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Introduction to the Physical Properties of Graphene
Introduction to the Physical Properties of Graphene Jean-No¨el FUCHS Mark Oliver GOERBIG Lecture Notes 2008 ii Contents 1 Introduction to Carbon Materials 1 1.1 TheCarbonAtomanditsHybridisations . 3 1.1.1 sp1 hybridisation ..................... 4 1.1.2 sp2 hybridisation – graphitic allotopes . 6 1.1.3 sp3 hybridisation – diamonds . 9 1.2 Crystal StructureofGrapheneand Graphite . 10 1.2.1 Graphene’s honeycomb lattice . 10 1.2.2 Graphene stacking – the different forms of graphite . 13 1.3 FabricationofGraphene . 16 1.3.1 Exfoliatedgraphene. 16 1.3.2 Epitaxialgraphene . 18 2 Electronic Band Structure of Graphene 21 2.1 Tight-Binding Model for Electrons on the Honeycomb Lattice 22 2.1.1 Bloch’stheorem. 23 2.1.2 Lattice with several atoms per unit cell . 24 2.1.3 Solution for graphene with nearest-neighbour and next- nearest-neighour hopping . 27 2.2 ContinuumLimit ......................... 33 2.3 Experimental Characterisation of the Electronic Band Structure 41 3 The Dirac Equation for Relativistic Fermions 45 3.1 RelativisticWaveEquations . 46 3.1.1 Relativistic Schr¨odinger/Klein-Gordon equation . ... 47 3.1.2 Diracequation ...................... 49 3.2 The2DDiracEquation. 53 3.2.1 Eigenstates of the 2D Dirac Hamiltonian . 54 3.2.2 Symmetries and Lorentz transformations . 55 iii iv 3.3 Physical Consequences of the Dirac Equation . 61 3.3.1 Minimal length for the localisation of a relativistic par- ticle ............................ 61 3.3.2 Velocity operator and “Zitterbewegung” . 61 3.3.3 Klein tunneling and the absence of backscattering . 61 Chapter 1 Introduction to Carbon Materials The experimental and theoretical study of graphene, two-dimensional (2D) graphite, is an extremely rapidly growing field of today’s condensed matter research. -
Recent Progress in Quantum Matter – Theories and Experiments January 7 – 9, 2019 Lecture Theatre T7, 1/F, Meng Wah Complex
Hong Kong Forum of Physics 2018: Recent Progress in Quantum Matter – Theories and Experiments January 7 – 9, 2019 Lecture Theatre T7, 1/F, Meng Wah Complex Poster Room Rm. 522, 5/F, Chong Yuet Ming Physics Building, HKU Organized by The Centre of Theoretical and Computational Physics, HKU Organizing Committee Gang Chen (HKU) DongKeun Ki (HKU) Chenjie Wang (HKU) Jian Wang (HKU) Zidan Wang (HKU) Wang Yao (HKU) Advisory Committee Anthony J. Leggett (Illinois, Urbana-Champaign) Philip Kim (Harvard University) Steve Louie (University of California, Berkeley) T. Maurice Rice (ETH Zurich) Fuchun Zhang (University of Chinese Academy of Science) Sponsored by K. C. Wong Education Foundation Department of Physics & The Centre of Theoretical and Computational Physics, HKU Abstract for Talks 2 Monday, January 7, 2019 (Day 1) 09:20 – 09:50 Topological Superfluid for 1D Spin-1 Bosons and Its Continuous Phase Transition Prof. Xiao-Gang Wen Massachusetts Institute of Technology Spin-1 bosons on a 1-dimensional chain, with anti-ferromagnetic spin interaction between neighboring bosons, may form a spin-1 boson condensed state that contains both gapless charge and spin excitations. We argue that the spin-1 boson condensed state is unstable, and will become one of two superfluids by opening a spin gap. One superfluid must have spin-1 ground state on a ring if it contains an odd number of bosons and has no degenerate states at the chain end. The other superfluid has spin-0 ground state on a ring for any numbers of bosons and has a spin-1/2 degeneracy at the chain end. -
ANDRE K. GEIM School of Physics and Astronomy, the University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom
RANDOM WALK TO GRAPHENE Nobel Lecture, December 8, 2010 by ANDRE K. GEIM School of Physics and Astronomy, The University of Manchester, Oxford Road, Manchester M13 9PL, United Kingdom. If one wants to understand the beautiful physics of graphene, they will be spoiled for choice with so many reviews and popular science articles now available. I hope that the reader will excuse me if on this occasion I recommend my own writings [1–3]. Instead of repeating myself here, I have chosen to describe my twisty scientific road that eventually led to the Nobel Prize. Most parts of this story are not described anywhere else, and its time- line covers the period from my PhD in 1987 to the moment when our 2004 paper, recognised by the Nobel Committee, was accepted for publication. The story naturally gets denser in events and explanations towards the end. Also, it provides a detailed review of pre-2004 literature and, with the benefit of hindsight, attempts to analyse why graphene has attracted so much inter- est. I have tried my best to make this article not only informative but also easy to read, even for non-physicists. ZOMBIE MANAGEMENT My PhD thesis was called “Investigation of mechanisms of transport relaxa- tion in metals by a helicon resonance method”. All I can say is that the stuff was as interesting at that time as it sounds to the reader today. I published five journal papers and finished the thesis in five years, the official duration for a PhD at my institution, the Institute of Solid State Physics. -
Modulation of Mechanical Resonance by Chemical Potential Oscillation in Graphene
Modulation of mechanical resonance by chemical potential oscillation in graphene The Harvard community has made this article openly available. Please share how this access benefits you. Your story matters Citation Chen, Changyao, Vikram V. Deshpande, Mikito Koshino, Sunwoo Lee, Alexander Gondarenko, Allan H. MacDonald, Philip Kim, and James Hone. 2015. “Modulation of Mechanical Resonance by Chemical Potential Oscillation in Graphene.” Nature Physics 12 (3) (December 7): 240–244. doi:10.1038/nphys3576. Published Version doi:10.1038/nphys3576 Citable link http://nrs.harvard.edu/urn-3:HUL.InstRepos:34309591 Terms of Use This article was downloaded from Harvard University’s DASH repository, and is made available under the terms and conditions applicable to Other Posted Material, as set forth at http:// nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of- use#LAA Modulation of mechanical resonance by chemical poten- tial oscillation in graphene 1 2 3 4 1 Changyao Chen †, Vikram V.Deshpande , Mikito Koshino , Sunwoo Lee , Alexander Gondarenko , 5 6 1 Allan H. MacDonald , Philip Kim & James Hone ⇤ 1Department of Mechanical Engineering, Columbia University, New York, NY 10027, USA 2Department of Physics and Astronomy, University of Utah, Salt Lake City, UT, 84112, USA 3Department of Physics, Tohoku University, Sendai 980-8578, Japan 4Department of Electrical Engineering, Columbia University, New York, NY 10027, USA 5Department of Physics, University of Texas, Austin, TX 78712, USA 6Department of Physics, Harvard University, Cambridge, MA, 02138, USA † Current address: Center for Nanoscale Materials, Argonne National Laboratory, Lemont, IL, 60439, USA ⇤ Corresponding email: [email protected] The classical picture of the force on a capacitor assumes a large density of electronic states, such that the electrochemical potential of charges added to the capacitor is given by the ex- ternal electrostatic potential and the capacitance is determined purely by geometry. -
Carbon Nanotube and Graphene Nanoelectromechanical Systems by Benjamın José Alemán a Dissertation Submitted in Partial Satisf
Carbon Nanotube and Graphene Nanoelectromechanical Systems by Benjam´ınJos´eAlem´an A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Physics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Alex Zettl, Chair Professor Carlos Bustamante Professor Tsu Jae King Liu Fall 2011 Carbon Nanotube and Graphene Nanoelectromechanical Systems Copyright 2011 by Benjam´ınJos´eAlem´an 1 Abstract Carbon Nanotube and Graphene Nanoelectromechanical Systems by Benjam´ınJos´eAlem´an Doctor of Philosophy in Physics University of California, Berkeley Professor Alex Zettl, Chair One-dimensional and two-dimensional forms of carbon are composed of sp2-hybridized carbon atoms arranged in a regular hexagonal, honeycomb lattice. The two-dimension- al form, called graphene, is a single atomic layer of hexagonally-bonded carbon atoms. The one-dimensional form, known as a carbon nanotube, can be conceptualized as a rectangular piece of graphene wrapped into a seamless, high-aspect-ratio cylinder or tube. This dissertation addresses the physics and applied physics of these one and two-dimensional carbon allotropes in nanoelectromechanical systems (NEMS). First, we give a theoretical background on the electrodynamics and mechanics of carbon nanotube NEMS. We then describe basic experimental techniques, such as electron and scanning probe microscopy, that we then use to probe static and dynamic mechanical and electronic behavior of the carbon nanotube NEMS. For ex- ample, we observe and control non-linear beam bending and single-electron quantum tunneling effects in carbon nanotube resonators. We then describe parametric ampli- fication, self-oscillation behavior, and dynamic, non-linear effects in carbon nanotube mechanical resonators. -
Probing Thermal Expansion of Graphene and Modal Dispersion at Low-Temperature Using Graphene NEMS Resonators
Probing thermal expansion of graphene and modal dispersion at low-temperature using graphene NEMS resonators Vibhor Singh1, Shamashis Sengupta1, Hari S. Solanki1, Rohan Dhall1, Adrien Allain1, Sajal Dhara1, Prita Pant2 and Mandar M. Deshmukh1 1Department of Condensed Matter Physics, TIFR, Homi Bhabha Road, Mumbai 400005 India 2Department of Metallurgical Engineering and Materials Science, IIT Bombay Powai, Mumbai : 400076, India E-mail: [email protected] Abstract. We use suspended graphene electromechanical resonators to study the variation of resonant frequency as a function of temperature. Measuring the change in frequency resulting from a change in tension, from 300 K to 30 K, allows us to extract information about the thermal expansion of monolayer graphene as a function of temperature, which is critical for strain engineering applications. We ¯nd that thermal expansion of graphene is negative for all temperatures between 300K and 30K. We also study the dispersion, the variation of resonant frequency with DC gate voltage, of the electromechanical modes and ¯nd considerable tunability of resonant frequency, desirable for applications like mass sensing and RF signal processing at room temperature. With lowering of temperature, we ¯nd that the positively dispersing electromechanical modes evolve to negatively dispersing ones. We quantitatively explain this crossover and discuss optimal electromechanical properties that are desirable for temperature compensated sensors. PACS numbers: 85.85.+j, 81.05.ue, 65.40.De, 65.60.+a Graphene NEMS resonator 2 1. Introduction Electronic properties of graphene have been studied extensively [1, 2] since the ¯rst experiments probing quantum Hall e®ect [3, 4]. In addition to the electronic properties, the remarkable mechanical properties of graphene include a high in-plane Young's modulus of »1 TPa probed using nanoindentation of suspended graphene[5], force extension measurements [6], and electromechanical resonators [7, 8, 9]. -
Spatially Resolved Electrical Transport In
SPATIALLY RESOLVED ELECTRICAL TRANSPORT IN CARBON-BASED NANOMATERIALS A Dissertation Presented to the Faculty of the Graduate School of Cornell University In Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy by Adam Wei Tsen January 2013 © 2013 Adam Wei Tsen SPATIALLY RESOLVED ELECTRICAL TRANSPORT IN CARBON-BASED NANOMATERIALS Adam Wei Tsen, Ph. D. Cornell University 2013 Nanoscale materials based on the element carbon have attracted tremendous attention over the years from a diverse array of scientific disciplines. There is particular interest in the development of such materials for electronic device applications, thus requiring comprehensive studies of their electrical transport properties. However, in systems with reduced dimensionality, the electrical behavior may show significant variation depending on the local physical structure. Therefore, it would be most ideal to understand these two facets simultaneously. In this thesis, we study electrical transport in carbon nanotubes, pentacene thin films, and graphene in a spatially resolved manner in combination with two different microscopy techniques. With photoelectrical microscopy, we first image the electrical conductance and transport barriers for individual carbon nanotubes. We then resolve the precise points where charge injection takes place in pentacene thin-film transistors and explicitly determine the resistance for each point contact using the same technique. Finally, we study the polycrystalline structure of large-area graphene films with transmission electron microscopy and measure the electrical properties of individual grain boundaries. BIOGRAPHICAL SKETCH Wei grew up in Palo Alto, California and attended Henry M. Gunn High School. In 2006, he received a B.S. in Electrical Engineering and Computer Sciences as well as in Engineering Physics from the University of California, Berkeley.