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Electronic properties of graphene: a perspective from scanning tunneling microscopy and

magnetotransport

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2012 Rep. Prog. Phys. 75 056501

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IOP PUBLISHING REPORTS ONPROGRESS INPHYSICS

Rep. Prog. Phys.75(2012) 056501 (47pp) doi:10.1088/0034-4885/75/5/056501

Electronic properties of graphene: aperspective from scanning tunnelingmicroscopy and magnetotransport

Eva Y Andrei1, Guohong Li1 and Xu Du2

1 Department of Physics and Astronomy, Rutgers University, Piscataway, NJ 08855, USA2 Department of Physics, SUNY at Stony Brook, NY, USA

Received 12 October 2011, in final form 5 February 2012Published 19 April 2012Online atstacks.iop.org/RoPP/75/056501

Abstract

This review covers recent experimental progress in probing the electronic properties ofgraphene and how they are influenced by various substrates, by the presence of a magneticfield and by the proximity to a superconductor. The focus is on results obtained using scanningtunneling microscopy, spectroscopy, transport and magnetotransport techniques.

(Some figures may appear in colour only in the online journal)

This article was invited by L H Greene.

Contents

1. Introduction 11.1. Historical note 21.2. Making graphene 21.3. Characterization 41.4. Structure and physical properties 61.5. Electronic properties 71.6. Effect of the substrate on the electronic

properties of graphene 112. Scanning tunneling microscopy and spectroscopy 13

2.1. Graphene on SiO2 132.2. Graphene on metallic substrates 14

2.3. Graphene on graphite 142.4. Twisted graphene layers 222.5. Graphene on chlorinated SiO2 252.6. Graphene on other substrates 29

3. Charge transport in graphene 293.1. Graphenesuperconductor Josephson junctions 313.2. Suspended graphene 343.3. Hot spots and the fractional quantum Hall effect 363.4. Magnetically induced insulating phase 40

Acknowledgments 41References 41

1. Introduction

In 2004 a Manchester University team led by Andre Geimdemonstrated a simple mechanical exfoliation process [1, 2]by which graphene, a one-atom thick two-dimensional (2D)crystal of carbon atoms arranged in a honeycomb lattice [38],could be isolated from graphite. The isolation of graphene andthe subsequent measurements which revealed its extraordinaryelectronic properties [9, 10]unleashed a frenzy of scientificactivity the magnitude of which was never seen. It quicklycrossed disciplinary boundaries and in May of 2010 the Nobelsymposium on graphene in Stockholm was brimming with

palpable excitement. At this historic event graphene wasthe centerpiece for lively interactions between players who

rarely share common ground: physicists, chemists, biologists,engineers and field-theorists. The excitement about grapheneextends beyond its unusual electronic properties. Everythingabout grapheneits chemical, mechanical, thermal andoptical propertiesis different in interesting ways.

This review focuses on the electronic properties of single-layer graphene that are accessible with scanning tunnelingmicroscopy (STM) and spectroscopy (STS) and with transportmeasurements. Section1 gives an overview starting with abrief history in section1.1followed by methods of producingand characterizing graphene in sections 1.2 and 1.3. Insection1.4the physical properties are discussed followed bya review of the electronic properties in section 1.5 and adiscussionofeffectsduetosubstrateinterferenceinsection 1.6.Section2is devoted to STM and STS measurements which

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Rep. Prog. Phys.75(2012) 056501 E Y Andreiet al

allow access to the atomic structure and to the electronicdensity of states. Sections2.1 and 2.2 focus on STM/STSmeasurements on graphene supported on standard SiO2 andon metallic substrates. Section2.3is devoted to graphenesupported above a graphite substrate and the observation oftheintrinsicelectronicpropertiesincludingthelineardensityof

states, Landau levels, the Fermi velocity and the quasiparticlelifetime. This section discusses the effects of electronphononinteractionsandof interlayercoupling. Section 2.4 isdedicatedto STM/STS studies of twisted graphene layers. Section2.5focuses on graphene on chlorinated SiO2substrates and thetransition between extended and localized electronic statesas the carrier density is swept across Landau levels. Abrief description of STM/STS work on epitaxial grapheneon SiC and on h-BN substrates is given in section 2.6.Section3 is devoted to transport measurements. Followinga discussion on graphene devices and substrate-inducedscattering sources for SiO2 substrates, section3.1 discussessuperconductorgraphenesuperconductor (SGS) Josephsonjunctions. Sections 3.2, 3.3 and 3.4 discuss suspendedgraphene devices, the fractional quantum Hall effect and themagnetically induced insulating phase.

List of abbreviations: atomic force microscopy (AFM),angular resolved photoemission spectroscopy (ARPES),charge neutrality point (CNP), chemical vapor deposition(CVD), density of states (DOS), Dirac point (DP), electronhole (eh), electronphonon (eph), highly oriented pyroliticgraphite (HOPG), Landau levels (LLs), lambda levels(Ls), multiple Andreev reflections (MARs), non-suspendedgraphene (NSG), quantum Hall effect (QHE), fractionalQHE (FQHE), suspended graphene (SG), scanning electron

microscopy (SEM), scanning tunneling microscopy (STM),scanning tunneling spectroscopy (STS), transmission electronmicroscopy (TEM).

1.1. Historical note

The story of graphene is both old and new. First postulatedin 1947 by Wallace [11]as a purely theoretical construct tohelp tackle the problem of calculating the band structure ofgraphite, this model of a 2D crystal arranged in a honeycomblattice, was now andagain dusted off and reused over the years[1215]. In 1984 Semenoff [12]resurrected it as a model for

a condensed matter realization of a 3D anomaly and in 1988Haldane [14]invoked it as model for a quantum Hall effect(QHE) without Landau levels (LLs). In the 1990s the modelwas used as a starting point forcalculating theband structureofcarbonnanotubes[16]. Butnobodyatthetimethoughtthatoneday it would be possible to fabricate a free standing materialrealization of this model. This skepticism stemmed from theinfluential MerminWagner theorem [17]which during thelatter part of the last century was loosely interpreted to meanthat2Dcrystalscannotexistinnature. Indeedonedoesnotfindnaturally occurring free standing 2D crystals, and computersimulations show that they do not form spontaneously becausethey are thermodynamically unstable against out of plane

fluctuations and roll-up [18]. It is with this backdrop that therealization of free standing graphene came as a huge surprise.

But on closer scrutiny it should not have been. The MerminWagner theorem does not preclude the existence of finite size2D crystals: its validity is limited to infinite systems with shortrange interactions in the ground state. While a finite size 2Dcrystal will be prone to developing topological defects at finitetemperatures, in line with the theorem, it is possible to prepare

suchacrystalinalong-livedmetastablestatewhichisperfectlyordered provided that the temperature is kept well below thecore energyof a topologicaldefect. How does oneachievesucha metastable state? It is clear that even though 2D crystals donot form spontaneously they can exist and are perfectly stablewhen stacked and held together by van der Waals forces aspart of a 3D structure such as graphite. The Manchester groupdiscovered that a single graphene layer can be dislodged fromitsgraphitecocoonbymechanicalexfoliationwithScotchtape.This was possible because thevan derWaals force between thelayersingraphiteismanytimesweakerthanthecovalentbondswithin the layer which help maintain the integrity of the 2Dcrystal during the exfoliation.

The exfoliated graphene layer can be supported on asubstrate or suspended from a supporting structure [1923].Although the question of whether free-standing graphene istruly 2D or contains tiny out-of-plane ripples [18](as wasobserved in suspended graphene (SG) membranes at roomtemperature [20]) is still under debate, there is no doubt aboutits having brought countless opportunities to explore newphysical phenomena and to implement novel devices.

1.2. Making graphene

We briefly describe some of the most widely used methods toproduce graphene, together with their range of applicability.

Exfoliation from graphite. Exfoliation from graphite,illustrated in figure1,is inexpensive and can yield small (upto 0.1 mm) high-quality research grade samples [1, 2]. Inthis method, which resembles writing with pencil on paper,the starting material is a graphite crystal such as naturalgraphite, Kish or highly oriented pyrolitic graphite (HOPG).Natural and Kish graphite tend to yield large graphene flakeswhile HOPG is more likely to be chemically pure. A thinlayer of graphite is removed from the crystal with Scotchtape or tweezers. The layer is subsequently pressed bymechanical pressure (or dry N2 jet for cleaner processing)

unto a substrate, typically a highly doped Si substrate cappedwith300nmofSiO2, which enables detection under an opticalmicroscope [1]as described in detail in the next section onoptical characterization [2426]. Often one follows up thisstep with an atomic force microscope (AFM) measurement ofthe height profile to determine the thickness (0.3 nm/layer)and/or Raman spectroscopy to confirm the number of layersand check the sample quality. Typical exfoliated grapheneflakes are several micrometers in size, but occasionally onecanfind largerflakes that canreach severalhundred m. Sinceexfoliation is facilitated by stacking defects, yields tend to belarger when starting with imperfect or turbostratic graphite butat the same time the sample size tends to be smaller. The

small size and labor intensive production of samples usingexfoliated graphene render them impractical for large scale

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Figure 1.Making exfoliated graphene. (a) HOPG flakes are deposited on Scotch tape shown with a cm ruler. (b) A Si/SiO2substrate ispressed onto flakes on the tape. (c) Optical micrograph of graphene deposited on SiO2showing flakes with various number of layers. Alarge flake of single-layer graphene, corresponding to the faintest contrast, is indicated by the arrow. Image credit: A Luican-Mayer.

Figure 2.Graphene grown by CVD. (a) Optical image of single crystal graphene flakes obtained by CVD growth on copper with AR/CH 4flow. Scale bar: 50 m. (Reprinted from A M B Goncalves and E Y Andrei 2012 unpublished). (b) Raman spectrum of graphene on copper

in panel(a). Inset: Raman spectrum of graphene on SiO2.

commercial applications. Nevertheless, exfoliated grapheneholds its own niche as a new platform for basic research.The high quality and large single crystal domains, so far notachieved with other methods of fabrication, have given accessto the intrinsic properties of the unusual charge carriers ingraphene, including ballistic transport and the fractional QHE,and opened a new arena of investigation into relativistic chiralquasiparticles [21, 2730].

Chemical vapor deposition (CVD) on metallic substrates. Aquick and relatively simple method to make graphene is CVD

by hydrocarbon decomposition on a metallic substrate [31].This method (figure2(a)) can produce large areas of graphenesuitable, after transfer to an insulating substrate, for largescale commercial applications. In this method a metallicsubstrate, which plays the role of catalyst, is placed in aheated furnace and is attached to a gas delivery system thatflows a gaseous carbon source downstream to the substrate.Carbon is adsorbed and absorbed into the metal surface athigh temperatures, where it is then precipitated out to formgraphene, typicallyat around 500800Cduringthecooldownto room temperature. The first examples of graphitic layerson metallic substrates were obtained simply by segregation ofcarbonimpuritieswhenthemetallicsinglecrystalswereheated

during the surface preparation. Applications of this methodusing the decomposition of ethylene on Ni surfaces [32]were

demonstrated in the 1970s. More recently graphene growthwas demonstrated on various metallic substrates includingRh [33], Pt [3436], Ir [37], Ru [3841], Pd [42]and Cufoil [4346]. The latter yields, at relatively low cost, single-layer graphene of essentially unlimited size and excellenttransport qualities characterized by mobility in excess of7000cm2 V1 s1 [47]. The hydrocarbon source is typicallya gas such as methane and ethylene but interestingly solidsources also seem to work, such as poly(methyl methacrylate)(PMMA) and even table sugar was recently demonstrated as aviable carbon source [48].

Surface graphitization and epitaxial growth on SiC crystals.

Heating of 6HSiC or 4HSiC crystals to temperatures inexcess of 1200 C causes sublimation of the silicon atomsfrom the surface [4951] and the remaining carbon atomsreconstruct into graphene sheets [52]. The number of layersand quality of the graphene depends on whether it grows onthe Si or C terminated face and on the annealing temperature[53]. The first carbon layer undergoes reconstruction due to itsinteractionwiththe substrateforming an insulatingbuffer layerwhile the next layers resemble graphene. C-face grapheneconsists of many layers, the first few being highly doped duethe field effect from the substrate. Growth on the Si-face

is more controlled and can yield single or bilayers. Usinghydrogen intercalation or thermal release tape [54, 55] one can

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Figure 3.(a) AFM image of epitaxial graphene grown on SiC shows micrometer-size terraces. (Reproduced with permission from K VEmtsevet al2009Nature Mater. 8203). (b) AFM scan (NT-MDT Integra Prima) of a single-layer graphene flake on an h-BN substrate.(c) The height profile shows a 0.7 nm step between the substrate and the flake surface. The bubble under the flake is 7 nm at its peak height.Image credit: B Kim, ND-MDT.

transfer these graphene layers to other substrates. Epitaxial

graphene can cover large areas, up to 4 , depending on thesize of the SiC crystal. Due to the lattice mismatch theselayers form terraces separated by grain boundaries whichlimit the size of crystal domains to several micrometers [56]as shown in figure3(a), and the electronic mobility to lessthan 3000cm2 V1 s1 which is significantly lower than inexfoliated graphene. The relatively large size and ease offabrication of epitaxial graphene make it possible to fabricatehigh-speed integrated circuits [57], but the high cost of the SiCcrystal starting material renders it impractical for large-scalecommercial applications.

Other methods. The success and commercial viability offuturegraphene-based devices rests on theability to synthesizeit efficiently, reliably and economically. CVD graphene is oneof the promising directions. Yet, in spite of the fast pace ofinnovation, CVD growth of graphene over large areas remainschallenging due to the need to operate at reduced pressuresor in controlled environments. The recent demonstration ofgraphene by open flame synthesis [58]offers the potential forhigh-volume continuous production at reduced cost. Manyother avenues are being explored in the race toward low cost,efficient and large scale synthesis of graphene. Solution-based exfoliation of graphite with organic solvents [59] ornon-covalentfunctionalization [60] followedby sonication can

be used in mass production of flakes for conducting coatingsor composites. Another promising approach is the use ofcolloidal suspensions [61]. The starting material is typicallya graphite oxide film which is then dispersed in a solvent andreduced. For example, reduction by hydrazine annealing inargon/hydrogen [62]produces large areas of graphene filmsfor use as transparent conducting coating, graphene paper orfilters.

1.3. Characterization

Optical. ForflakessupportedonSiO2afastandefficientwayto find and to identify graphene is using optical microscopy

as illustrated in figure1(c). Graphene is detected as a faintbut clearly visible shadow in the optical image whose contrast

increases with the number of layers in the flake. The shadow

is produced by the interference between light-beams reflectedfrom the graphene and the Si/SiO2 interface [2426]. Thequality of the contrast depends on the wavelength of the lightand thickness of the oxide. For a 300 nm thick SiO2oxidethe visibility is optimal for green light. Other sweet spotsoccur at90nm and500nm. This method allows us tovisualize micrometer-size flakes, and to distinguish betweensingle-layer,bilayer and multilayerflakes. Optical microscopyis also effective for identifying single-layer graphene flakesgrown by CVD on copper as illustrated in figure2(a).

Raman spectroscopy. Raman spectroscopy is a relativelyquick way to identify graphene and determine the number oflayers[63, 64]. Inorder tobeeffectivethe spatial resolution hasto bebetterthan thesample size; forsmall samples this requiresa companion high resolution optical microscope to find theflakes. The Raman spectrum of graphene, figure2(b), exhibitsthree main features: the G peak 1580 cm1 which is due toa first-order process involving the degenerate zone center E2goptical phonon; the 2D (G) peak at 2700 is a second orderpeak involving two A1zone-boundary optical phonons and theD peak, centered at1330cm1, involving one A 1phonon,which is attributed to disorder-induced first-order scattering.In pure single-layer graphene the 2D peak is typically3times larger than the G peak and the D peak is absent. With

increasing numberof layers, the2D peak becomes broader andloses its characteristic Lorentzian line shape. Since the G peakis attributed to intralayer effects, one finds that its intensityscales with the number of layers.

Atomic force microscopy (AFM). The AFM is a non-invasive and non-contaminating probe for characterizing thetopography of insulating as well as conducting surfaces.This makes it convenient to identify graphene flakes on anysurface and to determine the number of layers in the flakewithout damage, allowing the flake to be used in furtherprocessing or measurement. High-end commercial AFMmachines can produce topographical images of surfaces with

height resolution of 0.03nm. State-of-the-art machines haveeven demonstrated atomic resolution images of graphene. The

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Figure 4.STM and SEM on graphene. (a) Atomic resolution STM of graphene on a graphite substrate. (b), (c) SEM images on SG (FEISirion equipped with JC Nabity Lithography Systems). (b) SG flake supported on LOR polymer. Scale bar 1m. Image credit: J Meyerson.(c) SG flake (central area) held in place by a Ti/Au support. Scale bar 1 m. Image credit: A Luican-Mayer.

AFM image of epitaxial graphene on SiC shown in figure3(a)clearly illustrates the terraces in these samples. Figure 3 showsan AFM image of a graphene flake on an h-BN substrateobtained with the Integra Prima AFM by NT-MDT.

Scanning tunneling microscopy and spectroscopy (STM/STS)

STM, the technique of choice for atomic resolution images,employs the tunneling current between a sharp metallic tipand a conducting sample combined with a feedback loop toa piezoelectric motor. It provides access to the topographywith sub-atomic resolution, as illustrated in figure4(a). STScan give access to the electronic density of states (DOS) withenergy resolution as low as 0.1meV. TheDOS obtained withSTM is not limited by the position of the Fermi energyboth occupied and empty states are accessible. In addition,

measurements are not impeded by the presence of a magneticfield which made it possible to directly observe the uniquesequenceof LLsin graphene resultingfrom itsultra-relativisticcharge carriers [65, 66].

The high spatial resolution of STM necessarily limitsthe field of view so, unless optical access is available, it isusually quite difficult to locate small micrometer-size sampleswith a scanning tunneling microscope. A recently developedtechnique [67] which uses the STM tip as a capacitiveantenna allows locating sub-micrometer-size samples rapidlyand efficiently without the need for additional probes. A moredetailed discussion of STM/STS measurements on graphene

is presented in section2of this review.Scanning electron microscope and transmission electron

microscope. SEM is convenient for imaging large areas ofconducting samples. The electron beam directed at the sampletypically has an energy ranging from 0.5 to 40 keV, and aspot size of about 0.45nm in diameter. The image, which isformedby thedetectionof backscattered electronsor radiation,can achieve a resolution of10 nm in the best machines. Dueto the very narrow beam, SEM micrographs have a large depthof field yielding a characteristic 3D appearance. Examplesof SEM images of SG devices are shown in figures 4(b)and (c). A very useful feature available with SEM is the

possibility to write sub-micrometer-size patterns by exposingan electron-beam resist on the surface of a sample. The

disadvantage of using SEM for imaging is electron-beaminduced contamination due to the deposition of carbonaceousmaterial on the sample surface. This contamination is almostalways present after viewing by SEM, its extent dependingon the accelerating voltage and exposure. Contaminantdeposition rates can be as high as a few tens of nanometersper second.

In TEM the image is formed by detecting the transmittedelectrons that pass through an ultra-thin sample. Owing tothe small de Broglie wavelength of the electrons, transmissionelectron microscopes are capable of imaging at a significantlyhigher resolution than optical microscopes or SEM, and canachieve atomic resolution. Just as with SEM imaging TEMsuffers from electron-beam induced contamination.

Low-energy electron diffraction (LEED) and angular resolvedphotoemission spectroscopy (ARPES). These techniquesprovide reciprocal space information. LEED measures thediffraction pattern obtained by bombarding a clean crystallinesurface with a collimated beam of low-energy electrons, fromwhich one can determine the surface structure of crystallinematerials. Thetechnique requires theuse ofvery cleansamplesin ultra-high vacuum. It is useful for monitoring the thicknessof materials during growth. For example, LEED is used forinsitumonitoring of the formation of epitaxial graphene [68].

ARPES is used to obtain the band structure in zeromagnetic field as a function of both energy and momentum.

Since only occupied states can be accessed one is limitedto probing states below the Fermi energy. The typicalenergy resolution of ARPES machines is 0.2 eV for toroidalanalyzers. Recently, 0.025eV resolution was demonstratedwithalowtemperaturehemisphericalanalyzerattheAdvancedLight Source.

Other techniques. In situformation of graphitic layers onmetal surfaces was monitored in the early work by Augerelectron spectroscopy which shows a carbon peak [69]thatdisplays the characteristic fingerprint of graphite [70]. Inx-ray photoemission spectroscopy, which can also be usedduringthedeposition,graphiticcarbonis identifiedby a carbon

species with a C1s energy close to the bulk graphite value of284.5 eV [70].

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Figure 5.Graphene structure. (a) Hexagonal lattice. Red and green colors indicate the two triangular sublattices, labeled A and B. The grayarea subtended by the primitive translation vectors a1and a2marks the primitive unit cell and the vector markedconnects two adjacent Aand B atoms. (b) Brillouin zone showing the reciprocal lattice vectors G1and G2. Each zone corner coincides with a Dirac point found atthe apex of the Dirac cone excitation spectrum shown in figure6.Only two of these are inequivalent (any two which are not connected by areciprocal lattice vector) and are usually referred to as Kand K .

1.4. Structure and physical properties

Structurally, graphene is defined as a one-atom-thick planarsheet of sp2-bonded carbon atoms that are arranged in a hon-eycomb crystal lattice [3]as illustrated in figure5(a). Eachcarbon atom in graphene is bound to its three nearest neigh-bors by strong planarbonds that involve three of its valenceelectrons occupying the sp2-hybridized orbitals. In equilib-rium the carboncarbon bonds are 0.142nm long and are120apart. These bonds are responsible for the planar struc-ture of graphene andfor its mechanical and thermal properties.Thefourthvalenceelectronwhichremainsinthehalf-filled2pzorbital orthogonal to the graphene plane forms a weakbondby overlapping with other 2pzorbitals. These delocalized

electrons determine the transport properties of graphene.Mechanical properties. The covalent bonds which holdgraphene together and give it the planar structure are thestrongest chemical bonds known. This makes graphene oneof the strongest materials: its breaking strength is 200 timesgreater than steel, andits tensile strength, 130GPa [19, 71, 72],is larger than any measured so far. Bunch et al[72]were ableto inflate a graphene balloon and found that it is impermeableto gases [72], even to helium. They suggest that this propertymay be utilized in membrane sensors for pressure changes insmall volumes, as selective barriers for filtration of gases, asa platform for imaging of graphenefluid interfaces, and for

providing a physical barrier between two phases of matter.

Chemical properties. The strictly 2D structure together withthe unusual massless Dirac spectrum of the low-energyelectronic excitations in graphene (discussed below) giverise to exquisite chemical sensitivity. Shedin et al [73]demonstrated that the Hall resistivity of a micrometer-sizedgraphene flake is sensitive to the absorption or desorptionof a single gas molecule, producing step-like changes inthe resistance. This single molecule sensitivity, which wasattributed to theexceptionally low electronic noise in grapheneand to its linear electronic DOS, makes graphene a promisingcandidate for chemical detectors and for other applications

where local probes sensitive to external charge, magnetic fieldor mechanical strain are required.

Thermal properties. The strong covalent bonds between

the carbon atoms in graphene are also responsible forits exceptionally high thermal conductivity. For SGsamples the thermal conductivity reaches values as high as5000Wm1 K1 [74]at room temperature which is 2.5 timesgreater than that of diamond, therecord holderamongnaturallyoccurring materials. For graphene supported on a substrate,a configuration that is more likely to be found in usefulapplications and devices, the thermal conductivity (near roomtemperature) of single-layer graphene is about 600 Wm1 K1

[48]. Although this value is one order of magnitude lower thanfor SG, it is still about twice that of copper and 50 times largerthan for silicon.

Optical properties. Theopticalpropertiesofgraphenefollowdirectly from its 2D structure and gapless electronic spectrum(discussed below). For photon energies larger than thetemperature and Fermi energy the optical conductivity is auniversal constant independent of frequency: G = e2/4hwhere e is the electron charge and h the reduced Plankconstant [15, 75]. As a result all other measurable quantitiestransmittance T, reflectance Rand absorptance (or opacity)Pare also universal constants. In particular, the ratioof absorbed to incident light intensity for SG is simplyproportional to the fine structure constant = e2/hc =1/137 : P

=(1

T )

=2.3%. Here cis the speed of

light. This is one of the rare instances in which the propertiesof a condensed matter system are independent of materialparameters and can be expressed in terms of fundamentalconstants alone. Because the transmittance in graphene isreadily accessible by shining light on a SG membrane [76],it gives direct access in a simple bench-top experiment to afundamental constant, a quantity whose measurement usuallyrequires much more sophisticated techniques. The 2.3%opacity of graphene, which is a significant fraction of theincident light despite being only one atom thick, makes itpossible to seegraphenewith thenaked eye by looking througha glass slide covered with graphene. For a few layers of

graphene stacked on top of each other the opacity increasesin multiples of 2.3% for the first few layers.

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Figure 6.Graphene band structure. (a) Adapted with permission from [5]. (b) Zoom in to low-energy dispersion at one of the Kpointsshows the eh symmetric Dirac cone structure.

The combination of many desirable properties ingraphene: transparency, large conductivity, flexibility, andhigh chemical and thermal stability, make it [77, 78]a naturalcandidate for solar cells and other optoelectronic devices.

1.5. Electronic properties

Three ingredients go into producing the unusual electronicproperties of graphene: its 2D structure, the honeycomb latticeand the fact that all the sites on its honeycomb lattice areoccupied by the same atoms, which introduces inversionsymmetry. We note that the honeycomb lattice is not a Bravais

lattice. Instead, it canbe viewedas a bipartite lattice composedof two interpenetrating triangular sublattices, A and B witheach atom in the A sublattice having only B sublattice nearestneighbors and vice versa. In the case of graphene the atomsoccupying the two sublattices are identical and as we shall seethis has important implications to its electronic band structure.As shown in figure5(a), the carbon atoms in sublattice Aare located at positions R = ma1 + na2, where m, n areintegers anda1= a2 (3,

3),a2= a2 (3,

3)are the lattice

translation vectors for sublattice A. Atoms in sublattice B areatR + , where = (a2 + a1)/3. The reciprocal lattice vectors,G1= 23a (1,

3), G2= 23a (1,

3)and the first Brillouin

zone, a hexagon with corners at the so-called K points, areshowninfigure5(b). Onlytwoofthe Kpoints are inequivalent,the others being connected by reciprocal lattice vectors. Theelectronic properties of graphene are controlled by the low-energy conical dispersion around these Kpoints.

Tight binding Hamiltonian and band structure. The low-energy electronic states, which are determined by electronsoccupyingthe pzorbitals, canbe derived from thetightbindingHamiltonian [11]in the Huckel model for nearest neighborinteractions:

H= t| R(| R R+ | + | R R a1+ |

+ | R R a2+ | + h.c.). (1)

Here r| R = pz ( R r)is a wave function of the pzorbitalon an atom in sublattice A,r| R+is a similar state ona B sublattice atom and t is the hopping integral from astate on an A atom to a state on an adjacent B atom. Thehopping matrix element couples states on the A sublatticeto states on the B sublattice and vice versa. It is chosenas t 2.7eV so as to match the band structure near theK points obtained from first principle computations. Sincethere are two Bravais sublattices two sets of Bloch orbitals areneeded, one for each sublattice, to construct Bloch eigenstatesof the Hamiltonian: |kA = (1/N) Reik R| Rand |kB =(1/

N) Re

ik R| R+. These functions block-diagonalizethe one-electron Hamiltonian into 22 sub-blocks, withvanishing diagonal elements and with off-diagonal elementsgiven by: kA|H|kB = teik(1 + eika1 + eika2 )e(k).The single particle Bloch energies (k)= |e(k)|give theband structure plotted in figure 6(a), with (k) = |e(k)|corresponding to the conduction band and(k) = |e(k)|to the valence band . It is easy to see that (k) vanishes whenklies at a K point. For example, at K1= ( G1+ 2 G2)/3,e( K) = teik(1 +ei G1a1/3 +ei2 G2a2/3) = 0 where we usedGi aj= 2 ij. For reasons that will become clear, thesepoints are called Dirac points (DP). Everywhere else in k-space, the energy is finite and the splitting between the twobands is 2

|e(

k)

|.

Linear dispersion and spinor wavefunction. We now discussthe energy spectrum and eigenfunctions for k close to a DP.Since only two of the Kpointsalso known as valleysareinequivalent we need to focus only on those two. Followingconvention we label them K and K . For the K valley,it is convenient to define the (2D) vector q = Kk.Expanding aroundq= 0, and substitutingq ih(x , y )the eigenvalue equation becomes [35]

HK K= ihvF

0 x iyx+ iy 0

KAKB

=

KAKB

,

(2)

where vF= 32 (at/h) 106 m s1 is the Fermi velocity of thequasiparticles. The two components KA and KB give the

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amplitude of the wave function on the A and B sublattices.The operator couples KA to KB but not to itself, sincenearest-neighbor hopping on the honeycomb lattice couplesonly A-sites with B-sites. The eigenvalues are linear in themagnitude ofq and do not depend on its direction, (q)=hvF|q| producing the electronhole (eh) symmetric conicalband shown in figure6(b). The eh symmetry in the low-energydispersionofgrapheneisslightlymodifiedwhensecondorder and higher neighbor overlaps are included. But thedegeneracy at the DP remains unchanged even when higherorder corrections are added as discussed in the next section.The linear dispersion implies an energy independent groupvelocityvgroup= |E /hk| = |E/hq | = vFfor low-energyexcitations (|E| t ).

The eigenfunctions describing the low-energy excitationsnear pointKare

K (q ) =

KAKB

= 1

2

eiq /2

eiq /2

, q tan1(qx /qy ).

(3)Thistwo-component representation, whichformallyresemblesthat of a spin, corresponds to the projection of the electronwavefunction on each sublattice.

How robust is the Dirac point? A perfect undoped sheetof graphene has one electron per carbon in the band and,taking spin into account, this gives a half filled band at chargeneutrality. Therefore, the Fermi level lies between the twosymmetrical bands, with zero excitation energy needed toexcite an electron from just below the Fermi energy (holesector) to just above it (electron sector) at the DPs. The Fermisurface in graphene thus consists of the two Kand K pointsin the Brillouin zone where the and bands cross. Wenote that in the absence of degeneracy at the two K pointsgraphenewouldbe an insulator! Usually suchdegeneraciesareprevented by level repulsion opening a gap at crossing points.But in graphene the crossing points are protected by discretesymmetries [79]: C3, inversion and time reversal, so unlessone of these symmetries is broken the DP will remain intact.Density functional theory calculations [80]show that addingnext-nearest neighbor terms to the Hamiltonian removes theeh symmetry but leaves the degeneracy of the DPs. On theother hand the breaking of the symmetry between the A and Bsublattices, such as, for example, by a corrugated substrate,

is bound to lift the degeneracy at the DPs. The effect ofbreaking the (A, B) symmetry is directly seen in graphenessistercompound, h-BN.Just like grapheneh-BN isa 2Dcrystalwith a honeycomb lattice, but the two sublattices in h-BN areoccupiedby differentatomsand theresulting broken symmetryleaves the DP unprotected. Consequently h-BN is a bandinsulator with a gap of6 eV.

DiracWeyl Hamiltonian, masssles Dirac fermions and

chirality. A concise form of writing the Hamiltonian inequation (2) is

HK= hvF p,

where p = hq and the components of the operator =(x , y ) are the usual Pauli matrices, which now operate on the

sublattice degrees of freedom instead of spin, hence the termpseudospin. Formally, this is exactly the DiracWeyl equationin 2D, so the low-energy excitations are described not by theSchrodinger equation, but instead by an equation which wouldnormally be used to describe an ultra-relativistic (or massless)particle of spin 1/2 (such as a massless neutrino), with the

velocity of lightcreplaced by the Fermi velocityvF, which is300 times smaller. Therefore the low-energy quasiparticles ingraphene are often referred to as massless Dirac fermions.

The DiracWeyl equation in quantum electrodynamics(QED) follows from theDirac equation by setting the rest massof the particle to zero. This results in two equations describingparticlesof opposite helicity or chilarity (formassless particlesthe two are identical and the terms are used interchangeably).The chiral (helical) nature of the DiracWeyl equation is adirect consequence of the Hamiltonian being proportionalto the helicity operator: h = 12

p where

p is a unitvector in the direction of the momentum. Since hcommuteswith the Hamiltonian, the projection of the spin is a well-defined conserved quantity which can be either positive ornegative, corresponding to spin and momentum being parallelor antiparallel to each other.

In condensed matter physics hole excitations are oftenviewed as a condensed matter equivalent of positrons.However, electrons and holes are normally described byseparate Schrodinger equations, which are not in any wayconnected. In contrast, electron andhole states in graphene areinterconnected, exhibiting properties analogous to the charge-conjugation symmetry in QED. This is a consequence ofthe crystal symmetry which requires two-component wavefunctions to define the relative contributions of the A and B

sublattices in the quasiparticle make-up. The two-componentdescription for graphene is very similar to the spinor wavefunctions in QED, but the spin index for graphene indicatesthe sublattice rather than the real spin of the electrons. Thisallows one to introduce chirality in this problem as theprojection of pseudospin in the direction of the momentumwhich, in the Kvalley, is positive forelectronsandnegative forholes. Hence, just as in thecaseof neutrinos, each quasiparticleexcitation in graphene has its antiparticle. These particleantiparticle pairs correspond to eh pairs with the samemomentum but with opposite signs of the energy and withopposite chirality. InK the chirality of electrons and holes is

reversed, as we show below.Suppression of backscattering. The backscattering probabil-ity can be obtained from the projection of the wavefunctioncorresponding to a forward moving particle +K (q())on thewavefunction of the backscattered particle +K (q(+ )).Within thesame valley we have +K (q()) +K (q(+ )) =iK (q()) which gives +K (q())||K (q()) = 0 . In otherwords backscattering within a valley is suppressed. This se-lection rule follows from the fact that backscattering within thesame valley reverses the direction of the pseudospin.

We next consider backscattering between the two valleys.Expanding in

q

= K

knear the second DP yields HK

=hvF p(indicates complex conjugation) which is relatedto HK (q) by the time reversal symmetry operator, zC[5].

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The solution in the K valley is obtained by taking px pxin equation (2) resulting in

K (q ) = 1

2

eiq /2

eiq /2

.

Backscattering between valleys is also disallowed because it

entails the transformation+K (q ) +K (q+ ) = iK (q )which puts the particle in a state that is orthogonal to itsoriginal one. This selection rule follows from the fact thatbackscattering between valleys reverses the chirality of thequasiparticle.

The selection rules against backscattering in graphenehave important experimental consequences including ballistictransport at low temperature [21, 22], extremely large roomtemperature conductivity [81]and weak anti-localization [82].

Berry phase. Considering the quasiparticle wavefunction inequation (3),wenotethatitchangessignundera2rotationinreciprocal space:K (q )

= K (q+ 2 ). This sign change

is often used to argue that the wavefunctions in graphene havea Berry phase, of . A non-zero Berry phase [83] whichcan arise in systems that undergo a slow cyclic evolution inparameter space, can have far reaching physical consequencesthat canbe found in diversefields including atomic, condensedmatter, nuclear and elementary particle physics, and optics. IngraphenetheBerryphaseofis responsibleforthe zero energyLLs and the anomalous QHE discussed below.

On closer inspection, however, the definition of the Berryphase in terms of thewavefunction alone is ambiguousbecausethe sign change discussed above can be made to disappearsimply by multiplying the wavefunction by an overall phase

factor,eiq /2K (q ) =

12

eiq

1

.

For a less ambiguous result one should use a gaugeinvariant definition for the Berry phase [84] =

Cd()|i(/)|() where the integration is over a

closed path in parameter space and the wavefunction ()has to be single valued. Applying this definition to the singlevalued form of the wavefunction, i.e.

() = 12

eiq

1

and taking ; C 20 dover a contour that enclosesone of the DPs we find that the gauge invariant Berry phase ingraphene is= .

DOS and ambipolar gating. The linear DOS in graphene isa direct consequence of the conical dispersion and the ehsymmetry. It can be obtained by considering nK (q) = q2/2 ,thenumberofstatesinreciprocalspacewithinacircleofradius|q| = ||/hvFaround one of the DPs, say K , and taking intoaccount the spin degeneracy. The DOS associated with thispoint is (1/hvF)(dnK /dq). Since there are two DPs the totalDOS per unit area is

() = 2hvF

dnKdq

= 2

1(hvF)2

||. (4)

The DOS per unit cell is then ()Acwhere Ac= 3

3a2/2is the unit cell area. The DOS in graphene differs qualitativelyfrom that in non-relativistic 2D electron systems leading toimportant experimental consequences. It is linear in energy,eh symmetric and vanishes at the DPas opposed to aconstant value in the non-relativistic case where the energy

dispersion is quadratic. This makes it quite easy to dopegraphene with an externally applied gate. At zero doping,the lower half of the band is filled exactly up to the DPs.Applying a gate voltage induces a nonzero charge, which isequivalent to injecting (depending on the sign of the voltage)electrons in the upper half of Dirac cones or holes in thelower half. Due to the eh symmetry, the gating is ambipolarwith the gate induced charge changing sign at the DP. Thisis why the DP is commonly labeled as the charge neutralitypoint (CNP).

Cyclotron mass and LLs. Considering sucha doped graphene

device with carrier density per unit area, ns , at a low enoughtemperature so that the electrons form a degenerate Fermisea, one can then define a Fermi surface (in 2D a line).After taking into account the spin and valley degeneracies,the corresponding Fermi wave vector qFis qF= ( ns )1/2/2 .One can now define an effective mass min the usual way,m = hqF/vF = ( 1/2h/vF)n1/2s . In a 3D solid, the mostdirect way of measuringmis through the specific heat, but ina 2D system such as graphene this is not practical. Insteadone can use the fact that for an isotropic system the massmeasured in a cyclotronresonanceexperiment, mc , is identicalto mdefined above. This is because in thesemi-classical limitm

c=1/2[S/]F , where S()

= q2()

=(2/h2v2

F

),is the k space area enclosed by an orbit of energy , somc = hqF/vF = m. Cyclotron resonance experimentson graphene verify that m is indeed proportional ton1/2 [9].

The energy spectrum of 2D electron systems in thepresence of a magnetic field, B, normal to the plane breaksup into a sequence of discrete LLs. For the nonrelativisticcase realized in 2D electron system on helium [85] or insemiconductor heterostructures [86]the LL sequence consistsof a seriesof equally spaced levels similar to that of a harmonicoscillator EN= hc(N+1/2) with c= eB/m the cyclotronfrequency and a finite energy offset of 1/2hc. This spectrum

follows directly from the semi-classical Onsager quantizationcondition [87]for closed orbits in reciprocal space:

S() =

2 |e| Bh

(N+); N= 0, 1, . . .

and = 1/2 /2 , where is the Berry phase. Themagnetic field introduces a new length scale, the magneticlength lB=

h/eB, which is roughly the distance between

the flux quanta 0= h/e. The Onsager relation is equivalentto requiring that thecyclotron orbit encloses an integer numberof flux quanta.

For the case of non-relativistic electrons

=0 resulting

in the 1/2 sequence offset. In graphene, as a result of thelinear dispersion and Berry phase ofwhich gives = 0, the

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Figure 7.Low-energy dispersion and DOS. (a) Zero-field energydispersion of low-energy excitations illustrating the electron(red)hole (blue) symmetry. (b) The zero-field DOS is linear inenergy and vanishes at the DP. (c) Finite-field energy dispersionexhibits a discrete series of unevenly spaced LLs symmetricallyarranged about the zero-energy level,N= 0, at the DP. (d) DOS infinite magnetic field consists of a sequence of(EEN)functionswith gaps in between. All peaks have the same height, proportional

to the level degeneracy 4B/0.

LL spectrum is qualitatively different. Using the same semi-classical approximation, the quantization of the reciprocalspace orbit area, q2Fgives S()= q2F= ((2 |e|B)/h)N,which produces the LL energy sequence:

EN= hvFqF=

2ehv2FB|N|, N= 0, 1, . . . . (5)

Here the energy origin is taken to be the DP and refers toelectron and hole sectors, respectively.

Compared with the non-relativistic case the energy levels

are no longer equally spaced, the field dependence is nolonger linear and the sequence contains a level exactly at zeroenergy which is a direct manifestation of the Berry phase ingraphene [12].

We note that the LLs are highly degenerate, thedegeneracy/per unit area being equal to four times (for spinand valley) the orbital degeneracy (the density of flux lines):4(B/0).

The exact finite field solutions to this problem can beobtained [8891]from the Hamiltonian in equation (2), byreplacing i i +e A, where in the Landau gauge, thevector potential isA = B(y, 0) andB= A. The energysequence obtained in this approach is the same as above, but

now one can also obtain the explicit functional form of theeigenstates.

From bench-top quantum relativity to nanoelectronics.

Owing to the ultra-relativistic nature of its quasiparticles,graphene provides a platform which for the first time allowstesting in bench-top experiments some of the strange andcounterintuitive effects predicted by quantum relativity, butoften not yet seen experimentally, in a solid-state context.

One example is the so-called Klein paradox which predictsunimpeded penetration of relativistic particles through high[92] potential barriers. In graphene the transmissionprobability for scattering through a high potential barrier[93, 94]of widthDat an angle is

T= cos2( )

1 cos2(qx D) sin2( ).

In the forward direction the transmission probability is 1corresponding to perfect tunneling. Klein tunneling is oneof the most exotic and counterintuitive phenomena. It wasdiscussed in many contexts including in particle, nuclear and

astrophysics, but direct observation in these systems has sofar proved impossible. In graphene on the other hand it maybe observed [95]. Other examples of unusual phenomenaexpected due to the massless Dirac-like spectrum of thequasiparticles in graphene include electronic negative indexof refraction [96], zitterbewegung and atomic collapse [97].

Beyond these intriguing single-particle phenomenaelectronelectron interactions and correlation are expected toplay an important role in graphene [98104]because of itsweak screening and large effective fine structure constant = e2/hvF 2[3]. In addition, the interplay betweenspin and valley degrees of freedom is expected to show SU(4)

fractional quantum Hall physics in the presence of a strongmagnetic field which is qualitatively different from that inconventional 2D semiconductor structures [104, 105].

The excellent transport and thermal characteristics ofgraphene make it a promising material for nanoelectronicsapplications. Its high intrinsic carrier mobility [106], whichenables low operating power and fast time response, isparticularly attractive for high speed electronics [57]. Inaddition, the fact that graphene does not lose its electronicproperties down to nanometer length scales, is an invaluableasset in the quest to downscale devices for advancedintegration. These qualitieshave wongraphene a prime spot inthe race towards finding a material that can be used to resolvethe bottleneck problems currently encountered by Si-basedVLSI electronics.

Amongst the most exciting recent developments is the useof graphene in biological applications. The strong affinityof biomatter to graphene makes it an ideal interface forguiding and controlling biological processes. For example,graphene was found to be an excellent biosensor capableof differentiating between single and double stranded DNA[107]. New experiments report that graphene can enhance thedifferentiationofhumanneuralstemcellsforbrainrepair[108]and that it accelerates thedifferentiationof bone cell from stemcells [109]. Furthermore, graphene is a promising material

for building efficient DNA sequencing machines based onnanopores, or functionalized nanochannels [110].

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Is graphene special? The presence of eh symmetric Diraccones in the band structure of graphene endows it withextraordinary properties, such as ultra-high carrier mobilitywhich is extremely valuable for high speed electronics, highlyefficient ambipolar gating and exquisite chemical sensitivity.

One may ask why graphene is special. After all there

are many systems with Dirac cones in their band structure.Examples include transition metal dichalcogenites below thecharge density wave transition [111], cuprates below thesuperconducting transition [112]and pnictides below the spindensity wave transition [113]. However, in all the other casesthe effect of the DP on the electronic properties is drowned bystatesfrom other partsof theBrillouinzone which,nothaving aconical dispersion, make a much largercontributionto theDOSat the Fermi energy. In graphene, on the other hand, the effectof the DPs on the electronic properties is unmasked becausethey alonecontribute to theDOS at theFermienergy. In fact, asdiscussed above, had it not been for the DPs, graphene wouldbe a band insulator.

1.6. Effect of the substrate on the electronic properties of

graphene

The isolation of single-layer graphene by mechanical exfolia-tion was soon followedby theexperimental confirmation of theDirac-like nature of the low-energy excitations [9, 81]. Mea-surementsof the conductivity andHall coefficient on graphenefield-effect transistor (FET) devices demonstrated ambipolargating and a smooth transition from electron doping at posi-tive gate voltages to hole doping on the negative side. At thesame time the conductivity remained finite even at nominally

zero doping, consistent with the suppression of backscatteringexpected for massless Dirac fermions. Furthermore, magneto-transport measurements in high magnetic field which revealedthe QHE confirmed that the system is 2D and provided evi-dence for the chiral nature of the charge carriers through theabsence of a plateau at zero filling (anomalous QHE). Follow-ing these remarkable initial results, further attempts to probedeeper into the physics of the DP by measuring graphene de-posited on SiO2substrates, seemed to hit a hard wall. De-spite the fact that the QHE was readily observed, it was notpossible in these devices to approach the DP and to probe itsuniqueproperties such as ballistic transport [56, 114], specular

Andreev reflections (SARs) expected [63, 115]at graphenesuperconducting junctions[116, 117] or correlatedphenomenasuch as the fractional QHE [118]. Furthermore, STS measure-ments did not show the expected linear DOS or its vanishingat the CNP [119, 120].

The failure to probe the DP physics in graphene depositedon SiO2substrates was understood later, after applying sen-sitive local probes such as STM [119124]and SET (singleelectron transistor) microscopy [125], and attributed to thepresence of a random distribution of charge impurities associ-ated with the substrate. The electronic properties of grapheneare extremely sensitive to electrostatic potential fluctuationsbecause the carriers are at the surface and because of the

low carrier density at the DP. It is well known that insulat-ing substrates such as SiO2host randomly distributed charged

impurities, so that graphene deposited on their surface is sub-ject to spatially random gating and the DP energy (relative tothe Fermi level) displays random fluctuations, as illustrated infigure8(b). The random potential causes the charge to breakup into eh puddles: electron puddles when the local potentialis below the Fermi energy and hole puddles when it is above.

These puddles fill out the DOS near the DP (figures8(c) and(d)) making it impossible to attain the zero carrier density con-dition at the DP for any applied gate voltage as seen in the STSimage shown in figure8(e). Typically for graphene depositedonto SiO2the random potential causes DP smearing over anenergy range ER 30100 meV.When the Fermi energyis within ER of the DP, a gate voltage change transformselectrons into holes and vice versa but it leaves the net carrierdensity almost unchanged. As a result, close to the DP thegate voltage cannot affect significant changes in the net carrierdensity. This is directly seen as a broadening of order 110Vin the conductivity versus gate voltage curves, figures8(e) and(f), which corresponds to a minimum total carrier density inthese samples ofns 1011 cm2. The energy scale defined bythe random potential also defines a temperature kBT ERbelow which the electronic properties such as the conductivityare independent of temperature.

Integer and fractional QHE. The substrate induced randompotential which makes the DP inaccessible in graphenedeposited on SiO2explains the inability to observe in thesesamples the linear DOS and its vanishing at the CNP with STSmeasurements. As we show below this also helps understandwhy the integer QHE is readily observed in such samples butthe fractional QHE is not.

To observe the QHE in a 2Delectron systemone measuresthe Hall and longitudinal resistance while the Fermi energy isswept through the LLs, by changing either carrier density ormagnetic field [126]. The Fermi energy remains within an LLuntil all the available states, 4B/0 per unit area, are filledand then jumps across the gap to the next level unless, as isusually the case, there are localized impurity states availablewithin the gap which are populated first. In homogeneoussamples the LL energy is uniform in the bulk and divergesupwards (downwards) for electrons (holes) near the edges. Asa result, when the Fermi energy is placed within a bulk gapbetween two LLs, it must intersect all the filled LLs at theedge. This produces 1D ballistic edge channels, in whichthe quasiparticles on opposite sides of the sample move inopposite directions, as shown in figure9(a). These ballisticchannels lead to a vanishing longitudinal resistance and to aquantized Hall conductance:xy= e2h whereis the fillingfactor. counts the number of occupied ballistic channelswhich is the number of filled LLs multiplied by the (non-orbital) degeneracy, four in the case of graphene. TheN= 0level is special because half of its states are electron like ( Kvalley) and half hole like (K valley) so that its contributionconsists of only two states for each species. Therefore whenthe Fermi energy is in between levels Nand N+ 1, the numberof occupied states is 4N+2, corresponding to

=4(N+1/2).

The 12 offset, absent in the case of non-relativistic electrons,is a direct consequence of the chiral symmetry of the low

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strong correlations the system can lower its energy for certainfilling factors by forming composite fermions which consistof an electron bound together with an even number of fluxlines [128]. These composite fermions sense the remnantmagnetic field left after having swallowed the flux lines, andas a result their energy spectrum breaksup into lambda levels

(Ls) which are the equivalent of LLs but for the compositefermions in the smaller field. Just as the electrons display aninteger QHE whenever the Fermi energy is in a gap betweenLLs, so do the composite fermions when the Fermi energyis in a gap between the Ls. The filling factors for whichthis occurs take fractional values = p/(2mp 1), p =1, 2 . . . ; m= 1, 2. The characteristic spacing betweenthe Ls is controlled by the Coulomb energy, and is muchsmaller than the spacing between LLs:

E 0.1e2

lc 5meV (B[T])

1/2

,

where is the dielectric constant of the substrate. Thusthe criterion for decoupled edges in the fractional QHE casebecomes E > ER 30meV B > 50 T, whichis larger than any dc magnetic field attainable to date. Inother words, the fractional QHE is not observable in graphenedeposited on SiO2.

Therefore in order to access the intrinsic properties ofgraphene and correlation effect it is imperative to reduce thesubstrate-induced random potential. The rest of this reviewis devoted to the exploration of ways to reduce this randompotential and to access the intrinsic electronic properties ofgraphene.

2. Scanning tunneling microscopy and spectroscopy

In STM/STS experiments, one brings a sharp metallic tip veryclose to the surface of a sample, with a typical tipsampledistance of1 nm. For positive tipsample bias voltages,electronstunnel fromthe tipinto emptystatesin thesample; fornegative voltages, electrons tunneloutof theoccupied states inthe sample into tip. In the Bardeen tunneling formalism [129]the tunneling current is given by

I= 4 eh

+

[f (EF eV+) f (EF+)]S

(EF eV+)T(EF+)|M|2 d, (7)where eis the electron charge, f (x)is the Fermi function,EF the Fermi energy, V the sample bias voltage, T andS represent the DOS in the tip and sample, respectively.The tunneling matrix Mdepends strongly on the tipsampledistancez. When the tip DOS is constant and at sufficientlylow temperaturesthe tunnelingcurrent canbe approximatedbyI (r , z , V ) [eV S(r, ) d]expz(r), where 2m/his the inverse decay length and is the local barrier heightor average work function. The exponential dependence onheight makes it possible to obtain high resolution topographyof the surface at a given bias voltage. The image is obtained

by scanning the sample surface while maintaining a constanttunneling current with a feedback loop which adjusts the

tip height to follow the sample surface. We note that anSTM image not only reflects topography but also containsinformation about the local DOS which can be obtaineddirectly [130]by measuring the differential conductance:

dIdV

(V ) S( = eV). (8)

Here EFis set to be zero. In STS the tipsample distance isheld fixed by turning off thefeedback loop while measuringthetunneling currents as a function of bias voltage. Usually onecanusealock-intechniquetomeasuredifferentialconductancedirectly by applying a small ac modulation to the sample biasvoltage.

In practice, finite temperatures introduce thermalbroadening through the Fermi functions in equation (7),leading to reduced energy resolution in STS. For example,at 4.2 K the energy resolution cannot be better than 0.38meV.Correspondingly, the ac modulation of the sample bias shouldbe comparable to this broadening in order to achieve highest

resolution. The condition of a flat tip DOS is usuallyconsidered satisfied for common tips, such as PtIr, W or Au,as long as the sample bias voltage is not too high. Comparedwith a sharp tip, a blunt tip typically has a flatter DOS. In orderto have reliable STS, one should make sure a good vacuumtunneling is achieved. To this end, one can check the spatialandtemporal reproducibilityof thespectra andensure that theyare independent of tipsample distance [130].

2.1. Graphene on SiO2

As discussed in section 1, the insulating substrateof choiceandthemost convenient, SiO2, suffers from large randompotentialfluctuations which make it impossible to approach the DPdue to the formation of eh puddles [125]. STM topographyon these substrates does show a honeycomb structure forsingle-layer graphene and a triangular lattice for multi-layers[121, 131] with very few topologicaldefects whichis testimonyto the structural robustness of graphene. However, in contrastto the case of graphene on graphite [65], these samples showsignificant corrugation on various length scales ranging from132 nm due to the substrate, wrinkling during fabrication[100] and possibly intrinsic fluctuations. These corrugationscanlead to brokensublattice symmetry affecting both transportand the STM images and can lead, for example, to the

appearance of a triangular lattice instead of the honeycombstructure in unperturbed graphene [131, 132].In the presence of scattering centers, the electronic wave

functions can interfere to form standing wave patterns whichcanbe observed by measuring thespatial dependence of dI/dVat a fixed sample bias voltage. Using these interferencepatterns, it waspossible to discern individual scattering centersin the dI/dVmaps obtained at energies far from the CNPwhen the electron wavelength is small [133]. No correlationswerefoundbetweenthecorrugationsandthescatteringcenters,suggestingthelatterplayamoreimportantroleinthescatteringprocess. When the sample bias voltage is close to theCNP, the electron wavelength is so large that it covers many

scattering centers and the dI/dVmaps show coarse structures(figure10(b)) which are attributed to eh puddles.

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Figure 10.STM/STS of graphene on chlorinated SiO2. (a) STMtopography image of a typical 300 300 nm graphene area.Tunneling currentIt= 20 pA and bias voltageVbias= 190 mV.Legend shows height scale. (b) Differential conductance map overthe area in panel (a) taken close to the DP (140 mV), markedEDin (d). Legend shows differential conductance scale. (c) STMatomic resolution image (It= 20pA,Vbias= 300 mV) shows thehoneycomb structure. (d) Differential conductance averaged overthe area shown in (b). Adapted from [120].

The Fourier transform of the interference pattern providesinformation about the energy and momentum distribution ofquasiparticle scattering, which can be used to infer bandstructure [123]. While for unperturbed single-layer graphene,

the patterns should be absent or very weak [134], for grapheneon SiO2clear interference patterns arise [133]due to strongscattering centers which are believed to be trapped charges.The dispersionE(k)obtained from the interference pattern islinear with vF= (1.50.2)106 and (1.40.2)106 m s1for electron and hole states, respectively. It should be notedthat these values are for states with energies significantly farfrom the Fermi level and the CNP. At lower energies, transportmeasurements yieldedvF= 1.1 106 m s1 [9, 10].

2.2. Graphene on metallic substrates

As detailed in the introductory section epitaxial growth ofgraphitic layers can be achieved on a wide range of metalsubstrates by thermal decomposition of a hydrocarbon orby surface segregation of carbon atoms from the bulk metal[135, 136]. Graphene monolayers are relatively easy toprepare on metal surfaces and, with the right metal and growthconditions, thesizeof themonolayer flakesis almostunlimited.STM studies of graphitic flakes on metallic substrates havefocused mostly on the structure. On Ir(11 1) [46], Cu(11 1)[46, 137, 138] and on Ru(0001) [139] (figure 11(a)) theyrevealed structurally high-quality monolayer graphene andcontinuity which is not limited by the size of terraces inthe substrate, although the overall structure is often strongly

modulated by the mismatch with the lattice of the underlyingmetal which leads to Moire superstructures (figure 11(c)).

The electronic properties of these graphitic layers are stronglyaffected by the metallic substrates leading to significantdeviations from from the linear dispersion expected for freestanding graphene [139] (figure11(b)). Thus, in order toaccess the unique electronic properties of graphene while alsotaking advantage of the high quality and large scales achieved

on metallic substrates it is necessary to separate the graphiticlayer from its metallic substrate.

2.3. Graphene on graphite

The choice of a minimally invasive substrate for gainingaccess to the electronic properties of graphene is guided bythe following attributes: flat, uniform surface potential andchemically pure. Going down this list, the substrate thatmatches the requirements is graphite, the motherof graphene.Because it is a conductor, potential fluctuations are screenedand furthermore it is readily accessible to STM and STSstudies.

Almost ideal graphene seen by STM and STS. Duringexfoliation of a layered material, cleavage takes place betweenthe least coupled layers. Occasionally, when cleavage ispartial, a region in which the layers are separated can befound adjacent to one where they are still coupled. Thissituation in shown in figure 12(a) where partial cleavagecreates the boundaryseen as a diagonal dark ridgebetweenthe decoupled region marked G and a less coupled regionmarked W. The layer separation in these regions is obtainedfrom height profiles along lines and shown in the figure.In region W the layer separation,0.34nm, is close to theinterlayer spacing 0.335 nm of graphite, but in region G the

larger separation, 0.44 nm, means that the top layer is liftedby 30%. Atomic resolution STM images show a honeycombstructure in region G but a triangular one in region W. Thetriangular lattice in region W is consistent with the sublatticeasymmetry expected for Bernal stacked graphite. In thisstacking, which is the lowest energyconfiguration forgraphite,the atoms belonging to sublattice A in the topmost layer arestacked above B atoms in the second layer, while B atoms in thetopmostlayerareabovethehollowsitesofthecarbonhexagonsof the second layer. Ab initio band structure calculations[140] show that in the presence of interlayer coupling thissite asymmetry leads to a strong asymmetry in the local

DOS at the Fermi level with the B atoms having the largerDOS. This leads to STM images in which the B atoms ongraphite appear more prominent than the A atoms resultingin a triangular lattice [140, 141]. In the absence of interlayercoupling the DOS is symmetric between the two sublatticesand one would expect to observe a honeycomb structure asseen in region G. The observation of the honeycomb structureprovides an important first clue in the search for decoupledgraphene flakes on graphite, but it is not sufficient to establishdecoupling between the layers. This is because, even thoughthe atomic resolution topography of the surface of HOPG wasone of the first to be studied by STM, its interpretation is notunique and depends on other factors such as the bias voltage.

The triangular structure discussed above is commonly seen inatomic resolution topographic images of graphite at low bias

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Figure 11.STM/STS on graphene on Ru(0 0 0 1) and Cu(1 1 1). (a) Atomic-resolution image showing a graphene overlayer across a stepedge on the Ru substrate. (b) Differential conductance spectrum of a graphene layer on a Ru substrate. Adapted with permission from [139].(c) Atomic resolution STM topography image of graphene on Cu showing the Moir e pattern and the honeycomb structure. Adapted withpermission from [138].

Figure 12.Graphene flake on the surface of graphite. (a) Large area STM topography. Atomic steps are clearly visible at edges of graphenelayers. A diagonal ridge separates a region with honeycomb structure (region G), from a triangular structure (region B) below. The regionmarked C represents the surface of graphite surrounding the flake. (b), (c) Height profiles along cross sectional cuts markedand .(d), (e) Atomic resolution images show the honeycomb structure in region G and the triangular lattice in region W.

voltages, but there are also many reports of the appearanceof a honeycomb structure under various circumstances whichare often not reproducible [142149]. As we show below inorder to establish the degree of coupling of the top layer to the

layers underneath it is necessary to carry out spectroscopicmeasurements and in particular LL spectroscopy. In theearlier works only topographic measurements were reported[125132]and therefore it was not possible to correlate thestructure seen in STMwith the degree of coupling between thelayers.

We start in region C of figure12where atomic resolutiontopography imagesshowa triangular lattice forbias voltagesinthe range 100800 mV and for junction resistances exceeding1 Gas seen in figure13(a). Zero-field STS, figure13(b),shows finite differential conductance at the neutrality point,consistent with the finite DOS expected for bulk graphite. Thefinite field spectra shown in figure13(c) are again consistent

with bulk graphite: no LL sequence is observed consistentwith the energy dispersion normal to the surface. In summary,

the data in region C present the characteristic features of bulkgraphite.

The situation is qualitatively different in region G, wherean atomic resolution spectroscopy image (figure13(d)) shows

the honeycomb structure in the entire region, which extendsover 400 nm, with no visible vacancies or dislocations. STSin this region in zero field, figure13(e), shows that the DOS isV-shaped and vanishes at the DP which is 16 meV above theFermi energy (taken as zero) corresponding to unintentionalhole doping with a concentration ofns 2 1010 cm2. In thepresence of a magnetic field the DOS develops sharp LL peaks(figure13(f)). The results in figures11(d)(f) are consistentwith intrinsic graphene. In order to verify that the sequenceof peaks in figure13(f) does indeed correspond to massless-Dirac fermions, Liet al[65, 66]measured the dependence ofthe peak energies on field and level index and compared themwith the expected values (equation (5)):

En= ED

2ehv2F|N|B N= 0, 1 . . . , (9)

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Figure 13.Identifying a decoupled graphene layer. (a) Atomic resolution topography in region C of figure12(a) shows a triangular lattice.(b) STS in zero field and at T= 4.2 K in region C. (c) Finite field spectra (B= 3 T) in region C shows no LL peak sequence. (d) Atomicresolution topography in region G shows the honeycomb structure. (b) STS in zero field and at T= 4.2 K shows the V-shaped DOS thatvanishes at the DP expected for massless Dirac fermions. The Fermi energy is taken to be at zero. ( c) LLs are clearly seen in region G.Spectra atT= 4.2 K andB= 4 T . (ac modulation: 2 mV, junction resistance 6 G).

where ED is the energy at the DP. The N = 0 level isa consequence of the chirality of the Dirac fermions anddoes not exist in any other known 2D electron system. Thisfield-independent state at the DP together with a square-rootdependence on both field and level index are the hallmarks

of massless Dirac fermions. They are the criterion that isused for identifying graphene electronically decoupled fromthe environment or for determining the degree of couplingbetween coupled layers, as discussed below.

The field dependence of the STS spectra in region G,shown in figure14,exhibits an unevenly spaced sequence ofpeaks flanking symmetrically, in the electron and hole sectors,a peak at the DP. All the peaks, except the one at the DP, whichis identified with the N=0 LLs, fan out to higher energieswith increasing field. The peak heights increase with fieldconsistent with the increasing degeneracyof theLLs. To verifythat thesequenceis consistent with masslessDiracfermionswe

plot the peak positions as a function of the reduced parameter(|N|B)1/2 as shown in figure14(b). This scaling procedurecollapses all the data unto a straight line. Comparing withequation (9), the slope of the line gives a direct measure ofthe Fermi velocity, vF = 0.79106 m s1. This value is20% below that expected from single particle calculationsand, as discussed later, the reduction can be attributed to ephinteractions.

We conclude that the flake marked as region G iselectronically decoupled from the substrate.

LL spectroscopy. The technique described above, alsoknown as LL spectroscopy, was developed by Liet al[65, 66]

to probe the electronic properties of graphene on graphite.They showed that LL spectroscopy can be used to obtain

information about the intrinsic properties of graphene: theFermi velocity, the quasiparticle lifetime, the eph couplingand the degree of coupling to the substrate. LL spectroscopyis a powerful technique which gives access to the electronicproperties of Dirac fermions when they define the surface

electronic properties of a material and when it is possible totunnel into the surface states. The technique was adopted andsuccessfully implemented to probe massless Dirac fermionsin other systems including graphene on SiO2[120], epitaxialgraphene on SiC [150], graphene on Pt [151] and topologicalinsulators [152, 153].

An alternative method of accessing the LLs is to probe theallowed optical transitions between the LLs using cyclotronresonance measurements. This was demonstrated in earlyexperiments on SiO2[154, 155], epitaxial graphene [156]andmore recently on graphite [157].

Finding graphene on graphite. The flake in region G offigure 12 exhibits all the characteristicsof intrinsic graphenehoneycomb crystal structure, a V-shaped DOS which vanishesat the DP, an LL sequence which displays the characteristicsquare root dependence on field and level index, and containsan N= 0 level. One can use these criteria to develop arecipe for finding decoupled graphene flakes on graphite. Fora successful search one needs the following. (1) STM witha coarse motor that allows scanning large areas in search ofstacking faults or atomic steps. Decoupledgraphene is usuallyfound covering such faults as shown in figure 12. (2) Afine motor to zoom into subatomic length scales after havingidentifieda region of interest. If theatomic resolution image in

this region shows a honeycomb structure as in figure 13(d)onecontinues to the next step. (3) STS. If the region is completely

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-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

-150

-100

-50

0

50

100

150

2002 T4 T6 T8 T10 T

Energy(meV)

sgn(N) (|N|B)1/2

-200 -100 0 100 2000

100

200

300

400

500(a) (b)

0+

0- 10 T

8 T

6 T

4 T

2 TdI/dV(p

A/V)

Sample bias (mV)

T = 4.4 K

0 T

-1 1-2 2-3 3

Figure 14.LL spectroscopy of graphene. (a) Evolution of LLs with field at 4.4K and indicated values of field. (b) LL energies plottedagainst the reduced parameter (NB)1/2 collapse onto a straight line indicating the square root dependence on level index and field. Symbols

represent the peak positions obtained from (a) and the solid line is a fit to equation (9).

decoupled from the substrate the STS will produce a V-shapedspectrum as in figure13(e). (4) The last and crucial step is LLspectroscopy. A completely decoupled layer will exhibit thecharacteristic single-layer sequence and scaling as shown infigure13(f). In the presence of coupling to the substrate theLL sequence is modified. Importantly LL spectroscopy canbe used to quantify the degree of coupling to the substrate, asdiscussed later in the section on multi-layers.

LLs linewidth and electronelectron interactions. Com-

paring the LL spectra in figure 13(f) with the idealizedsequence of equal height delta peaks in figure 7,it is clearthat the spectrum is strongly modified by a finite linewidth.The data in figure14are resolution limited so in order toaccess the intrinsic broadening of the LL high resolutionspectra are obtained by decreasing the ac modulation until thespectrum becomes independent of the modulation amplitude.The peculiar V-shaped lower envelope of the spectrum is adirect consequence of the square root dependence on energyas we show below. Similarly, the down-sloping of the upperenvelope is a direct consequence of the linear increase inlinewidth with energy.

The sharpness and large signal-to noise ratio of the LLpeaks make it possible to extract the energy dependence of thequasiparticle lifetime from the spectrum. The level sequencecan be fit to high accuracy with a sum of peak functionscentered at the measured peak energies and with the linewidth of each peak left as free parameters. Comparing fitswith various line shapes, Li et al found that Lorentziansgive far better fits than Gaussians. This suggests that thelinewidthreflects the intrinsicquasiparticle lifetime rather thanimpurity broadening. From the measured energy dependenceof the linewidth (figure 15(b)) they found that the inversequasiparticle lifetime is

1

=|E|

+ 10

, (10)

where E is the LL energy in units of eV, 9 fs eV1 and0 0.5 ps at the Fermi level. The linear energy dependenceof the first term is attributed to the intrinsic lifetime of theDirac fermion quasiparticles. It was shown theoretically[158] that graphene should display marginal Fermi liquidcharacteristics leading to a linear energy dependence of theinverse quasiparticle lifetime arising from electronelectroninteractions, as opposed to the quadratic dependence in Fermiliquids. Theoretical estimates of the life time in zero field give

20fseV1. Since the electronelectron interactions are

enhanced in magnetic field, it is possible that the agreementwould be even better if calculations were made in finite field.The energy independent term in equation (10)correspondsto an extrinsic scattering mechanism with characteristic meanfree path oflmfp= vF0 400 nm. This is comparable to thesample size indicating that the extrinsic scattering is primarilydue to the boundaries and that inside the sample the motion isessentially ballistic. Note that hadthis been a diffusive sample,with the same carrier density (ns= 3 1010 cm2 ) and meanfree path, its transport mobility would be =evFlmfp/EF=220000cm2 V1 s1.

Line shape and LL spectrum. Several factors contribute toproduce the envelope of the LL spectra figure13(f): the finitelifetime of the quasiparticles which is inversely proportionalto the linewidth, the uneven spacing of the LL and the energydependence of the linewidths. Figure16illustrates how theV-shaped lower envelope of the spectrum builds up as theindividual LLs become broader when each level has the samewidth and the same degeneracy (or peak area). Since the peaksare unevenly spaced (equation (9)), the overlap between peaksincreases at higher energies, hence the increasing backgroundin the overall DOS with increasing energy away from the CNP(which here coincides with the Fermi energy). Comparingwith the spectrum in figure13(c) we note that that the N

=1

peak is higher than the N= 2 peak which is not the case inthe simulated spectra.

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-100 -80 -60 -40 -20

0.0

0.4

0.8

dI/dV(a.u.)

(a) (b)

sample bias (mV)

-100 -50 00

5

10

Wid

th(meV)

Energy (meV)

Figure 15.Quasiparticle lifetime in graphene. (a) High resolution tunneling spectrum of graphene in 4T at 4.4 K on the hole side(symbols). The solid line represents a fit with Lorentzian peaks. (b) Energy dependence of the peak width.

Figure 16.The origin of the V-shaped background in the DOS. Left panels: illustration of the levels and their increased overlap as thelinewidth is increased from the top to the bottom panel. The area under each peak is kept constant. Right panels: overall DOS. The unevenlyspaced peaks overlap to produce the V-shaped background. Energy unit: E1=

2ehv2FB.

In order to simulate the down-turn of the upper envelopeaway from zero energy seen in the high resolution spectra offigure13(f) one has to require the peak width to increase withenergy, as shown in figure17(d).

eph interaction and velocity renormalization. The single-electron physics of the carriers in graphene is captured in atight-binding model [11]. However, many-body effects areoften not negligible. Ab initiodensity functional calculations[159] show that eph interactions introduceadditional featuresin the electron self-energy, leading to a renormalized velocityat the Fermi energy vF

= vF0 (1 + )1, where vF0 is the

bare velocity and is the eph coupling constant. Awayfrom the Fermi energy, two dips are predicted in the velocity

renormalization factor,(vF vF0)/vF,at energiesEF hph,where ph is the characteristic phonon energy. Such dipsgive rise to shoulders in the zero-field DOS at the energy ofthe relevant phonons, and can provide a clear signature ofthe eph interactions in STS measurements. The tunnelingspectra measured on a decoupled graphene flake on graphiteexhibit two shoulders that flank the Fermi energy seen around150meV (figure18)which are independent of tipsampledistance for tunneling junction resistances in the range 3.850 G.

Further analysis of these features requires a calibrationof the zero-field DOS. This is done using the information

obtained from LL spectroscopy to ascertain massless Diracfermion nature of the excitations in this area, and to obtain

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Figure 17.(a) Simulated overall DOS including the energy dependence of the lifetime. (b) Individual peaks used to obtain the spectrumin (a).

Figure 18.Zero-field tunneling spectra at 4.4 K. Thick line is theDOS calculated according to equation (4). Thin lines are tunnelingspectra taken with different tunneling junction settings. Circles

highlight the shoulder features signaling deviations from thelinear DOS.

the average value ofvFfor energies up to 150 meV. The nextstep is to compare the expected DOS per unit cell with themeasured spectrum in order to calibrate the arbitrary units ofdI/dV. Since dI/dV is proportional to the DOS, (E), thelinear spectra at low energies in figure 13(f) together withequation (11) give

c(E) = 33/2a2

|E ED|h2v2F

= 0.123|E ED|. (11)

For an isotropic band (a good approximation for the relevantenergies E < 150meV), the dispersion is related to theDOS by

k(E) = 33/2a2

EED

() d

1/2

. (12)

The result, obtained by integrating the spectrum infigure13(f), is shown in figure19(a). Now the shouldersin figure18 appear as kinks in the dispersion. The energydependent velocity obtained from the dispersion

vF= dEh dk

(13)

plotted in figure 19(b) resembles that obtained by densityfunctional theory: it exhibits two dips at the energy of the

opticalbreathingphononA1, 150meV, suggesting that thisphonon, which couples the K and K valleys and undergoesa Kohn anomaly, is an important player in the velocityrenormalization. Incidentally, this same phonon is involved

in producing the D and 2D peaks in the Raman spectra ofgraphene.TheA1phonon has very large line width for single-layer

graphene, indicating strong eph coupling. However, theline width decreases significantly for bilayer graphene anddecreases even more for graphite [160, 161]. Therefore ephcoupling through the A1phonon is suppressed by interlayercoupling and the eph induced velocity renormalization isonly observed in single-layer graphene decoupled from thesubstrate. Consequently and paradoxically the Fermi velocityin multilayer graphene will be closer to the bare value, asdiscussed in the next section.

Multi-layersfrom weak to strong coupling. Unlike inconventional layered mat

electronic properties of graphene__a perspective from scanning tunneling microscopy and...

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