martin
- Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA
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