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The electronic properties of graphene

A. H. Castro Neto

Department of Physics, Boston University, 590 Commonwealth Avenue, Boston,Massachusetts 02215, USA

F. Guinea

Instituto de Ciencia de Materiales de Madrid, CSIC, Cantoblanco, E-28049 Madrid, Spain

N. M. R. Peres

Center of Physics and Department of Physics, Universidade do Minho, P-4710-057,Braga, Portugal

K. S. Novoselov and A. K. Geim

Department of Physics and Astronomy, University of Manchester, Manchester, M13 9PL,United Kingdom

Published 14 January 2009

This article reviews the basic theoretical aspects of graphene, a one-atom-thick allotrope of carbon,with unusual two-dimensional Dirac-like electronic excitations. The Dirac electrons can be controlledby application of external electric and magnetic fields, or by altering sample geometry and/or topology.The Dirac electrons behave in unusual ways in tunneling, confinement, and the integer quantum Halleffect. The electronic properties of graphene stacks are discussed and vary with stacking order andnumber of layers. Edge surface states in graphene depend on the edge termination zigzag orarmchair and affect the physical properties of nanoribbons. Different types of disorder modify theDirac equation leading to unusual spectroscopic and transport properties. The effects ofelectron-electron and electron-phonon interactions in single layer and multilayer graphene are alsopresented.

DOI: 10.1103/RevModPhys.81.109 PACS numbers: 81.05.Uw, 73.20.r, 03.65.Pm, 82.45.Mp

CONTENTS

I. Introduction 110II. Elementary Electronic Properties of Graphene 112

A. Single layer: Tight-binding approach 1121. Cyclotron mass 1132. Density of states 114

B. Dirac fermions 1141. Chiral tunneling and Klein paradox 1152. Confinement and Zitterbewegung 117

C. Bilayer graphene: Tight-binding approach 118D. Epitaxial graphene 119E. Graphene stacks 120

1. Electronic structure of bulk graphite 121F. Surface states in graphene 122G. Surface states in graphene stacks 124H. The spectrum of graphene nanoribbons 124

1. Zigzag nanoribbons 1252. Armchair nanoribbons 126

I. Dirac fermions in a magnetic field 126J. The anomalous integer quantum Hall effect 128K. Tight-binding model in a magnetic field 128L. Landau levels in graphene stacks 130M. Diamagnetism 130N. Spin-orbit coupling 131

III. Flexural Phonons, Elasticity, and Crumpling 132IV. Disorder in Graphene 134

A. Ripples 135

B. Topological lattice defects 136

C. Impurity states 137

D. Localized states near edges, cracks, and voids 137

E. Self-doping 138

F. Vector potential and gauge field disorder 139

1. Gauge field induced by curvature 140

2. Elastic strain 140

3. Random gauge fields 141

G. Coupling to magnetic impurities 141

H. Weak and strong localization 142

I. Transport near the Dirac point 143

J. Boltzmann equation description of dc transport in

doped graphene 144

K. Magnetotransport and universal conductivity 145

1. The full self-consistent Born approximation

FSBA 146

V. Many-Body Effects 148

A. Electron-phonon interactions 148

B. Electron-electron interactions 150

1. Screening in graphene stacks 152

C. Short-range interactions 152

1. Bilayer graphene: Exchange 153

2. Bilayer graphene: Short-range interactions 154

D. Interactions in high magnetic fields 154

VI. Conclusions 154

Acknowledgments 155

References 155

REVIEWS OF MODERN PHYSICS, VOLUME 81, JANUARY–MARCH 2009

0034-6861/2009/811/10954 ©2009 The American Physical Society109

http://dx.doi.org/10.1103/RevModPhys.81.109

I. INTRODUCTION

Carbon is the materia prima for life and the basis of allorganic chemistry. Because of the flexibility of its bond-ing, carbon-based systems show an unlimited number ofdifferent structures with an equally large variety ofphysical properties. These physical properties are, ingreat part, the result of the dimensionality of thesestructures. Among systems with only carbon atoms,graphene—a two-dimensional 2D allotrope ofcarbon—plays an important role since it is the basis forthe understanding of the electronic properties in otherallotropes. Graphene is made out of carbon atoms ar-ranged on a honeycomb structure made out of hexagonssee Fig. 1, and can be thought of as composed of ben-zene rings stripped out from their hydrogen atomsPauling, 1972. Fullerenes Andreoni, 2000 are mol-ecules where carbon atoms are arranged spherically, andhence, from the physical point of view, are zero-dimensional objects with discrete energy states.Fullerenes can be obtained from graphene with the in-troduction of pentagons that create positive curvaturedefects, and hence, fullerenes can be thought aswrapped-up graphene. Carbon nanotubes Saito et al.,1998; Charlier et al., 2007 are obtained by rollinggraphene along a given direction and reconnecting thecarbon bonds. Hence carbon nanotubes have only hexa-gons and can be thought of as one-dimensional 1D ob-jects. Graphite, a three dimensional 3D allotrope ofcarbon, became widely known after the invention of thepencil in 1564 Petroski, 1989, and its usefulness as aninstrument for writing comes from the fact that graphiteis made out of stacks of graphene layers that are weaklycoupled by van der Waals forces. Hence, when onepresses a pencil against a sheet of paper, one is actuallyproducing graphene stacks and, somewhere amongthem, there could be individual graphene layers. Al-though graphene is the mother for all these differentallotropes and has been presumably produced everytime someone writes with a pencil, it was only isolated440 years after its invention Novoselov et al., 2004. Thereason is that, first, no one actually expected grapheneto exist in the free state and, second, even with the ben-

efit of hindsight, no experimental tools existed to searchfor one-atom-thick flakes among the pencil debris cov-ering macroscopic areas Geim and MacDonald, 2007.Graphene was eventually spotted due to the subtle op-tical effect it creates on top of a chosen SiO2 substrateNovoselov et al., 2004 that allows its observation withan ordinary optical microscope Abergel et al., 2007;Blake et al., 2007; Casiraghi et al., 2007. Hence,graphene is relatively straightforward to make, but notso easy to find.

The structural flexibility of graphene is reflected in itselectronic properties. The sp2 hybridization between ones orbital and two p orbitals leads to a trigonal planarstructure with a formation of a bond between carbonatoms that are separated by 1.42 Å. The band is re-sponsible for the robustness of the lattice structure in allallotropes. Due to the Pauli principle, these bands havea filled shell and, hence, form a deep valence band. Theunaffected p orbital, which is perpendicular to the pla-nar structure, can bind covalently with neighboring car-bon atoms, leading to the formation of a band. Sinceeach p orbital has one extra electron, the band is halffilled.

Half-filled bands in transition elements have playedan important role in the physics of strongly correlatedsystems since, due to their strong tight-binding charac-ter, the Coulomb energies are large, leading to strongcollective effects, magnetism, and insulating behaviordue to correlation gaps or Mottness Phillips, 2006. Infact, Linus Pauling proposed in the 1950s that, on thebasis of the electronic properties of benzene, grapheneshould be a resonant valence bond RVB structurePauling, 1972. RVB states have become popular in theliterature of transition-metal oxides, and particularly instudies of cuprate-oxide superconductors Maple, 1998.This point of view should be contrasted with contempo-raneous band-structure studies of graphene Wallace,1947 that found it to be a semimetal with unusual lin-early dispersing electronic excitations called Dirac elec-trons. While most current experimental data ingraphene support the band structure point of view, therole of electron-electron interactions in graphene is asubject of intense research.

It was P. R. Wallace in 1946 who first wrote on theband structure of graphene and showed the unusualsemimetallic behavior in this material Wallace, 1947.At that time, the thought of a purely 2D structure wasnot reality and Wallace’s studies of graphene served himas a starting point to study graphite, an important mate-rial for nuclear reactors in the post–World War II era.During the following years, the study of graphite culmi-nated with the Slonczewski-Weiss-McClure SWM bandstructure of graphite, which provided a description ofthe electronic properties in this material McClure, 1957;Slonczewski and Weiss, 1958 and was successful in de-scribing the experimental data Boyle and Nozières1958; McClure, 1958; Spry and Scherer, 1960; Soule etal., 1964; Williamson et al., 1965; Dillon et al., 1977.From 1957 to 1968, the assignment of the electron andhole states within the SWM model were opposite to

FIG. 1. Color online Graphene top left is a honeycomblattice of carbon atoms. Graphite top right can be viewed asa stack of graphene layers. Carbon nanotubes are rolled-upcylinders of graphene bottom left. Fullerenes C60 are mol-ecules consisting of wrapped graphene by the introduction ofpentagons on the hexagonal lattice. From Castro Neto et al.,2006a.

110 Castro Neto et al.: The electronic properties of graphene

Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009

what is accepted today. In 1968, Schroeder et al.Schroeder et al., 1968 established the currently ac-cepted location of electron and hole pockets McClure,1971. The SWM model has been revisited in recentyears because of its inability to describe the van derWaals–like interactions between graphene planes, aproblem that requires the understanding of many-bodyeffects that go beyond the band-structure descriptionRydberg et al., 2003. These issues, however, do notarise in the context of a single graphene crystal but theyshow up when graphene layers are stacked on top ofeach other, as in the case, for instance, of the bilayergraphene. Stacking can change the electronic propertiesconsiderably and the layering structure can be used inorder to control the electronic properties.

One of the most interesting aspects of the grapheneproblem is that its low-energy excitations are massless,chiral, Dirac fermions. In neutral graphene, the chemicalpotential crosses exactly the Dirac point. This particulardispersion, that is only valid at low energies, mimics thephysics of quantum electrodynamics QED for masslessfermions except for the fact that in graphene the Diracfermions move with a speed vF, which is 300 timessmaller than the speed of light c. Hence, many of theunusual properties of QED can show up in graphene butat much smaller speeds Castro Neto et al., 2006a;Katsnelson et al., 2006; Katsnelson and Novoselov,2007. Dirac fermions behave in unusual ways whencompared to ordinary electrons if subjected to magneticfields, leading to new physical phenomena Gusynin andSharapov, 2005; Peres, Guinea, and Castro Neto, 2006asuch as the anomalous integer quantum Hall effectIQHE measured experimentally Novoselov, Geim,Morozov, et al., 2005a; Zhang et al., 2005. Besides beingqualitatively different from the IQHE observed in Siand GaAlAs heterostructures devices Stone, 1992,the IQHE in graphene can be observed at room tem-perature because of the large cyclotron energies for“relativistic” electrons Novoselov et al., 2007. In fact,the anomalous IQHE is the trademark of Dirac fermionbehavior.

Another interesting feature of Dirac fermions is theirinsensitivity to external electrostatic potentials due tothe so-called Klein paradox, that is, the fact that Diracfermions can be transmitted with probability 1 through aclassically forbidden region Calogeracos and Dombey,1999; Itzykson and Zuber, 2006. In fact, Dirac fermionsbehave in an unusual way in the presence of confiningpotentials, leading to the phenomenon of Zitter-bewegung, or jittery motion of the wave function Itzyk-son and Zuber, 2006. In graphene, these electrostaticpotentials can be easily generated by disorder. Since dis-order is unavoidable in any material, there has been agreat deal of interest in trying to understand how disor-der affects the physics of electrons in graphene and itstransport properties. In fact, under certain conditions,Dirac fermions are immune to localization effects ob-served in ordinary electrons Lee and Ramakrishnan,1985 and it has been established experimentally thatelectrons can propagate without scattering over large

distances of the order of micrometers in graphene No-voselov et al., 2004. The sources of disorder in grapheneare many and can vary from ordinary effects commonlyfound in semiconductors, such as ionized impurities inthe Si substrate, to adatoms and various molecules ad-sorbed in the graphene surface, to more unusual defectssuch as ripples associated with the soft structure ofgraphene Meyer, Geim, Katsnelson, Novoselov, Booth,et al., 2007a. In fact, graphene is unique in the sensethat it shares properties of soft membranes Nelson etal., 2004 and at the same time it behaves in a metallicway, so that the Dirac fermions propagate on a locallycurved space. Here analogies with problems of quantumgravity become apparent Fauser et al., 2007. The soft-ness of graphene is related with the fact that it has out-of-plane vibrational modes phonons that cannot befound in 3D solids. These flexural modes, responsiblefor the bending properties of graphene, also account forthe lack of long range structural order in soft mem-branes leading to the phenomenon of crumpling Nelsonet al., 2004. Nevertheless, the presence of a substrate orscaffolds that hold graphene in place can stabilize a cer-tain degree of order in graphene but leaves behind theso-called ripples which can be viewed as frozen flexuralmodes.

It was realized early on that graphene should alsopresent unusual mesoscopic effects Peres, Castro Neto,and Guinea, 2006a; Katsnelson, 2007a. These effectshave their origin in the boundary conditions required forthe wave functions in mesoscopic samples with varioustypes of edges graphene can have Nakada et al., 1996;Wakabayashi et al., 1999; Peres, Guinea, and CastroNeto, 2006a; Akhmerov and Beenakker, 2008. Themost studied edges, zigzag and armchair, have drasticallydifferent electronic properties. Zigzag edges can sustainedge surface states and resonances that are not presentin the armchair case. Moreover, when coupled to con-ducting leads, the boundary conditions for a grapheneribbon strongly affect its conductance, and the chiralDirac nature of fermions in graphene can be used forapplications where one can control the valley flavor ofthe electrons besides its charge, the so-called valleytron-ics Rycerz et al., 2007. Furthermore, when supercon-ducting contacts are attached to graphene, they lead tothe development of supercurrent flow and Andreev pro-cesses characteristic of the superconducting proximityeffect Heersche et al., 2007. The fact that Cooper pairscan propagate so well in graphene attests to the robustelectronic coherence in this material. In fact, quantuminterference phenomena such as weak localization, uni-versal conductance fluctuations Morozov et al., 2006,and the Aharonov-Bohm effect in graphene rings havealready been observed experimentally Recher et al.,2007; Russo, 2007. The ballistic electronic propagationin graphene can be used for field-effect devices such asp-n Cheianov and Fal’ko, 2006; Cheianov, Fal’ko, andAltshuler, 2007; Huard et al., 2007; Lemme et al., 2007;Tworzydlo et al., 2007; Williams et al., 2007; Fogler,Glazman, Novikov, et al., 2008; Zhang and Fogler, 2008and p-n-p Ossipov et al., 2007 junctions, and as “neu-

111Castro Neto et al.: The electronic properties of graphene


trino” billiards Berry and Modragon, 1987; Miao et al.,2007. It has also been suggested that Coulomb interac-tions are considerably enhanced in smaller geometries,such as graphene quantum dots Milton Pereira et al.,2007, leading to unusual Coulomb blockade effectsGeim and Novoselov, 2007 and perhaps to magneticphenomena such as the Kondo effect. The transportproperties of graphene allow for their use in a plethoraof applications ranging from single molecule detectionSchedin et al., 2007; Wehling et al., 2008 to spin injec-tion Cho et al., 2007; Hill et al., 2007; Ohishi et al., 2007;Tombros et al., 2007.

Because of its unusual structural and electronic flex-ibility, graphene can be tailored chemically and/or struc-turally in many different ways: deposition of metal at-oms Calandra and Mauri, 2007; Uchoa et al., 2008 ormolecules Schedin et al., 2007; Leenaerts et al., 2008;Wehling et al., 2008 on top; intercalation as done ingraphite intercalated compounds Dresselhaus et al.,1983; Tanuma and Kamimura, 1985; Dresselhaus andDresselhaus, 2002; incorporation of nitrogen and/orboron in its structure Martins et al., 2007; Peres,Klironomos, Tsai, et al., 2007 in analogy with what hasbeen done in nanotubes Stephan et al., 1994; and usingdifferent substrates that modify the electronic structureCalizo et al., 2007; Giovannetti et al., 2007; Varchon etal., 2007; Zhou et al., 2007; Das et al., 2008; Faugeras etal., 2008. The control of graphene properties can beextended in new directions allowing for the creation ofgraphene-based systems with magnetic and supercon-ducting properties Uchoa and Castro Neto, 2007 thatare unique in their 2D properties. Although thegraphene field is still in its infancy, the scientific andtechnological possibilities of this new material seem tobe unlimited. The understanding and control of this ma-terial’s properties can open doors for a new frontier inelectronics. As the current status of the experiment andpotential applications have recently been reviewedGeim and Novoselov, 2007, in this paper we concen-trate on the theory and more technical aspects of elec-tronic properties with this exciting new material.

II. ELEMENTARY ELECTRONIC PROPERTIES OFGRAPHENE

A. Single layer: Tight-binding approach

Graphene is made out of carbon atoms arranged inhexagonal structure, as shown in Fig. 2. The structurecan be seen as a triangular lattice with a basis of twoatoms per unit cell. The lattice vectors can be written as

a1 =a

23,3, a2 =

a

23,− 3 , 1

where a1.42 Å is the carbon-carbon distance. Thereciprocal-lattice vectors are given by

b1 =2

3a1,3, b2 =

2

3a1,− 3 . 2

Of particular importance for the physics of graphene arethe two points K and K at the corners of the grapheneBrillouin zone BZ. These are named Dirac points forreasons that will become clear later. Their positions inmomentum space are given by

K = 2

3a,

2

33a, K = 2

3a,−

2

33a . 3

The three nearest-neighbor vectors in real space aregiven by

1 =a

21,3 2 =

a

21,− 3 3 = − a1,0 4

while the six second-nearest neighbors are located at1= ±a1, 2= ±a2, 3= ± a2−a1.

The tight-binding Hamiltonian for electrons ingraphene considering that electrons can hop to bothnearest- and next-nearest-neighbor atoms has the formwe use units such that =1

H = − t i,j,

a,i† b,j + H.c.

− t i,j,

a,i† a,j + b,i

† b,j + H.c. , 5

where ai, ai,† annihilates creates an electron with

spin = ↑ , ↓ on site Ri on sublattice A an equiva-lent definition is used for sublattice B, t2.8 eV is thenearest-neighbor hopping energy hopping between dif-ferent sublattices, and t is the next nearest-neighborhopping energy1 hopping in the same sublattice. Theenergy bands derived from this Hamiltonian have theform Wallace, 1947

E±k = ± t3 + fk − tfk ,

1The value of t is not well known but ab initio calculationsReich et al., 2002 find 0.02t t0.2t depending on the tight-binding parametrization. These calculations also include theeffect of a third-nearest-neighbors hopping, which has a valueof around 0.07 eV. A tight-binding fit to cyclotron resonanceexperiments Deacon et al., 2007 finds t0.1 eV.

a

a

1

2

b

b

1

2

KΓ

k

k

x

y

1

2

3

M

δ δ

δ

A B

K’

FIG. 2. Color online Honeycomb lattice and its Brillouinzone. Left: lattice structure of graphene, made out of two in-terpenetrating triangular lattices a1 and a2 are the lattice unitvectors, and i, i=1,2 ,3 are the nearest-neighbor vectors.Right: corresponding Brillouin zone. The Dirac cones are lo-cated at the K and K points.



fk = 2 cos3kya + 4 cos32

kyacos32

kxa , 6

where the plus sign applies to the upper * and theminus sign the lower band. It is clear from Eq. 6that the spectrum is symmetric around zero energy if t=0. For finite values of t, the electron-hole symmetry isbroken and the and * bands become asymmetric. InFig. 3, we show the full band structure of graphene withboth t and t. In the same figure, we also show a zoom inof the band structure close to one of the Dirac points atthe K or K point in the BZ. This dispersion can beobtained by expanding the full band structure, Eq. 6,close to the K or K vector, Eq. 3, as k=K+q, withq K Wallace, 1947,

E±q ± vFq + Oq/K2 , 7

where q is the momentum measured relatively to theDirac points and vF is the Fermi velocity, given by vF=3ta /2, with a value vF 1106 m/s. This result wasfirst obtained by Wallace 1947.

The most striking difference between this result andthe usual case, q=q2 / 2m, where m is the electronmass, is that the Fermi velocity in Eq. 7 does not de-pend on the energy or momentum: in the usual case wehave v=k /m=2E /m and hence the velocity changessubstantially with energy. The expansion of the spectrumaround the Dirac point including t up to second orderin q /K is given by

E±q 3t ± vFq − 9ta2

4±

3ta2

8sin3qq2, 8

where

q = arctanqx

qy 9

is the angle in momentum space. Hence, the presence oft shifts in energy the position of the Dirac point andbreaks electron-hole symmetry. Note that up to orderq /K2 the dispersion depends on the direction in mo-mentum space and has a threefold symmetry. This is theso-called trigonal warping of the electronic spectrumAndo et al., 1998, Dresselhaus and Dresselhaus, 2002.

1. Cyclotron mass

The energy dispersion 7 resembles the energy of ul-trarelativistic particles; these particles are quantum me-chanically described by the massless Dirac equation seeSec. II.B for more on this analogy. An immediate con-sequence of this massless Dirac-like dispersion is a cy-clotron mass that depends on the electronic density as itssquare root Novoselov, Geim, Morozov, et al., 2005;Zhang et al., 2005. The cyclotron mass is defined, withinthe semiclassical approximation Ashcroft and Mermin,1976, as

m* =1

2 AE

E

E=EF

, 10

with AE the area in k space enclosed by the orbit andgiven by

AE = qE2 = E2

vF2 . 11

Using Eq. 11 in Eq. 10, one obtains

m* =EF

vF2 =

kF

vF. 12

The electronic density n is related to the Fermi momen-tum kF as kF

2 /=n with contributions from the twoDirac points K and K and spin included, which leads to

m* =vF

n . 13

Fitting Eq. 13 to the experimental data see Fig. 4provides an estimation for the Fermi velocity and the

FIG. 3. Color online Electronic dispersion in the honeycomblattice. Left: energy spectrum in units of t for finite values oft and t, with t=2.7 eV and t=−0.2t. Right: zoom in of theenergy bands close to one of the Dirac points.

FIG. 4. Color online Cyclotron mass of charge carriers ingraphene as a function of their concentration n. Positive andnegative n correspond to electrons and holes, respectively.Symbols are the experimental data extracted from the tem-perature dependence of the SdH oscillations; solid curves arethe best fit by Eq. 13. m0 is the free-electron mass. Adaptedfrom Novoselov, Geim, Morozov, et al., 2005.



hopping parameter as vF106 ms−1 and t3 eV, respec-tively. Experimental observation of the n dependenceon the cyclotron mass provides evidence for the exis-tence of massless Dirac quasiparticles in graphene No-voselov, Geim, Morozov, et al., 2005; Zhang et al., 2005;Deacon et al., 2007; Jiang, Henriksen, Tung, et al.,2007—the usual parabolic Schrödinger dispersion im-plies a constant cyclotron mass.

2. Density of states

The density of states per unit cell, derived from Eq.5, is given in Fig. 5 for both t=0 and t0, showing inboth cases semimetallic behavior Wallace, 1947; Benaand Kivelson, 2005. For t=0, it is possible to derive ananalytical expression for the density of states per unitcell, which has the form Hobson and Nierenberg, 1953

E =4

2

Et2

1Z0

F2

,Z1

Z0 ,

Z0 = 1 + E

t2

−E/t2 − 12

4, − t E t

4E

t , − 3t E − t ∨ t E 3t ,

Z1 = 4E

t , − t E t

1 + E

t2

−E/t2 − 12

4, − 3t E − t ∨ t E 3t , 14

where F /2 ,x is the complete elliptic integral of thefirst kind. Close to the Dirac point, the dispersion is ap-proximated by Eq. 7 and the density of states per unitcell is given by with a degeneracy of 4 included

E =2Ac

EvF

2 , 15

where Ac is the unit cell area given by Ac=33a2 /2. It isworth noting that the density of states for graphene isdifferent from the density of states of carbon nanotubesSaito et al., 1992a, 1992b. The latter shows 1/E singu-larities due to the 1D nature of their electronic spec-trum, which occurs due to the quantization of the mo-mentum in the direction perpendicular to the tube axis.From this perspective, graphene nanoribbons, whichalso have momentum quantization perpendicular to theribbon length, have properties similar to carbon nano-tubes.

B. Dirac fermions

We consider the Hamiltonian 5 with t=0 and theFourier transform of the electron operators,

an =1

Nck

e−ik·Rnak , 16

where Nc is the number of unit cells. Using this transfor-mation, we write the field an as a sum of two terms,coming from expanding the Fourier sum around K andK. This produces an approximation for the representa-tion of the field an as a sum of two new fields, written as

an e−iK·Rna1,n + e−iK·Rna2,n,

bn e−iK·Rnb1,n + e−iK·Rnb2,n, 17

-4 -2 0 20

1

2

3

4

5

ρ(ε)

t’=0.2t

0 0.2 0.4 0.6 0.8 10

0.05

0.1

0.15

0.2

-2 0 2ε /t

0

0.2

0.4

0.6

0.8

1

ρ(ε)

t’=0

-0.8 -0.4 0 0.4 0.8ε /t

0

0.1

0.2

0.3

0.4

FIG. 5. Density of states per unit cell as a function of energyin units of t computed from the energy dispersion 5, t=0.2t top and t=0 bottom. Also shown is a zoom-in of thedensity of states close to the neutrality point of one electronper site. For the case t=0, the electron-hole nature of thespectrum is apparent and the density of states close to theneutrality point can be approximated by .



where the index i=1 i=2 refers to the K K point.These new fields, ai,n and bi,n, are assumed to varyslowly over the unit cell. The procedure for derivinga theory that is valid close to the Dirac point con-sists in using this representation in the tight-

binding Hamiltonian and expanding the opera-tors up to a linear order in . In the derivation, oneuses the fact that e

±iK·=e±iK·=0. After some

straightforward algebra, we arrive at Semenoff,1984

H − t dxdy1†r 0 3a1 − i3/4

− 3a1 + i3/4 0x + 0 3a− i − 3/4

− 3ai − 3/4 0y1r

+ 2†r 0 3a1 + i3/4

− 3a1 − i3/4 0x + 0 3ai − 3/4

− 3a− i − 3/4 0y2r

= − ivF dxdy1†r · 1r + 2

†r* · 2r , 18

with Pauli matrices = x ,y, *= x ,−y, and i†

= ai† ,bi

† i=1,2. It is clear that the effective Hamil-tonian 18 is made of two copies of the massless Dirac-like Hamiltonian, one holding for p around K and theother for p around K. Note that, in first quantized lan-guage, the two-component electron wave function r,close to the K point, obeys the 2D Dirac equation,

− ivF · r = Er . 19

The wave function, in momentum space, for the mo-mentum around K has the form

±,Kk =12

e−ik/2

±eik/2 20

for HK=vF ·k, where the signs correspond to theeigenenergies E= ±vFk, that is, for the * and bands,respectively, and k is given by Eq. 9. The wave func-tion for the momentum around K has the form

±,Kk =12

eik/2

±e−ik/2 21

for HK=vF* ·k. Note that the wave functions at K andK are related by time-reversal symmetry: if we set theorigin of coordinates in momentum space in the M pointof the BZ see Fig. 2, time reversal becomes equivalentto a reflection along the kx axis, that is, kx ,ky→ kx ,−ky. Also note that if the phase is rotated by2, the wave function changes sign indicating a phase of in the literature this is commonly called a Berry’sphase. This change of phase by under rotation is char-acteristic of spinors. In fact, the wave function is a two-component spinor.

A relevant quantity used to characterize the eigen-functions is their helicity defined as the projection of themomentum operator along the pseudospin direction.The quantum-mechanical operator for the helicity hasthe form

h =12

·pp

. 22

It is clear from the definition of h that the states Krand Kr are also eigenstates of h,

hKr = ± 12Kr , 23

and an equivalent equation for Kr with inverted sign.Therefore, electrons holes have a positive negativehelicity. Equation 23 implies that has its two eigen-values either in the direction of ⇑ or against ⇓ themomentum p. This property says that the states of thesystem close to the Dirac point have well defined chiral-ity or helicity. Note that chirality is not defined in regardto the real spin of the electron that has not yet ap-peared in the problem but to a pseudospin variable as-sociated with the two components of the wave function.The helicity values are good quantum numbers as longas the Hamiltonian 18 is valid. Therefore, the existenceof helicity quantum numbers holds only as anasymptotic property, which is well defined close to theDirac points K and K. Either at larger energies or dueto the presence of a finite t, the helicity stops being agood quantum number.

1. Chiral tunneling and Klein paradox

In this section, we address the scattering of chiral elec-trons in two dimensions by a square barrier Katsnelsonet al., 2006; Katsnelson, 2007b. The one-dimensionalscattering of chiral electrons was discussed earlier in thecontext on nanotubes Ando et al., 1998; McEuen et al.,1999.

We start by noting that by a gauge transformation thewave function 20 can be written as



Kk =12

1

±eik . 24

We further assume that the scattering does not mix themomenta around K and K points. In Fig. 6, we depictthe scattering process due to the square barrier of widthD.

The wave function in the different regions can be writ-ten in terms of incident and reflected waves. In region I,we have

Ir =12

1

sei eikxx+kyy +r

2 1

sei− ei−kxx+kyy,

25

with =arctanky /kx, kx=kF cos , ky=kF sin , and kFthe Fermi momentum. In region II, we have

IIr =a2

1

sei eiqxx+kyy +b2

1

sei− ei−qxx+kyy,

26

with =arctanky /qx and

qx = V0 − E2/vF2 − ky

2, 27

and finally in region III we have a transmitted wave only,

IIIr =t

2 1

sei eikxx+kyy, 28

with s=sgnE and s=sgnE−V0. The coefficients r, a,b, and t are determined from the continuity of the wavefunction, which implies that the wave function has toobey the conditions Ix=0,y=IIx=0,y and IIx=D ,y=IIIx=D ,y. Unlike the Schrödinger equation,we only need to match the wave function but not itsderivative. The transmission through the barrier is ob-tained from T= tt* and has the form

T =cos2 cos2

cosDqxcos cos 2 + sin2Dqx1 − ss sin sin 2 . 29

This expression does not take into account a contribu-tion from evanescent waves in region II, which is usuallynegligible, unless the chemical potential in region II is atthe Dirac energy see Sec. IV.A.

Note that T=T−, and for values of Dqx satisfy-ing the relation Dqx=n, with n an integer, the barrierbecomes completely transparent since T=1, indepen-dent of the value of . Also, for normal incidence →0 and →0 and any value of Dqx, one obtains T0=1, and the barrier is again totally transparent. This re-sult is a manifestation of the Klein paradox Calogeracosand Dombey, 1999; Itzykson and Zuber, 2006 and doesnot occur for nonrelativistic electrons. In this latter caseand for normal incidence, the transmission is alwayssmaller than 1. In the limit V0 E, Eq. 29 has thefollowing asymptotic form:

T cos2

1 − cos2Dqxsin2 . 30

In Fig. 7, we show the angular dependence of T fortwo different values of the potential V0; it is clear thatthere are several directions for which the transmission is

1. Similar calculations were done for a graphene bilayerKatsnelson et al., 2006 with the absence of tunneling inthe forward ky=0 direction its most distinctive behav-ior.

The simplest example of a potential barrier is a squarepotential discussed previously. When intervalley scatter-ing and the lack of symmetry between sublattices areneglected, a potential barrier shows no reflection forelectrons incident in the normal direction Katsnelson etal., 2006. Even when the barrier separates regionswhere the Fermi surface is electronlike on one side andholelike on the other, a normally incident electron con-tinues propagating as a hole with 100% efficiency. Thisphenomenon is another manifestation of the chirality ofthe Dirac electrons within each valley, which preventsbackscattering in general. The transmission and reflec-tion probabilities of electrons at different angles dependon the potential profile along the barrier. A slowly vary-ing barrier is more efficient in reflecting electrons atnonzero incident angles Cheianov and Fal’ko, 2006.

Electrons moving through a barrier separating p- andn-doped graphene, a p-n junction, are transmitted as

D

V0E

x

Energy

x

y

θφ

φ

D

III III

FIG. 6. Color online Klein tunneling in graphene. Top: sche-matic of the scattering of Dirac electrons by a square potential.Bottom: definition of the angles and used in the scatteringformalism in regions I, II, and III.



holes. The relation between the velocity and the mo-mentum for a hole is the inverse of that for an electron.This implies that, if the momentum parallel to the bar-rier is conserved, the velocity of the quasiparticle is in-verted. When the incident electrons emerge from asource, the transmitting holes are focused into an imageof the source. This behavior is the same as that of pho-tons moving in a medium with negative reflection indexCheianov, Fal’ko, and Altshuler, 2007. Similar effectscan occur in graphene quantum dots, where the innerand outer regions contain electrons and holes, respec-tively Cserti, Palyi, and Peterfalvi, 2007. Note that thefact that barriers do not impede the transmission of nor-mally incident electrons does not preclude the existenceof sharp resonances, due to the confinement of electronswith a finite parallel momentum. This leads to the pos-sibility of fabricating quantum dots with potential barri-ers Silvestrov and Efetov, 2007. Finally, at half filling,due to disorder graphene can be divided in electron andhole charge puddles Katsnelson et al., 2006; Martin etal., 2008. Transport is determined by the transmissionacross the p-n junctions between these puddles Che-ianov, Fal’ko, Altshuler, et al., 2007; Shklovskii, 2007.There is much progress in the measurement of transportproperties of graphene ribbons with additional top gatesthat play the role of tunable potential barriers Han etal., 2007; Huard et al., 2007; Lemme et al., 2007; Özyil-maz et al., 2007; Williams et al., 2007.

A magnetic field and potential fluctuations break boththe inversion symmetry of the lattice and time-reversalsymmetry. The combination of these effects also breaksthe symmetry between the two valleys. The transmissioncoefficient becomes valley dependent, and, in general,electrons from different valleys propagate along differ-ent paths. This opens the possibility of manipulating thevalley index Tworzydlo et al., 2007 valleytronics in away similar to the control of the spin in mesoscopic de-vices spintronics. For large magnetic fields, a p-n junc-tion separates regions with different quantized Hall con-

ductivities. At the junction, chiral currents can flow atboth edges Abanin and Levitov, 2007, inducing back-scattering between the Hall currents at the edges of thesample.

The scattering of electrons near the Dirac point bygraphene-superconductor junctions differs from An-dreev scattering process in normal metals Titov andBeenakker, 2006. When the distance between the Fermienergy and the Dirac energy is smaller than the super-conducting gap, the superconducting interaction hybrid-izes quasiparticles from one band with quasiholes in theother. As in the case of scattering at a p-n junction, thetrajectories of the incoming electron and reflected holenote that hole here is meant as in the BCS theory ofsuperconductivity are different from those in similarprocesses in metals with only one type of carrier Bhat-tacharjee and Sengupta, 2006; Maiti and Sengupta,2007.

2. Confinement and Zitterbewegung

Zitterbewegung, or jittery motion of the wave functionof the Dirac problem, occurs when one tries to confinethe Dirac electrons Itzykson and Zuber, 2006. Local-ization of a wave packet leads, due to the Heisenbergprinciple, to uncertainty in the momentum. For a Diracparticle with zero rest mass, uncertainty in the momen-tum translates into uncertainty in the energy of the par-ticle as well this should be contrasted with the nonrela-tivistic case, where the position-momentum uncertaintyrelation is independent of the energy-time uncertaintyrelation. Thus, for an ultrarelativistic particle, a par-ticlelike state can have holelike states in its time evolu-tion. Consider, for instance, if one tries to construct awave packet at some time t=0, and assume, for simplic-ity, that this packet has a Gaussian shape of width w withmomentum close to K,

0r =e−r2/2w2

weiK·r , 31

where is spinor composed of positive energy statesassociated with +,K of Eq. 20. The eigenfunction ofthe Dirac equation can be written in terms of the solu-tion 20 as

r,t = d2k

22 a=±1

a,ka,Kke−iak·r+vFkt, 32

where ±,k are Fourier coefficients. We can rewrite Eq.31 in terms of Eq. 32 by inverse Fourier transformand find that

±,k = we−k2w2/2±,K† k . 33

Note that the relative weight of positive energy stateswith respect to negative energy states +/−, given byEq. 20 is 1, that is, there are as many positive energystates as negative energy states in a wave packet. Hence,these will cause the wave function to be delocalized atany time t0. Thus, a wave packet of electronlike stateshas holelike components, a result that puzzled many re-

-90 -75 -60 -45 -30 -15 0 15 30 45 60 75 900

0.2

0.4

0.6

0.8

1T

(φ)

V0 = 200 meV

V0 = 285 meV

-90 -75 -60 -45 -30 -15 0 15 30 45 60 75 90angle φ

0

0.2

0.4

0.6

0.8

1

T(φ

)

FIG. 7. Color online Angular behavior of T for two dif-ferent values of V0: V0=200 meV, dashed line; V0=285 meV,solid line. The remaining parameters are D=110 nm top, D=50 nm bottom E=80 meV, kF=2 /, and =50 nm.



searchers in the early days of QED Itzykson and Zuber,2006.

Consider the tight-binding description Peres, CastroNeto, and Guinea, 2006b; Chen, Apalkov, andChakraborty, 2007 of Sec. II.A when a potential Vi onsite Ri is added to the problem,

He = i

Vini, 34

where ni is the local electronic density. For simplicity, weassume that the confining potential is 1D, that is, that Vivanishes in the bulk but becomes large at the edge of thesample. We assume a potential that decays exponentiallyaway from the edges into the bulk with a penetrationdepth . In Fig. 8, we show the electronic spectrum for agraphene ribbon of width L=600a, in the presence of aconfining potential,

Vx = V0e−x−L/2/ + e−L/2−x/ , 35

where x is the direction of confinement and V0 is thestrength of the potential. One can see that in the pres-ence of the confining potential the electron-hole symme-try is broken and, for V00, the hole part of the spec-trum is distorted. In particular, for k close to the Diracpoint, we see that the hole dispersion is given byEn,=−1k−nk2−nk4, where n is a positive integer,and n0 n0 for nN* nN*. Hence, at n=N*

the hole effective mass diverges N* =0 and, by tuningthe chemical potential via a back gate to the holeregion of the spectrum 0 one should be able toobserve an anomaly in the Shubnikov–de Haas SdHoscillations. This is how Zitterbewegung could manifestitself in magnetotransport.

C. Bilayer graphene: Tight-binding approach

The tight-binding model developed for graphite canbe easily extended to stacks with a finite number ofgraphene layers. The simplest generalization is a bilayerMcCann and Fal’ko, 2006. A bilayer is interesting be-cause the IQHE shows anomalies, although differentfrom those observed in a single layer Novoselov et al.,2006, and also a gap can open between the conductionand valence band McCann and Fal’ko, 2006. The bi-layer structure, with the AB stacking of 3D graphite, isshown in Fig. 9.

The tight-binding Hamiltonian for this problem canbe written as

Ht.b. = − 0 i,jm,

am,i,† bm,j, + H.c.

− 1j,

a1,j,† a2,j, + H.c.,

− 4j,

a1,j,† b2,j, + a2,j,

† b1,j, + H.c.

− 3j,

b1,j,† b2,j, + H.c. , 36

where am,i, bm,i annihilates an electron with spin ,on sublattice A B, in plane m=1,2, at site Ri. Here weuse the graphite nomenclature for the hopping param-eters: 0= t is the in-plane hopping energy and 1 1= t0.4 eV in graphite Brandt et al., 1988; Dresselhausand Dresselhaus, 2002 is the hopping energy betweenatom A1 and atom A2 see Fig. 9, 4 40.04 eV ingraphite Brandt et al., 1988; Dresselhaus and Dressel-haus, 2002 is the hopping energy between atom A1A2 and atom B2 B1, and 3 30.3 eV in graphiteBrandt et al., 1988; Dresselhaus and Dresselhaus, 2002connects B1 and B2.

In the continuum limit, by expanding the momentumclose to the K point in the BZ, the Hamiltonian reads

H = kk

† · HK · k, 37

where ignoring 4 for the time being

HK − V vFk 0 33ak*

vFk* − V 1 0

0 1 V vFk

33ak 0 vFk* V , 38

and k=kx+ iky is a complex number; we have added V,which is here half the shift in electrochemical potentialbetween the two layers this term will appear if a poten-tial bias is applied between the layers, and

k† = „b1

†k,a1†k,a2

†k,b2†k… 39

is a four-component spinor.

Π2

kya

30.2

0.20.40.60.8

1

EeV

FIG. 8. Color online Energy spectrum in units of t for agraphene ribbon 600a wide, as a function of the momentum kalong the ribbon in units of 1 /3a, in the presence of confin-ing potential with V0=1 eV, =180a.

FIG. 9. Color online Lattice structure of bilayer graphene, itsrespective electronic hopping energies, and Brillouin zone. aLattice structure of the bilayer with the various hopping pa-rameters according to the SWM model. The A sublattices areindicated by darker spheres. b Brillouin zone. Adapted fromMalard et al., 2007.



If V=0 and 3 ,vFk1, one can eliminate the high-energy states perturbatively and write an effectiveHamiltonian,

HK 0vF

2k2

1+ 33ak*

vF2k*2

1+ 33ak 0 . 40

The hopping 4 leads to a k-dependent coupling be-tween the sublattices or a small renormalization of 1.The same role is played by the inequivalence betweensublattices within a layer.

For 3=0, Eq. 40 gives two parabolic bands, k,±

±vF2k2 / t, which touch at =0 as shown in Fig. 10.

The spectrum is electron-hole symmetric. There are twoadditional bands that start at ±t. Within this approxi-mation, the bilayer is metallic, with a constant density ofstates. The term 3 changes qualitatively the spectrum atlow energies since it introduces a trigonal distortion, orwarping, of the bands note that this trigonal distortion,unlike the one introduced by large momentum in Eq.8, occurs at low energies. The electron-hole symmetryis preserved but, instead of two bands touching at k=0,we obtain three sets of Dirac-like linear bands. OneDirac point is at =0 and k=0, while the three otherDirac points, also at =0, lie at three equivalent pointswith a finite momentum. The stability of points wherebands touch can be understood using topological argu-ments Mañes et al., 2007. The winding number of aclosed curve in the plane around a given point is aninteger representing the total number of times that thecurve travels counterclockwise around the point so thatthe wave function remains unaltered. The winding num-ber of the point where the two parabolic bands cometogether for 3=0 has winding number +2. The trigonalwarping term 3 splits it into a Dirac point at k=0 andwinding number −1, and three Dirac points at k0 andwinding numbers +1. An in-plane magnetic field, or asmall rotation of one layer with respect to the other,splits the 3=0 degeneracy into two Dirac points withwinding number +1.

The term V in Eq. 38 breaks the equivalence of thetwo layers, or, alternatively, inversion symmetry. In thiscase, the dispersion relation becomes

±,k2 = V2 + vF

2k2 + t2 /2 ± 4V2vF

2k2 + t2vF2k2 + t

4 /4,

41

giving rise to the dispersion shown in Fig. 11, and to theopening of a gap close to, but not directly at, the Kpoint. For small momenta, and V t, the energy of theconduction band can be expanded,

k V − 2VvF2k2/t + vF

4k4/2t2 V . 42

The dispersion for the valence band can be obtained byreplacing k by −k. The bilayer has a gap at k2

2V2 /vF2 . Note, therefore, that the gap in the biased

bilayer system depends on the applied bias and hencecan be measured experimentally McCann, 2006; Mc-Cann and Fal’ko, 2006; Castro, Novoselov, Morozov, etal., 2007. The ability to open a gap makes bilayergraphene interesting for technological applications.

D. Epitaxial graphene

It has been known for a long time that monolayers ofgraphene could be grown epitaxially on metal surfacesusing catalytic decomposition of hydrocarbons or carbonoxide Shelton et al., 1974; Eizenberg and Blakely, 1979;Campagnoli and Tosatti, 1989; Oshima and Nagashima,1997; Sinitsyna and Yaminsky, 2006. When such sur-faces are heated, oxygen or hydrogen desorbs, and thecarbon atoms form a graphene monolayer. The resultinggraphene structures could reach sizes up to a microme-ter with few defects, and they were characterized by dif-ferent surface-science techniques and local scanningprobes Himpsel et al., 1982. For example, graphenegrown on ruthenium has zigzag edges and also ripplesassociated with a 1010 reconstruction Vázquez deParga et al., 2008.

Graphene can also be formed on the surface of SiC.Upon heating, the silicon from the top layers desorbs,and a few layers of graphene are left on the surfaceBommel et al., 1975; Forbeaux et al., 1998; Coey et al.,2002; Berger et al., 2004; Rollings et al., 2006; Hass,Feng, Millán-Otoya, et al., 2007; de Heer et al., 2007.The number of layers can be controlled by limiting timeor temperature of the heating treatment. The qualityand the number of layers in the samples depend on theSiC face used for their growth de Heer et al., 2007 thecarbon-terminated surface produces few layers but witha low mobility, whereas the silicon-terminated surfaceproduces several layers but with higher mobility. Epi-taxially grown multilayers exhibit SdH oscillations with

FIG. 10. Color online Band structure of bilayer graphene ofV=0 and 3=0.

FIG. 11. Color online Band structure of bilayer graphene forV0 and 3=0.



a Berry phase shift of Berger et al., 2006, which is thesame as the phase shift for Dirac fermions observed in asingle layer as well as for some subbands present inmultilayer graphene and graphite Luk’yanchuk and Ko-pelevich, 2004. The carbon layer directly on top of thesubstrate is expected to be strongly bonded to it, and itshows no bands Varchon et al., 2007. The next layershows a 6363 reconstruction due to the substrate,and has graphene properties. An alternate route to pro-duce few layers of graphene is based on synthesis fromnanodiamonds Affoune et al., 2001.

Angle-resolved photoemission experiments ARPESshow that epitaxial graphene grown on SiC has linearlydispersing quasiparticles Dirac fermions Zhou,Gweon, et al., 2006; Bostwick, Ohta, Seyller, et al., 2007;Ohta et al., 2007, in agreement with the theoretical ex-pectation. Nevertheless, these experiments show thatthe electronic properties can change locally in space, in-dicating a certain degree of inhomogeneity due to thegrowth method Zhou et al., 2007. Similar inhomogene-ities due to disorder in the c-axis orientation of grapheneplanes are observed in graphite Zhou, Gweon, andLanzara, 2006. Moreover, graphene grown this way isheavily doped due to the charge transfer from the sub-strate to the graphene layer with the chemical potentialwell above the Dirac point and therefore all sampleshave strong metallic character with large electronic mo-bilities Berger et al., 2006; de Heer et al., 2007. There isalso evidence for strong interaction between a substrateand the graphene layer leading to the appearance ofgaps at the Dirac point Zhou et al., 2007. Indeed, gapscan be generated by the breaking of the sublattice sym-metry and, as in the case of other carbon-based systemssuch as polyacethylene Su et al., 1979, 1980, it can leadto solitonlike excitations Jackiw and Rebbi 1976; Hou etal., 2007. Multilayer graphene grown on SiC have alsobeen studied with ARPES Bostwick, Ohta, McChesney,et al., 2007; Ohta et al., 2007 and the results seem toagree quite well with band-structure calculations Mat-tausch and Pankratov, 2007. Spectroscopy measure-ments also show the transitions associated with Landaulevels Sadowski et al., 2006 and weak-localization ef-fects at low magnetic fields, also expected for Dirac fer-mions Wu et al., 2007. Local probes reveal a rich struc-ture of terraces Mallet et al., 2007 and interferencepatterns due to defects at or below the graphene layersRutter et al., 2007.

E. Graphene stacks

In stacks with more than one graphene layer, twoconsecutive layers are normally oriented in such a waythat the atoms in one of the two sublattices An ofthe honeycomb structure of one layer are directlyabove one-half of the atoms in the neighboring layer,sublattice An±1. The second set of atoms in one layer sitson top of the empty center of a hexagon in the otherlayer. The shortest distance between carbon atoms indifferent layers is dAnAn±1

=c=3.4 Å. The next distance is

dAnBn±1=c2+a2. This is the most common arrangement

of nearest-neighbor layers observed in nature, althougha stacking order in which all atoms in one layer occupypositions directly above the atoms in the neighboringlayers hexagonal stacking has been considered theo-retically Charlier et al., 1991 and appears in graphiteintercalated compounds Dresselhaus and Dresselhaus,2002.

The relative position of two neighboring layers allowsfor two different orientations of the third layer. If welabel the positions of the first two atoms as 1 and 2, thethird layer can be of type 1, leading to the sequence 121,or it can fill a third position different from 1 and 2 seeFig. 12, labeled 3. There are no more inequivalent po-sitions where a new layer can be placed, so that thickerstacks can be described in terms of these three orienta-tions. In the most common version of bulk graphite, thestacking order is 1212… Bernal stacking. Regions withthe stacking 123123… rhombohedral stacking havealso been observed in different types of graphite Bacon,1950; Gasparoux, 1967. Finally, samples with no dis-cernible stacking order turbostratic graphite are alsocommonly reported.

Beyond two layers, the stack ordering can be arbi-trarily complex. Simple analytical expressions for theelectronic bands can be obtained for perfect Bernal1212… and rhombohedral 123123… stacking Guineaet al., 2006; Partoens and Peeters, 2006. Even if we con-sider one interlayer hopping t=1, the two stacking or-ders show different band structures near =0. A Bernalstack with N layers, N even, has N /2 electronlike andN /2 holelike parabolic subbands touching at =0. WhenN is odd, an additional subband with linear Dirac dis-persion emerges. Rhombohedral systems have only twosubbands that touch at =0. These subbands disperse askN, and become surface states localized at the top andbottom layers when N→. In this limit, the remaining2N−2 subbands of a rhombohedral stack become Dirac-like, with the same Fermi velocity as a single graphenelayer. The subband structure of a trilayer with the Ber-nal stacking includes two touching parabolic bands, andone with Dirac dispersion, combining the features of bi-layer and monolayer graphene.

The low-energy bands have different weights on thetwo sublattices of each graphene layer. The states at a

FIG. 12. Color online Sketch of the three inequivalent orien-tations of graphene layers with respect to each other.



site directly coupled to the neighboring planes arepushed to energies ± t. The bands near =0 are lo-calized mostly at the sites without neighbors in the nextlayers. For the Bernal stacking, this feature implies thatthe density of states at =0 at sites without nearestneighbors in the contiguous layers is finite, while it van-ishes linearly at the other sites. In stacks with rhombo-hedral stacking, all sites have one neighbor in anotherplane, and the density of states vanishes at =0 Guineaet al., 2006. This result is consistent with the well knownfact that only one of the two sublattices at a graphitesurface can be resolved by scanning tunneling micros-copy STM Tománek et al., 1987.

As in the case of a bilayer, an inhomogeneous chargedistribution can change the electrostatic potential in thedifferent layers. For more than two layers, this breakingof the equivalence between layers can take place even inthe absence of an applied electric field. It is interestingto note that a gap can open in a stack with Bernal or-dering and four layers if the electronic charge at the twosurface layers is different from that at the two innerones. Systems with a higher number of layers do notshow a gap, even in the presence of charge inhomoge-neity. Four representative examples are shown in Fig. 13.The band structure analyzed here will be modified bythe inclusion of the trigonal warping term, 3. Experi-mental studies of graphene stacks have showed that,with an increasing number of layers, the system becomesincreasingly metallic concentration of charge carriers atzero energy gradually increases, and there appear sev-eral types of electronlike and holelike carries No-voselov et al., 2004; Morozov et al., 2005. An inhomo-geneous charge distribution between layers becomesvery important in this case, leading to 2D electron andhole systems that occupy only a few graphene layersnear the surface, and can completely dominate transportproperties of graphene stacks Morozov et al., 2005.

The degeneracies of the bands at =0 can be studiedusing topological arguments Mañes et al., 2007. Multi-layers with an even number of layers and Bernal stack-ing have inversion symmetry, leading to degeneracieswith winding number +2, as in the case of a bilayer. Thetrigonal lattice symmetry implies that these points canlead, at most, to four Dirac points. In stacks with an oddnumber of layers, these degeneracies can be completelyremoved. The winding number of the degeneraciesfound in stacks with N layers and orthorhombic orderingis ±N. The inclusion of trigonal warping terms will leadto the existence of many weaker degeneracies near =0.

Furthermore, it is well known that in graphite, theplanes can be rotated relative to each other giving rise toMoiré patterns that are observed in STM of graphitesurfaces Rong and Kuiper, 1993. The graphene layerscan be rotated relative to each other due to the weakcoupling between planes that allows for the presence ofmany different orientational states that are quasidegen-erate in energy. For certain angles, the graphene layersbecome commensurate with each other leading to a low-ering of the electronic energy. Such a phenomenon is

quite similar to the commensurate-incommensuratetransitions observed in certain charge-density-wave sys-tems or adsorption of gases on graphite Bak, 1982.This kind of electronic structure dependence on therelative rotation angle between graphene layers leads towhat is called superlubricity in graphite Dienwiebel etal., 2004, namely, the vanishing of the friction betweenlayers as a function of the rotation angle. In the case ofbilayer graphene, a rotation by a small commensurateangle leads to the effective decoupling between layersand recovery of the linear Dirac spectrum of the singlelayer, albeit with a modification on the value of theFermi velocity Lopes dos Santos et al., 2007.

1. Electronic structure of bulk graphite

The tight-binding description of graphene describedearlier can be extended to systems with an infinite num-ber of layers. The coupling between layers leads to hop-ping terms between orbitals in different layers, leadingto the so-called Slonczewski-Weiss-McClure modelSlonczewski and Weiss, 1958. This model describes theband structure of bulk graphite with the Bernal stacking

(b)

(a)

(c)

(d)

FIG. 13. Color online Electronic bands of graphene multilay-ers. a Biased bilayer. b Trilayer with Bernal stacking. cTrilayer with orthorhombic stacking. d Stack with four layerswhere the top and bottom layers are shifted in energy withrespect to the two middle layers by +0.1 eV.



order in terms of seven parameters: 0, 1, 2, 3, 4, 5,and . The parameter 0 describes the hopping withineach layer, and it has been considered previously. Thecoupling between orbitals in atoms that are nearestneighbors in successive layers is 1, which we called t

earlier. The parameters 3 and 4 describe the hoppingbetween orbitals at next nearest neighbors in successivelayers and were discussed in the case of the bilayer. Thecouplings between orbitals at next-nearest-neighbor lay-ers are 2 and 5. Finally, is an on-site energy thatreflects the inequivalence between the two sublattices ineach graphene layer once the presence of neighboringlayers is taken into account. The values of these param-eters, and their dependence with pressure, or, equiva-lently, the interatomic distances, have been extensivelystudied McClure, 1957, Nozières, 1958; Dresselhaus andMavroides, 1964; Soule et al., 1964; Dillon et al., 1977;Brandt et al., 1988. A representative set of values isshown in Table I.

The unit cell of graphite with Bernal stacking includestwo layers, and two atoms within each layer. The tight-binding Hamiltonian described previously can be repre-sented as a 44 matrix. In the continuum limit, the twoinequivalent corners of the BZ can be treated separately,and the in-plane terms can be described by the Diracequation. The next terms of importance for the low-energy electronic spectrum are the nearest-neighborcouplings 1 and 3. The influence of the parameter 4on the low-energy bands is much smaller, as discussedbelow. Finally, the fine details of the spectrum of bulkgraphite are determined by , which breaks theelectron-hole symmetry of the bands preserved by 0, 1,and 3. It is usually assumed to be much smaller than theother terms.

We label the two atoms from the unit cell in one layeras 1 and 2, and 3 and 4 correspond to the second layer.Atoms 2 and 3 are directly on top of each other. Then,the matrix elements of the Hamiltonian can be writtenas

H11K = 22 cos2kz/c ,

H12K = vFkx + iky ,

H13K =

34a

21 + eikzckx + iky ,

H14K =

33a

21 + eikzckx − iky ,

H22K = + 25 cos2kz/c ,

H23K = 11 + eikzc ,

H24K =

34a

21 + eikzckx + iky ,

H33K = + 25 cos2kz/c ,

H34K = vFkx + iky ,

H44K = 22 cos2kz/c , 43

where c is the lattice constant in the out-of-plane direc-tion, equal to twice the interlayer spacing. The matrixelements of HK can be obtained by replacing kx by −kxother conventions for the unit cell and the orientationof the lattice lead to different phases. Recent ARPESexperiments Ohta et al., 2006; Zhou, Gweon, and Lan-zara, 2006; Zhou, Gweon, et al., 2006; Bostwick, Ohta,Seyller, et al., 2007 performed in epitaxially growngraphene stacks Berger et al., 2004 confirm the mainfeatures of this model, formulated on the basis of Fermisurface measurements McClure, 1957; Soule et al.,1964. The electronic spectrum of the model can also becalculated in a magnetic field de Gennes, 1964; Nakao,1976, and the results are also consistent with STM ongraphite surfaces Kobayashi et al., 2005; Matsui et al.,2005; Niimi et al., 2006; Li and Andrei, 2007, epitaxiallygrown graphene stacks Mallet et al., 2007, and opticalmeasurements in the infrared range Li et al., 2006.

F. Surface states in graphene

So far we have discussed the basic bulk properties ofgraphene. Nevertheless, graphene has very interestingsurface edge states that do not occur in other systems.A semi-infinite graphene sheet with a zigzag edge has aband of zero-energy states localized at the surfaceFujita et al., 1996; Nakada et al., 1996; Wakabayashi etal., 1999. In Sec. II.H, we discuss the existence of edgestates using the Dirac equation. Here we discuss thesame problem using the tight-binding Hamiltonian. Tosee why these edge states exist, we consider the ribbongeometry with zigzag edges shown in Fig. 14. The ribbonwidth is such that it has N unit cells in the transversecross section y direction. We assume that the ribbonhas infinite length in the longitudinal direction x direc-tion.

We rewrite Eq. 5, with t=0, in terms of the integerindices m and n, introduced in Fig. 14, and labeling theunit cells,

TABLE I. Band-structure parameters of graphite Dressel-haus and Dresselhaus, 2002.

0 3.16 eV1 0.39 eV2 −0.020 eV3 0.315 eV4 0.044 eV5 0.038 eV −0.008 eV



H = − t m,n,

a†m,nbm,n + a

†m,nbm − 1,n

+ a†m,nbm,n − 1 + H.c. . 44

Given that the ribbon is infinite in the a1 direction, onecan introduce a Fourier decomposition of the operatorsleading to

H = − t dk

2n,

a†k,nbk,n + eikaa

†k,nbk,n

+ a†k,nbk,n − 1 + H.c. , 45

where c†k ,n0= c , ,k ,n and c=a ,b. The one-

particle Hamiltonian can be written as

H1p = − t dkn,

1 + eikaa,k,n,b,k,n,

+ a,k,n,b,k,n − 1, + H.c. . 46

The solution of the Schrödinger equation H1p ,k ,=E,k ,k , can be generally expressed as

,k, = n

k,na,k,n, + k,nb,k,n, ,

47

where the coefficients and satisfy the followingequations:

E,kk,n = − t1 + eikak,n + k,n − 1 , 48

E,kk,n = − t1 + e−ikak,n + k,n + 1 . 49

As the ribbon has a finite width, we have to be carefulwith the boundary conditions. Since the ribbon only ex-ists between n=0 and n=N−1 at the boundary, Eqs. 48and 49 read

E,kk,0 = − t1 + eikak,0 , 50

E,kk,N − 1 = − t1 + e−ikak,N − 1 . 51

The surface edge states are solutions of Eqs. 48–51with E,k=0,

0 = 1 + eikak,n + k,n − 1 , 52

0 = 1 + e−ikak,n + k,n + 1 , 53

0 = k,0 , 54

0 = k,N − 1 . 55

Equations 52 and 53 are easily solved, giving

k,n = − 2 coska/2neika/2nk,0 , 56

k,n = − 2 coska/2N−1−n

e−ika/2N−1−nk,N − 1 . 57

Consider, for simplicity, a semi-infinite system with asingle edge. We must require the convergence condition−2 coska /21 in Eq. 57 because otherwise thewave function would diverge in the semi-infinitegraphene sheet. Therefore, the semi-infinite system hasedge states for ka in the region 2 /3ka4 /3, whichcorresponds to 1/3 of the possible momenta. Note thatthe amplitudes of the edge states are given by

k,n = 2

ke−n/k, 58

k,n = 2

ke−N−1−n/k, 59

where the penetration length is given by

k = − 1/ln2 coska/2 . 60

It is easily seen that the penetration length divergeswhen ka approaches the limits of the region2 /3 ,4 /3.

Although the boundary conditions defined by Eqs.54 and 55 are satisfied for solutions 56 and 57 inthe semi-infinite system, they are not in the ribbon ge-ometry. In fact, Eqs. 58 and 59 are eigenstates only inthe semi-infinite system. In the graphene ribbon the twoedge states, which come from both sides of the edge, willoverlap with each other. The bonding and antibondingstates formed by the two edge states will then be theribbon eigenstates Wakabayashi et al., 1999 note thatat zero energy there are no other states with which theedge states could hybridize. As bonding and antibond-ing states result in a gap in energy, the zero-energy flatbands of edge states will become slightly dispersive, de-pending on the ribbon width N. The overlap betweenthe two edge states is larger as ka approaches 2 /3 and4 /3. This means that deviations from zero-energy flat-ness will be stronger near these points.

Edge states in graphene nanoribbons, as in carbonnanotubes, are predicted to be Luttinger liquids, that is,interacting one-dimensional electron systems CastroNeto et al., 2006b. Hence, clean nanoribbons must have1D square root singularities in their density of statesNakada et al., 1996 that can be probed by Raman spec-troscopy. Disorder may smooth out these singularities,however. In the presence of a magnetic field, when thebulk states are gapped, the edge states are responsiblefor the transport of spin and charge Abanin et al., 2006;Abanin, Lee, and Levitov, 2007; Abanin and Levitov,2007; Abanin, Novoselov, Zeitler, et al., 2007.

x

y

m− m m+

n+

m+

n+

n

n−

1

1

1

1

2

2

1a

2a

A

B

FIG. 14. Color online Ribbon geometry with zigzag edges.



G. Surface states in graphene stacks

Single-layer graphene can be considered a zero gapsemiconductor, which leads to the possibility of midgapstates, at =0, as discussed in the previous section. Themost studied such states are those localized near agraphene zigzag edge Fujita et al., 1996; Wakayabashiand Sigrist, 2000. It can be shown analytically Castro,Peres, Lopes dos Santos, et al., 2008 that a bilayer zig-zag edge, like that shown in Fig. 15, analyzed within thenearest-neighbor tight-binding approximation describedbefore, has two bands of localized states, one completelylocalized in the top layer and indistinguishable fromsimilar states in single-layer graphene, and another bandthat alternates between the two layers. These states, at=0, have finite amplitudes on one-half of the sites only.

These bands, as in single-layer graphene, occupy one-third of the BZ of a stripe bounded by zigzag edges.They become dispersive in a biased bilayer. As graphitecan be described in terms of effective bilayer systems,one for each value of the perpendicular momentum kz,bulk graphite with a zigzag termination should show onesurface band per layer.

H. The spectrum of graphene nanoribbons

The spectrum of graphene nanoribbons depends onthe nature of their edges: zigzag or armchair Brey andFertig, 2006a, 2006b; Nakada et al., 1996. In Fig. 16, weshow a honeycomb lattice having zigzag edges along thex direction and armchair edges along the y direction. Ifwe choose the ribbon to be infinite in the x direction, weproduce a graphene nanoribbon with zigzag edges; con-versely, choosing the ribbon to be macroscopically largealong the y but finite in the x direction, we produce agraphene nanoribbon with armchair edges.

In Fig. 17, we show 14 energy levels, calculated in thetight-binding approximation, closest to zero energy for ananoribbon with zigzag and armchair edges and of widthN=200 unit cells. We show that both are metallic, andthat the zigzag ribbon presents a band of zero-energymodes that is absent in the armchair case. This band at

zero energy is the surface states living near the edge ofthe graphene ribbon. More detailed ab initio calcula-tions of graphene nanoribbon spectra show that interac-tion effects can lead to electronic gaps Son et al., 2006band magnetic states close to the graphene edges, inde-pendent of their nature Son et al., 2006a; Yang, Cohen,and Louie, 2007; Yang, Park, Son, et al., 2007.

From the experimental point of view, however,graphene nanoribbons currently have a high degree ofroughness at the edges. Such edge disorder can changesignificantly the properties of edge states Areshkin andWhite, 2007; Gunlycke et al., 2007, leading to Andersonlocalization and anomalies in the quantum Hall effectCastro Neto et al., 2006b; Martin and Blanter, 2007 aswell as Coulomb blockade effects Sols et al., 2007. Sucheffects have already been observed in lithographicallyengineered graphene nanoribbons Han et al., 2007;

x

y

m+1 m+2m- 1 m

n+1

n

2a

1a

B1

A2B2

A1

FIG. 15. Sketch of a zigzag termination of a graphene bilayer.As discussed by Castro, Peres, Lopes dos Santos, et al. 2008,there is a band of surface states completely localized in thebottom layer, and another surface band which alternates be-tween the two.

zigzag edge

arm

chai

red

ge

y

x

a

A

B

FIG. 16. Color online A piece of a honeycomb lattice dis-playing both zigzag and armchair edges.

0 1 2 3 4 5 6 7-1

0

1

ener

gy/t

0 0.1 0.2 0.3 0.4-0.2

0

0.2

0 1 2 3 4 5 6 7momentum ka

-1

0

1

ener

gy/t

1.5 2 2.5 3 3.5 4 4.5momentum ka

-0.2

0

0.2

FIG. 17. Electronic dispersion for graphene nanoribbons. Left:energy spectrum, as calculated from the tight-binding equa-tions, for a nanoribbon with armchair top and zigzag bot-tom edges. The width of the nanoribbon is N=200 unit cells.Only 14 eigenstates are depicted. Right: zoom of the low-energy states shown on the right.



Özyilmaz et al., 2007. Furthermore, the problem of edgepassivation by hydrogen or other elements is not cur-rently understood experimentally. Passivation can bemodeled in the tight-binding approach by modificationsof the hopping energies Novikov, 2007 or via addi-tional phases in the boundary conditions Kane andMele, 1997. Theoretical modeling of edge passivationindicate that those have a strong effect on the electronicproperties at the edge of graphene nanoribbons Baroneet al., 2006; Hod et al., 2007.

In what follows, we derive the spectrum for both zig-zag and armchair edges directly from the Dirac equa-tion. This was originally done both with and without amagnetic field Nakada et al., 1996; Brey and Fertig,2006a, 2006b.

1. Zigzag nanoribbons

In the geometry of Fig. 16, the unit-cell vectors area1=a01,0 and a2=a01/2 ,3/2, which generate theunit vectors of the BZ given by b1

=4 / a033/2 ,−1/2 and b2=4 / a030,1. Fromthese two vectors, we find two inequivalent Dirac pointsgiven by K= 4 /3a0 ,0= K ,0 and K= −4 /3a0 ,0= −K ,0, with a0=3a. The Dirac Hamil-tonian around the Dirac point K reads in momentumspace

HK = vF 0 px − ipy

px + ipy 0 , 61

and around the K as

HK = vF 0 px + ipy

px − ipy 0 . 62

The wave function, in real space, for the sublattice A isgiven by

Ar = eiK·rAr + eiK·rA r , 63

and for sublattice B is given by

Br = eiK·rBr + eiK·rB r , 64

where A and B are the components of the spinor wavefunction of Hamiltonian 61 and A and B have iden-tical meaning but relative to Eq. 62. Assume that theedges of the nanoribbons are parallel to the x axis. Inthis case, the translational symmetry guarantees that thespinor wave function can be written as

r = eikxxAyBy

, 65

and a similar equation for the spinor of Hamiltonian62. For zigzag edges, the boundary conditions at theedge of the ribbon located at y=0 and y=L, where L isthe ribbon width are

Ay = L = 0, By = 0 = 0, 66

leading to

0 = eiKxeikxxAL + e−iKxeikxxA L , 67

0 = eiKxeikxxB0 + e−iKxeikxxB 0 . 68

The boundary conditions 67 and 68 are satisfied forany x by the choice

AL = A L = B0 = B 0 = 0. 69

We need now to find out the form of the envelope func-tions. The eigenfunction around the point K has theform

0 kx − y

kx + y 0Ay

By = Ay

By , 70

with = /vF and the energy eigenvalue. The eigen-problem can be written as two linear differential equa-tions of the form

kx − yB = A,

kx + yA = B. 71

Applying the operator kx+y to the first of Eqs. 71leads to

− y2 + kx

2B = 2B, 72

with A given by

A =1

kx − yB. 73

The solution of Eq. 72 has the form

B = Aezy + Be−zy, 74

leading to an eigenenergy 2=kx2−z2. The boundary con-

ditions for a zigzag edge require that Ay=L=0 andBy=0=0, leading to

By = 0 = 0 ⇔ A + B = 0,

Ay = L = 0 ⇔ kx − zAezL + kx + zBe−zL = 0,

75

which leads to an eigenvalue equation of the form

e−2zL =kx − z

kx + z. 76

Equation 76 has real solutions for z, whenever kx ispositive; these solutions correspond to surface wavesedge states existing near the edge of the graphene rib-bon. In Sec. II.F, we discussed these states from thepoint of view of the tight-binding model. In addition toreal solutions for z, Eq. 76 also supports complex ones,of the form z= ikn, leading to

kx =kn

tanknL. 77

The solutions of Eq. 77 correspond to confined modesin the graphene ribbon.



If we apply the same procedure to the Dirac equationaround the Dirac point K, we obtain a different eigen-value equation given by

e−2zL =kx + z

kx − z. 78

This equation supports real solutions for z if kx is nega-tive. Therefore, we have edge states for negative valueskx, with momentum around K. As in the case of K, thesystem also supports confined modes, given by

kx = −kn

tanknL. 79

One should note that the eigenvalue equations for Kare obtained from those for K by inversion, kx→−kx.

We finally note that the edge states for zigzag nano-ribbons are dispersionless localized in real space whent=0. When electron-hole symmetry is broken t0,these states become dispersive with a Fermi velocity ve ta Castro Neto et al., 2006b.

2. Armchair nanoribbons

We now consider an armchair nanoribbon with arm-chair edges along the y direction. The boundary condi-tions at the edges of the ribbon located at x=0 and x=L, where L is the width of the ribbon,

Ax = 0 = Bx = 0 = Ax = L = Bx = L = 0.

80

Translational symmetry guarantees that the spinor wavefunction of Hamiltonian 61 can be written as

r = eikyyAxBx

, 81

and a similar equation for the spinor of the Hamiltonian62. The boundary conditions have the form

0 = eikyyA0 + eikyyA 0 , 82

0 = eikyyB0 + eikyyB 0 , 83

0 = eiKLeikyyAL + e−iKLeikyyA L , 84

0 = eiKLeikyyBL + e−iKLeikyyB L , 85

and are satisfied for any y if

0 + 0 = 0 86

and

eiKLL + e−iKL L = 0, 87

with =A ,B. It is clear that these boundary conditionsmix states from the two Dirac points. Now we must findthe form of the envelope functions obeying the bound-ary conditions 86 and 87. As before, the functions B

and B obey the second-order differential equation 72with y replaced by x, and the functions A and A are

determined from Eq. 73. The solutions of Eq. 72 havethe form

B = Aeiknx + Be−iknx, 88

B = Ceiknx + De−iknx. 89

Applying the boundary conditions 86 and 87, one ob-tains

0 = A + B + C + D , 90

0 = Aeikn+KL + De−ikn+KL + Be−ikn−KL + Ceikn−KL.

91

The boundary conditions are satisfied with the choice

A = − D, B = C = 0, 92

which leads to sinkn+KL=0. Therefore, the allowedvalues of kn are given by

kn =n

L−

4

3a0, 93

and the eigenenergies are given by

2 = ky2 + kn

2 . 94

No surface states exist in this case.

I. Dirac fermions in a magnetic field

We now consider the problem of a uniform magneticfield B applied perpendicular to the graphene plane.2

We use the Landau gauge: A=B−y ,0. Note that themagnetic field introduces a new length scale in the prob-lem,

B = c

eB, 95

which is the magnetic length. The only other scale in theproblem is the Fermi-Dirac velocity. Dimensional analy-sis shows that the only quantity with dimensions of en-ergy we can make is vF /B. In fact, this determines thecyclotron frequency of the Dirac fermions,

c = 2vF

B96

the 2 factor comes from the quantization of the prob-lem, see below. Equations 95 and 96 show that thecyclotron energy scales like B, in contrast with the non-relativistic problem where the cyclotron energy is linearin B. This implies that the energy scale associated withthe Dirac fermions is rather different from the onefound in the ordinary 2D electron gas. For instance, forfields of the order of B10 T, the cyclotron energy inthe 2D electron gas is of the order of 10 K. In contrast,

2The problem of transverse magnetic and electric fields canalso be solved exactly. See Lukose et al. 2007 and Peres andCastro 2007.



for the Dirac fermion problem and the same fields, thecyclotron energy is of the order of 1000 K, that is, twoorders of magnitude larger. This has strong implicationsfor the observation of the quantum Hall effect at roomtemperature Novoselov et al., 2007. Furthermore, forB=10 T the Zeeman energy is relatively small, gBB5 K, and can be disregarded.

We now consider the Dirac equation in more detail.Using the minimal coupling in Eq. 19 i.e., replacing−i by −i +eA /c, we find

vF · − i + eA/cr = Er , 97

in the Landau gauge the generic solution for the wavefunction has the form x ,y=eikxy, and the Diracequation reads

vF 0 y − k + Bey/c

− y − k + Bey/c 0y = Ey ,

98

which can be rewritten as

c 0 O

O† 0 = E , 99

or equivalently

O+ + O†− = 2E/c , 100

where ±=x± iy, and we have defined the dimension-less length scale

=y

B− Bk 101

and 1D harmonic-oscillator operators

O =12

+ ,

O† =12

− + , 102

which obey canonical commutation relations O ,O†=1.The number operator is simply N=O†O.

First, we note that Eq. 100 allows for a solution withzero energy,

O+ + O†−0 = 0, 103

and since the Hilbert space generated by is of dimen-sion 2, and the spectrum generated by O† is boundedfrom below, we just need to ensure that

O0 = 0,

−0 = 0, 104

in order for Eq. 103 to be fulfilled. The obvious zero-mode solution is

0 = 0 ⇓ , 105

where ⇓ indicates the state localized on sublattice A,and ⇑ indicates the state localized on sublattice B. Fur-thermore,

O0 = 0 106

is the ground states of the 1D harmonic oscillator. Allthe solutions can now be constructed from the zeromode,

N,± = N−1 ⇑ ± N ⇓ = N−1±N

,

107

and their energy is given by McClure, 1956

E±N = ± cN , 108

where N=0,1 ,2 , . . . is a positive integer and N is thesolution of the 1D harmonic oscillator explicitly, N=2−N/2N!−1/2 exp−2 /2HN, where HN is a Her-mite polynomial. The Landau levels at the oppositeDirac point K have exactly the same spectrum andhence each Landau level is doubly degenerate. Of par-ticular importance for the Dirac problem discussed hereis the existence of a zero-energy state N=0, which isresponsible for the anomalies observed in the quantumHall effect. This particular Landau level structure hasbeen observed by many different experimental probes,from Shubnikov–de Haas oscillations in single layergraphene see Fig. 18 Novoselov, Geim, Morozov, etal., 2005; Zhang et al., 2005, to infrared spectroscopy

FIG. 18. Color online SdH oscillations observed in longitu-dinal resistivity xx of graphene as a function of the chargecarrier concentration n. Each peak corresponds to the popula-tion of one Landau level. Note that the sequence is not inter-rupted when passing through the Dirac point, between elec-trons and holes. The period of oscillations is n=4B /0,where B is the applied field and 0 is the flux quantum No-voselov, Geim, Morozov, et al., 2005.



Jiang, Henriksen, Tung, et al., 2007, and to scanningtunneling spectroscopy Li and Andrei, 2007 STS on agraphite surface.

J. The anomalous integer quantum Hall effect

In the presence of disorder, Landau levels get broad-ened and mobility edges appear Laughlin, 1981. Notethat there will be a Landau level at zero energy thatseparates states with hole character 0 from stateswith electron character 0. The components of theresistivity and conductivity tensors are given by

xx =xx

xx2 + xy

2 ,

xy =xy

xx2 + xy

2 , 109

where xx xx is the longitudinal component and xyxy is the Hall component of the conductivity resistiv-ity. When the chemical potential is inside a region oflocalized states, the longitudinal conductivity vanishes,xx=0, and hence xx=0, xy=1/xy. On the other hand,when the chemical potential is in a region of delocalizedstates, when the chemical potential is crossing a Landaulevel, we have xx0 and xy varies continuously Shenget al., 2006, 2007.

The value of xy in the region of localized states canbe obtained from Laughlin’s gauge invariance argumentLaughlin, 1981: one imagines making a graphene rib-bon such as shown in Fig. 19 with a magnetic field Bnormal through its surface and a current I circling itsloop. Due to the Lorentz force, the magnetic field pro-duces a Hall voltage VH perpendicular to the field andcurrent. The circulating current generates a magneticflux that threads the loop. The current is given by

I = cE

, 110

where E is the total energy of the system. The localizedstates do not respond to changes in , only the delocal-ized ones. When the flux is changed by a flux quantum=0=hc /e, the extended states remain the same by

gauge invariance. If the chemical potential is in the re-gion of localized states, all the extended states below thechemical potential will be filled both before and afterthe change of flux by 0. However, during the change offlux, an integer number of states enter the cylinder atone edge and leave at the opposite edge.

The question is: How many occupied states are trans-ferred between edges? We consider a naive and, asshown further, incorrect calculation in order to show theimportance of the zero mode in this problem. Each Lan-dau level contributes with one state times its degeneracyg. In the case of graphene, we have g=4 since there aretwo spin states and two Dirac cones. Hence, we expectthat when the flux changes by one flux quantum, thechange in energy would be Einc= ±4NeVH, where N isan integer. The plus sign applies to electron statescharge +e and the minus sign to hole states charge −e.Hence, we conclude that Iinc= ±4e2 /hVH and hencexy,inc=I /VH= ±4Ne2 /h, which is the naive expectation.The problem with this result is that when the chemicalpotential is exactly at half filling, that is, at the Diracpoint, it would predict a Hall plateau at N=0 withxy,inc=0, which is not possible since there is an N=0Landau level, with extended states at this energy. Thesolution for this paradox is rather simple: because of thepresence of the zero mode that is shared by the twoDirac points, there are exactly 2 2N+1 occupiedstates that are transferred from one edge to another.Hence, the change in energy is E= ±22N+1eVH for achange of flux of =hc /e. Therefore, the Hall conduc-tivity is Schakel, 1991; Gusynin and Sharapov, 2005;Herbut, 2007; Peres, Guinea, and Castro Neto, 2006a,2006b

xy =I

VH=

c

VH

E

= ± 22N + 1

e2

h, 111

without any Hall plateau at N=0. This result has beenobserved experimentally Novoselov, Geim, Morozov, etal., 2005; Zhang et al., 2005 as shown in Fig. 20.

K. Tight-binding model in a magnetic field

In the tight-binding approximation, the hopping inte-grals are replaced by a Peierls substitution,

eieRRA·drtR,R = ei2/0R

RA·drtR,R, 112

where tR,R represents the hopping integral between thesites R and R, with no field present. The tight-bindingHamiltonian for a single graphene layer, in a constantmagnetic field perpendicular to the plane, is conve-niently written as

H = − t m,n,

ei/0n1+z/2a†m,nbm,n

+ e−i/0na†m,nb„m − 1,n − 1 − z/2…

+ ei/0nz−1/2a†m,nbm,n − z + H.c. ,

113

FIG. 19. Color online Geometry of Laughlin’s thought ex-periment with a graphene ribbon: a magnetic field B is appliednormal to the surface of the ribbon; a current I circles the loop,generating a Hall voltage VH and a magnetic flux .



holding for a graphene stripe with a zigzag z=1 andarmchair z=−1 edges oriented along the x direction.Fourier transforming along the x direction gives

H = − t k,n,

ei/0n1+z/2a†k,nbk,n

+ e−i/0neikaa†k,nb„k,n − 1 − z/2…

+ ei/0nz−1/2a†k,nbk,n − z + H.c. .

We now consider the case of zigzag edges. The eigen-problem can be rewritten in terms of Harper’s equationsHarper, 1955, and for zigzag edges we obtain Rammal,1985

E,kk,n = − teika/22 cos

0n −

ka

2k,n

+ k,n − 1 , 114

E,kk,n = − te−ika/22 cos

0n −

ka

2k,n

+ k,n + 1 , 115

where the coefficients k ,n and k ,n show up inHamiltonian’s eigenfunction k written in terms oflattice-position-state states as

k = n,

k,na ;k,n, + k,nb ;k,n, .

116

Equations 114 and 115 hold in the bulk. Consideringthat the zigzag ribbon has N unit cells along its width,from n=0 to n=N−1, the boundary conditions at theedges are obtained from Eqs. 114 and 115, and read

E,kk,0 = − teika/22 coska

2k,0 , 117

E,kk,N − 1 = − 2te−ika/2 cos

0N − 1 −

ka

2

k,N − 1 . 118

Similar equations hold for a graphene ribbon with arm-chair edges.

In Fig. 21, we show 14 energy levels, around zero en-ergy, for both the zigzag and armchair cases. The forma-tion of the Landau levels is signaled by the presence offlat energy bands, following the bulk energy spectrum.From Fig. 21, it is straightforward to obtain the value ofthe Hall conductivity in the quantum Hall effect regime.We assume that the chemical potential is in between twoLandau levels at positive energies, shown by the dashedline in Fig. 21. The Landau level structure shows twozero-energy modes; one of them is electronlike hole-like, since close to the edge of the sample its energy isshifted upwards downwards. The other Landau levelsare doubly degenerate. The determination of the valuesfor the Hall conductivity is done by counting how manyenergy levels of electronlike nature are below thechemical potential. This counting produces the value2N+1, with N=0,1 ,2 , . . . for the case of Fig. 21 one has

FIG. 20. Color online Quantum Hall effect in graphene as afunction of charge-carrier concentration. The peak at n=0shows that in high magnetic fields there appears a Landau levelat zero energy where no states exist in zero field. The fielddraws electronic states for this level from both conduction andvalence bands. The dashed lines indicate plateaus in xy de-scribed by Eq. 111. Adapted from Novoselov, Geim, Moro-zov, et al., 2005.

0 1 2 3 4 5 6 7-1

-0.5

0

0.5

1

ener

gy/t

zigzag: N=200, φ/φ0=1/701

0 1 2 3 4 5 6 7-1

-0.5

0

0.5

1

armchair: N=200, φ/φ0=1/701

2 3 4 5 6momentum ka

-0.4

-0.2

0

0.2

0.4

ener

gy/t

0.5 1 1.5 2momentum ka

-0.4

-0.2

0

0.2

0.4

µ

µ

FIG. 21. Color online Fourteen energy levels of tight-bindingelectrons in graphene in the presence of a magnetic flux =0 /701, for a finite stripe with N=200 unit cells. The bottompanels are zoom-in images of the top ones. The dashed linerepresents the chemical potential .



N=2. From this counting, the Hall conductivity is given,including an extra factor of 2 accounting for the spindegree of freedom, by

xy = ± 2e2

h2N + 1 = ± 4

e2

hN +

12 . 119

Equation 119 leads to the anomalous integer quantumHall effect discussed previously, which is the hallmark ofsingle-layer graphene.

L. Landau levels in graphene stacks

The dependence of the Landau level energies on theLandau index N roughly follows the dispersion relation

of the bands, as shown in Fig. 22. Note that, in a trilayerwith Bernal stacking, two sets of levels have a N de-pendence, while the energies of other two depend lin-early on N. In an infinite rhombohedral stack, the Lan-dau levels remain discrete and quasi-2D Guinea et al.,2006. Note that, even in an infinite stack with the Ber-nal structure, the Landau level closest to E=0 forms aband that does not overlap with the other Landau levels,leading to the possibility of a 3D integer quantum Halleffect Luk’yanchuk and Kopelevich, 2004; Kopelevichet al., 2006; Bernevig et al., 2007.

The optical transitions between Landau levels canalso be calculated. The selection rules are the same asfor a graphene single layer, and only transitions betweensubbands with Landau level indices M and N such thatN= M±1 are allowed. The resulting transitions, withtheir respective spectral strengths, are shown in Fig. 23.The transitions are grouped into subbands, which giverise to a continuum when the number of layers tends toinfinity. In Bernal stacks with an odd number of layers,the transitions associated with Dirac subbands with lin-ear dispersion have the largest spectral strength, andthey give a significant contribution to the total absorp-tion even if the number of layers is large, NL30–40Sadowski et al., 2006.

M. Diamagnetism

Back in 1952, Mrozowski Mrozowski, 1952 studieddiamagnetism of polycrystalline graphite and othercondensed-matter molecular-ring systems. It was con-cluded that in such ring systems diamagnetism has twocontributions: i diamagnetism from the filled bands ofelectrons, and ii Landau diamagnetism of free elec-trons and holes. For graphite the second source of dia-magnetism is by far the largest of the two.

McClure 1956 computed diamagnetism of a 2D hon-eycomb lattice using the theory introduced by Wallace1947, a calculation he later generalized to three-dimensional graphite McClure, 1960. For the honey-comb plane, the magnetic susceptibility in units ofemu/g is

= −0.0014

T0

2 sech2

2kBT , 120

where is the Fermi energy, T is the temperature, andkB is the Boltzmann constant. For graphite, the magneticsusceptibility is anisotropic and the difference between

(b)

(a)

(c)

(d)

FIG. 22. Color online Landau levels of graphene stacksshown in Fig. 13. The applied magnetic field is 1 T.

FIG. 23. Color online Relative spectral strength of the low energy optical transitions between Landau levels in graphene stackswith Bernal ordering and an odd number of layers. The applied magnetic field is 1 T. Left: 3 layers. Middle: 11 layers. Right: 51layers. The large circles are the transitions in a single layer.



the susceptibility parallel to the principal axis and thatperpendicular to the principal axis is −21.510−6 emu/g. The susceptibility perpendicular to theprincipal axis is equal to about the free-atom suscepti-bility, −0.510−6 emu/g. In the 2D model, the suscepti-bility turns out to be large due to the presence of fastmoving electrons, with a velocity of the order of vF 106 m/s, which in turn is a consequence of the largevalue of the hopping parameter 0. In fact, the suscepti-bility turns out to be proportional to the square of 0.Later Sharma et al. extended the calculation of McClurefor graphite by including the effect of trigonal warpingand showed that the low-temperature diamagnetism in-creases Sharma et al., 1974.

Safran and DiSalvo 1979, interested in the suscepti-bility of graphite intercalation compounds, recalculated,in a tight-binding model, the susceptibility perpendicularto a graphite plane using Fukuyama’s theory Fuku-yama, 1971, which includes interband transitions. Theresults agree with those by McClure 1956. Later, Saf-ran computed the susceptibility of a graphene bilayershowing that this system is diamagnetic at small valuesof the Fermi energy, but there appears a paramagneticpeak when the Fermi energy is of the order of the inter-layer coupling Safran, 1984.

The magnetic susceptibility of other carbon-based ma-terials, such as carbon nanotubes and C60 molecular sol-ids, was measured Heremans et al., 1994 showing a dia-magnetic response at finite magnetic fields differentfrom that of graphite. Studying the magnetic response ofnanographite ribbons with both zigzag and armchairedges was done by Wakabayashi et al. using a numericaldifferentiation of the free energy Wakabayashi et al.,1999. From these two systems, the zigzag edge state isof particular interest since the system supports edgestates even in the presence of a magnetic field, leading tovery high density of states near the edge of the ribbon.The high-temperature response of these nanoribbonswas found to be diamagnetic, whereas the low-temperature susceptibility is paramagnetic.

The Dirac-like nature of the electronic quasiparticlesin graphene led Ghosal et al. 2007 to consider in gen-eral the problem of the diamagnetism of nodal fermions,and Nakamura to study the orbital magnetism of Diracfermions in weak magnetic fields Nakamura, 2007. Ko-shino and Ando considered the diamagnetism of disor-dered graphene in the self-consistent Born approxima-tion, with a disorder potential of the form Ur=1uir−R Koshino and Ando, 2007. At the neutrality pointand zero temperature, the susceptibility of disorderedgraphene is given by

0 = −gvgs

32 e2022W

!0, 121

where gs=gv=2 is the spin and valley degeneracies, W isa dimensionless parameter for the disorder strength, de-fined as W=niui

2 /402, ni is the impurity concentration,

and !0=c exp−1/ 2W, with c a parameter defining acutoff function used in the theory. At finite Fermi energy

F and zero temperature, the magnetic susceptibility isgiven by

F = −gvgs

3e20

2 W

F. 122

In the clean limit, the susceptibility is given by Mc-Clure, 1956; Safran and DiSalvo, 1979; Koshino andAndo, 2007

F = −gvgs

6e20

2F . 123

N. Spin-orbit coupling

Spin-orbit coupling describes a process in which anelectron changes simultaneously its spin and angularmomentum or, in general, moves from one orbital wavefunction to another. The mixing of the spin and the or-bital motion is a relativistic effect, which can be derivedfrom Dirac’s model of the electron. The mixing is largein heavy ions, where the average velocity of the elec-trons is higher. Carbon is a light atom, and the spin orbitinteraction is expected to be weak.

The symmetries of the spin-orbit interaction, however,allow the formation of a gap at the Dirac points in cleangraphene. The spin-orbit interaction leads to a spin-dependent shift of the orbitals, which is of a differentsign for the two sublattices, acting as an effective masswithin each Dirac point Dresselhaus and Dresselhaus,1965; Kane and Mele, 2005; Wang and Chakraborty,2007a. The appearance of this gap leads to a nontrivialspin Hall conductance, similar to other models thatstudy the parity anomaly in relativistic field theory in 2+1 dimensions Haldane, 1988. When the inversionsymmetry of the honeycomb lattice is broken, either be-cause the graphene layer is curved or because an exter-nal electric field is applied Rashba interaction, addi-tional terms, which modulate the nearest-neighborhopping, are induced Ando, 2000. The intrinsic andextrinsic spin-orbit interactions can be written asDresselhaus and Dresselhaus, 1965; Kane and Mele,2005

HSO;int so d2r†rszz"zr ,

HSO;ext R d2r†r− sxy + syx"zr , 124

where and " are Pauli matrices that describe the sub-lattice and valley degrees of freedom and s are Paulimatrices acting on actual spin space. so is the spin-orbitcoupling and R is the Rashba coupling.

The magnitude of the spin-orbit coupling in graphenecan be inferred from the known spin-orbit coupling inthe carbon atom. This coupling allows for transitions be-tween the pz, px, and py orbitals. An electric field alsoinduces transitions between the pz and s orbitals. Theseintra-atomic processes mix the and bands in



graphene. Using second-order perturbation theory, oneobtains a coupling between the low-energy states in the band. Tight-binding Huertas-Hernando et al., 2006;Zarea and Sandler, 2007 and band-structure calcula-tions Min et al., 2006; Yao et al., 2007 give estimates forthe intrinsic and extrinsic spin-orbit interactions in therange 0.01−0.2 K, and hence much smaller than theother energy scales in the problem kinetic, interaction,and disorder.

III. FLEXURAL PHONONS, ELASTICITY, ANDCRUMPLING

Graphite, in the Bernal stacking configuration, is alayered crystalline solid with four atoms per unit cell. Itsbasic structure is essentially a repetition of the bilayerstructure discussed earlier. The coupling between layers,as discussed, is weak and, therefore, graphene has al-ways been the starting point for the discussion ofphonons in graphite Wirtz and Rubio, 2004. Graphenehas two atoms per unit cell, and if we consider grapheneas strictly 2D it should have two acoustic modes withdispersion ack k as k→0 and two optical modeswith dispersion opk const, as k→0 solely due tothe in-plane translation and stretching of the graphenelattice. Nevertheless, graphene exists in the 3D spaceand hence the atoms can oscillate out-of-plane leadingto two out-of-plane phonons one acoustic and anotheroptical called flexural modes. The acoustic flexuralmode has dispersion flexk k2 as k→0, which repre-sents the translation of the whole graphene plane essen-tially a one-atom-thick membrane in the perpendiculardirection free-particle motion. The optical flexuralmode represents the out-of-phase, out-of-plane oscilla-tion of the neighboring atoms. In first approximation, ifwe neglect the coupling between graphene planes,graphite has essentially the same phonon modes, al-though they are degenerate. The coupling betweenplanes has two main effects: i it lifts the degeneracy ofthe phonon modes, and ii it leads to a strong suppres-sion of the energy of the flexural modes. Theoretically,flexural modes should become ordinary acoustic and op-tical modes in a fully covalent 3D solid, but in practicethe flexural modes survive due to the fact that the planesare coupled by weak van der Waals–like forces. Thesemodes have been measured experimentally in graphiteWirtz and Rubio, 2004. Graphene can also be obtainedas a suspended membrane, that is, without a substrate,supported only by a scaffold or bridging micrometer-scale gaps Bunch et al., 2007; Meyer, Geim, Katsnelson,Novoselov, Booth, et al., 2007; Meyer, Geim, Katsnelson,Novoselov, Obergfell, et al., 2007. Figure 24 shows asuspended graphene sheet and an atomic resolution im-age of its crystal lattice.

Because the flexural modes disperse like k2, theydominate the behavior of structural fluctuations ingraphene at low energies low temperatures and longwavelengths. It is instructive to understand how thesemodes appear from the point of view of elasticity theoryChaikin and Lubensky, 1995; Nelson et al., 2004. Con-

sider, for instance, a graphene sheet in 3D and param-etrize the position of the sheet relative of a fixed coor-dinate frame in terms of the in-plane vector r and theheight variable hr so that a position in the graphene isgiven by the vector R= „r ,hr…. The unit vector normalto the surface is given by

N =z − h

1 + h2, 125

where = x ,y is the 2D gradient operator and z is theunit vector in the third direction. In a flat graphene con-figuration, all the normal vectors are aligned and there-fore ·N=0. Deviations from the flat configuration re-quire misalignment of the normal vectors and costelastic energy. This elastic energy can be written as

E0 =#

2 d2r · N2

#

2 d2r2h2, 126

where # is the bending stiffness of graphene, and theexpression in terms of hr is valid for smooth distor-tions of the graphene sheet h21. The energy 126is valid in the absence of a surface tension or a substratethat breaks the rotational and translational symmetry ofthe problem, respectively. In the presence of tension,

(b)

(a)

FIG. 24. Color online Suspended graphene sheet. a Bright-field transmission-electron-microscope image of a graphenemembrane. Its central part homogeneous and featureless re-gion is monolayer graphene. Adapted from Meyer, Geim,Katsnelson, Novoselov, Booth, et al., 2007. b Despite onlyone atom thick, graphene remains a perfect crystal at thisatomic resolution. The image is obtained in a scanning trans-mission electron microscope. The visible periodicity is given bythe lattice of benzene rings. Adapted from Booth et al., 2008.



there is an energy cost for solid rotations of thegraphene sheet h0 and hence a new term has to beadded to the elastic energy,

ET =

2 d2rh2, 127

where is the interfacial stiffness. A substrate, de-scribed by a height variable sr, pins the graphene sheetthrough van der Waals and other electrostatic potentialsso that the graphene configuration tries to follow thesubstrate hrsr. Deviations from this configurationcost extra elastic energy that can be approximated by aharmonic potential Swain and Andelman, 1999,

ES =v2 d2rsr − hr2, 128

where v characterizes the strength of the interaction po-tential between substrate and graphene.

First, consider the free floating graphene problem126 that we can rewrite in momentum space as

E0 =#

2 k

k4h−khk. 129

We now canonically quantize the problem by introduc-ing a momentum operator Pk that has the followingcommutator with hk:

hk,Pk = ik,k, 130

and we write the Hamiltonian as

H = kP−kPk

2+#k4

2h−khk , 131

where is graphene’s 2D mass density. From theHeisenberg equations of motion for the operators, it istrivial to find that hk oscillates harmonically with a fre-quency given by

flexk = #1/2

k2, 132

which is the long-wavelength dispersion of flexuralmodes. In the presence of tension, it is easy to see thatthe dispersion is modified to

k = k#

k2 +

, 133

indicating that the dispersion of flexural modes becomeslinear in k, as k→0, under tension. That is what happensin graphite, where the interaction between layers breaksthe rotational symmetry of the graphene layers.

Equation 132 also allows us to relate the bendingenergy of graphene with the Young modulus Y of graph-ite. The fundamental resonance frequency of a macro-scopic graphite sample of thickness t is given by Bunchet al., 2007

$k = Y

1/2

tk2, 134

where = / t is the 3D mass density. Assuming that Eq.134 works down to the single plane level, that is, whent is the distance between planes, we find

# = Yt3, 135

which provides a simple relationship between the bend-ing stiffness and the Young modulus. Given that Y1012 N/m and t3.4 Å we find #1 eV. This result isin good agreement with ab initio calculations of thebending rigidity Lenosky et al., 1992; Tu and Ou-Yang,2002 and experiments in graphene resonators Bunch etal., 2007.

The problem of structural order of a “free-floating”graphene sheet can be fully understood from the exis-tence of the flexural modes. Consider, for instance, thenumber of flexural modes per unit of area at a certaintemperature T,

Nph = d2k22nk =

1

2

0

dkk

e#/k2− 1

, 136

where nk is the Bose-Einstein occupation number =1/kBT. For T0, the above integral is logarithmicallydivergent in the infrared k→0 indicating a divergentnumber of phonons in the thermodynamic limit. For asystem with finite size L, the smallest possible wave vec-tor is of the order of kmin2 /L. Using kmin as a lowercutoff in the integral 136, we find

Nph =

LT2 ln 1

1 − e−LT2 /L2 , 137

where

LT =2

kBT#1/4

138

is the thermal wavelength of flexural modes. Note thatwhen LLT, the number of flexural phonons in Eq.137 diverges logarithmically with the size of the sys-tem,

Nph 2

LT2 ln L

LT , 139

indicating that the system cannot be structurally orderedat any finite temperature. This is nothing but the crum-pling instability of soft membranes Chaikin and Luben-sky, 1995; Nelson et al., 2004. For LLT, one finds thatNph goes to zero exponentially with the size of the sys-tem indicating that systems with finite size can be flat atsufficiently low temperatures. Note that for #1 eV, 2200 kg/m3, t=3.4 Å 7.510−7 kg/m2, and T300 K, we find LT1 Å, indicating that free-floatinggraphene should always crumple at room temperaturedue to thermal fluctuations associated with flexuralphonons. Nevertheless, the previous discussion involvesonly the harmonic quadratic part of the problem. Non-linear effects such as large bending deformations Peliti



and Leibler, 1985, the coupling between flexural andin-plane modes or phonon-phonon interactions LeDoussal and Radzihovsky, 1992; Bonini et al., 2007, andthe presence of topological defects Nelson and Peliti,1987 can lead to strong renormalizations of the bendingrigidity, driving the system toward a flat phase at lowtemperatures Chaikin and Lubensky, 1995. This situa-tion has been confirmed in numerical simulations of freegraphene sheets Adebpour et al., 2007; Fasolino et al.,2007.

The situation is rather different if the system is undertension or in the presence of a substrate or scaffold thatcan hold the graphene sheet. In fact, static rippling ofgraphene flakes suspended on scaffolds has been ob-served for single layer as well as bilayers Meyer, Geim,Katsnelson, Novoselov, Booth, et al., 2007; Meyer, Geim,Katsnelson, Novoselov, Obergfell, et al., 2007. In thiscase the dispersion, in accordance with Eq. 133, is atleast linear in k, and the integral in Eq. 136 convergesin the infrared k→0, indicating that the number offlexural phonons is finite and graphene does notcrumple. We should note that these thermal fluctuationsare dynamic and hence average to zero over time, there-fore the graphene sheet is expected to be flat underthese circumstances. Obviously, in the presence of a sub-strate or scaffold described by Eq. 128, static deforma-tions of the graphene sheet are allowed. Also, hydrocar-bon molecules that are often present on top of freehanging graphene membranes could quench flexuralfluctuations, making them static.

Finally, one should note that in the presence of a me-tallic gate the electron-electron interactions lead to thecoupling of the phonon modes to the electronic excita-tions in the gate. This coupling could be partially re-sponsible for the damping of the phonon modes due todissipative effects Seoanez et al., 2007 as observed ingraphene resonators Bunch et al., 2007.

IV. DISORDER IN GRAPHENE

Graphene is a remarkable material from an electronicpoint of view. Because of the robustness and specificityof the sigma bonding, it is very hard for alien atoms toreplace the carbon atoms in the honeycomb lattice. Thisis one of the reasons why the electron mean free path ingraphene can be so long, reaching up to 1 m in theexisting samples. Nevertheless, graphene is not immuneto disorder and its electronic properties are controlledby extrinsic as well as intrinsic effects that are unique tothis system. Among the intrinsic sources of disorder,highlight are surface ripples and topological defects. Ex-trinsic disorder can come about in many different forms:adatoms, vacancies, charges on top of graphene or in thesubstrate, and extended defects such as cracks andedges.

It is easy to see that from the point of view of singleelectron physics that is, terms that can be added to Eq.5, there are two main terms to which disorder couples.The first one is a local change in the single site energy,

Hdd = i

Viai†ai + bi

†bi , 140

where Vi is the strength of the disorder potential on siteRi, which is diagonal in the sublattice indices and hence,from the point of view of the Dirac Hamiltonian 18,can be written as

Hdd = d2r a=1,2

Vara†rar , 141

which acts as a chemical potential shift for the Diracfermions, it that is, shifts locally the Dirac point.

Because the density of states vanishes in single-layergraphene, and as a consequence the lack of electrostaticscreening, charge potentials may be rather important indetermining the spectroscopic and transport propertiesAndo, 2006b; Adam et al., 2007; Nomura and Mac-Donald, 2007. Of particular importance is the Coulombimpurity problem, where

Var =e2

0

1

r, 142

with 0 the dielectric constant of the medium. The solu-tion of the Dirac equation for the Coulomb potential in2D has been studied analytically DiVincenzo and Mele,1984; Biswas et al., 2007; Novikov, 2007a; Pereira, Nils-son, and Castro Neto, 2007; Shytov et al., 2007. Its so-lution has many of the features of the 3D relativistichydrogen atom problem Baym, 1969. As in the case ofthe 3D problem, the nature of the eigenfunctions de-pends strongly on graphene’s dimensionless couplingconstant,

g =Ze2

0vF. 143

Note that the coupling constant can be varied by eitherchanging the charge of the impurity Z or modifying thedielectric environment and changing 0. For ggc= 1

2 ,the solutions of this problem are given in terms of Cou-lomb wave functions with logarithmic phase shifts. Thelocal density of states LDOS is affected close to theimpurity due the electron-hole asymmetry generated bythe Coulomb potential. The local charge density decayslike 1/r3 plus fast oscillations of the order of the latticespacing in the continuum limit this gives rise to a Diracdelta function for the density Kolezhuk et al., 2006. Asin 3D QED, the 2D problem becomes unstable for ggc= 1

2 leading to supercritical behavior and the so-called fall of electron to the center Landau and Lifshitz,1981. In this case, the LDOS is strongly affected by thepresence of the Coulomb impurity with the appearanceof bound states outside the band and scattering reso-nances within the band Pereira, Nilsson, and CastroNeto, 2007 and the local electronic density decaysmonotonically like 1/r2 at large distances.

Schedin et al. 2007 argued that without a highvacuum environment, these Coulomb effects can bestrongly suppressed by large effective dielectric con-stants due to the presence of a nanometer thin layer of



absorbed water Sabio et al., 2007. In fact, experimentsin ultrahigh vacuum conditions Chen, Jang, Fuhrer, etal., 2008 display scattering features in the transport thatcan be associated with charge impurities. Screening ef-fects that affect the strength and range of the Coulombinteraction are rather nontrivial in graphene Fogler,Novikov, and Shklovskii, 2007; Shklovskii, 2007 and,therefore, important for the interpretation of transportdata Bardarson et al., 2007; Nomura et al., 2007; San-Jose et al., 2007; Lewenkopf et al., 2008.

Another type of disorder is the one that changes thedistance or angles between the pz orbitals. In this case,the hopping energies between different sites are modi-fied, leading to a new term to the original Hamiltonian5,

Hod = i,j

tijabai

†bj + H.c. + tijaaai

†aj + bi†bj ,

144

or in Fourier space,

Hod = k,k

ak†bk

i,ab

tiabeik−k·Ri−iaa·k + H.c.

+ ak†ak + bk

†bk i,aa

tiaaeik−k·Ri−iab·k, 145

where tijab tij

aa is the change of the hopping energybetween orbitals on lattice sites Ri and Rj on the same

different sublattices we have written Rj=Ri+ , where

ab is the nearest-neighbor vector and aa is the next-nearest-neighbor vector. Following the procedure ofSec. II.B, we project out the Fourier components of theoperators close to the K and K points of the BZ usingEq. 17. If we assume that tij is smooth over the latticespacing scale, that is, it does not have a Fourier compo-nent with momentum K−K so the two Dirac cones arenot coupled by disorder, we can rewrite Eq. 145 inreal space as

Hod = d2rAra1†rb1r + H.c.

+ ra1†ra1r + b1

†rb1r , 146

with a similar expression for cone 2 but with A replacedby A*, where

Ar = ab

tabre−iab·K, 147

r = aa

taare−iaa·K. 148

Note that whereas r=*r, because of the inversionsymmetry of the two triangular sublattices that make upthe honeycomb lattice, A is complex because of a lack ofinversion symmetry for nearest-neighbor hopping.Hence,

Ar = Axr + iAyr . 149

In terms of the Dirac Hamiltonian 18, we can rewriteEq. 146 as

Hod = d2r1†r · A r1r + r1

†r1r ,

150

where A = Ax ,Ay. This result shows that changes in the

hopping amplitude lead to the appearance of vector Aand scalar potentials in the Dirac Hamiltonian. Thepresence of a vector potential in the problem indicates

that an effective magnetic field B = c /evFA shouldalso be present, naively implying a broken time-reversalsymmetry, although the original problem was time-reversal invariant. This broken time-reversal symmetryis not real since Eq. 150 is the Hamiltonian aroundonly one of the Dirac cones. The second Dirac cone isrelated to the first by time-reversal symmetry, indicatingthat the effective magnetic field is reversed in the secondcone. Therefore, there is no global broken symmetry buta compensation between the two cones.

A. Ripples

Graphene is a one-atom-thick system, the extremecase of a soft membrane. Hence, like soft membranes, itis subject to distortions of its structure due to eitherthermal fluctuations as we discussed in Sec. III or in-teraction with a substrate, scaffold, and absorbandsSwain and Andelman, 1999. In the first case, the fluc-tuations are time dependent although with time scalesmuch longer than the electronic ones, while in the sec-ond case the distortions act as quenched disorder. Inboth cases, disorder occurs because of the modificationof the distance and relative angle between the carbonatoms due to the bending of the graphene sheet. Thistype of off-diagonal disorder does not exist in ordinary3D solids, or even in quasi-1D or quasi-2D systems,where atomic chains and atomic planes, respectively, areembedded in a 3D crystalline structure. In fact,graphene is also different from other soft membranesbecause it is semimetallic, while previously studiedmembranes were insulators.

The problem of bending graphitic systems and its ef-fect on the hybridization of the orbitals has been stud-ied in the context of classical minimal surfaces Lenoskyet al., 1992 and applied to fullerenes and carbon nano-tubes Tersoff, 1992; Kane and Mele, 1997; Zhong-can etal., 1997; Xin et al., 2000; Tu and Ou-Yang, 2002. Inorder to understand the effect of bending on graphene,consider the situation shown in Fig. 25. The bending ofthe graphene sheet has three main effects: the decreaseof the distance between carbon atoms, a rotation ofthe pZ orbitals compression or dilation of the latticeare energetically costly due to the large spring con-stant of graphene 57 eV/Å2 Xin et al., 2000, and arehybridization between and orbitals Eun-Ah Kim



and Castro Neto, 2008. Bending by a radius R de-creases the distance between the orbitals from to d=2R sin / 2R−3 /24R2 for R. The decrease inthe distance between the orbitals increases the overlapbetween the two lobes of the pZ orbital Harrison, 1980:VppaVppa

0 1+2 / 12R2, where a=, , and Vppa0 is the

overlap for a flat graphene sheet. The rotation of the pZorbitals can be understood within the Slater-Koster for-malism, namely, the rotation can be decomposed into apz−pz bond plus a px−px bond hybridizationwith energies Vpp and Vpp, respectively Harrison,1980: V=Vpp cos2−Vpp sin2Vpp− Vpp+Vpp / 2R2, leading to a decrease in the overlap.Furthermore, the rotation leads to rehybridization be-tween and orbitals leading to a further shift in en-ergy of the order of Eun-Ah Kim and Castro Neto,2008 Vsp

2 +Vpp2 / −a.

In the presence of a substrate, as discussed in Sec. III,elasticity theory predicts that graphene can be expectedto follow the substrate in a smooth way. Indeed, by mini-mizing the elastic energy 126–128 with respect to theheight h, we get Swain and Andelman, 1999

#4hr − 2hr + vhr = vsr , 151

which can be solved by Fourier transform,

hk =sk

1 + tk2 + ck4 , 152

where

t = v1/2

,

c = #v1/4

. 153

Equation 152 gives the height configuration in terms ofthe substrate profile, and t and c provide the lengthscales for elastic distortion of graphene on a substrate.Hence, disorder in the substrate translates into disorderin the graphene sheet albeit restricted by elastic con-straints. This picture has been confirmed by STM mea-surements on graphene Ishigami et al., 2007; Stolyarovaet al., 2007 in which strong correlations were found be-tween the roughness of the substrate and the graphene

topography. Ab initio band-structure calculations alsogive support to this scenario Dharma-Wardana, 2007.

The connection between the ripples and the electronicproblem comes from the relation between the heightfield hr and the local curvature of the graphene sheetR,

2

Rr 2hr , 154

hence we see that due to bending the electrons are sub-ject to a potential that depends on the structure of agraphene sheet Eun-Ah Kim and Castro Neto, 2008,

Vr V0 − 2hr2, 155

where 10 eV Å2 is the constant that depends onmicroscopic details. The conclusion from Eq. 155 isthat Dirac fermions are scattered by ripples of thegraphene sheet through a potential that is proportionalto the square of the local curvature. The coupling be-tween geometry and electron propagation is unique tographene, and results in additional scattering and resis-tivity due to ripples Katsnelson and Geim, 2008.

B. Topological lattice defects

Structural defects of the honeycomb lattice like pen-tagons, heptagons, and their combinations such asStone-Wales defect a combination of two pentagon-heptagon pairs are also possible in graphene and canlead to scattering Cortijo and Vozmediano, 2007a,2007b. These defects induce long-range deformations,which modify the electron trajectories.

We consider first a disclination. This defect is equiva-lent to the deletion or inclusion of a wedge in the lattice.The simplest one in the honeycomb lattice is the absenceof a 60° wedge. The resulting edges can be glued in sucha way that all sites remain threefold-coordinated. Thehoneycomb lattice is recovered everywhere, except atthe apex of the wedge, where a fivefold ring, a pentagon,is formed. One can imagine a situation in which thenearest-neighbor hoppings are unchanged. Nevertheless,the existence of a pentagon implies that the two sublat-tices of the honeycomb structure can no longer be de-fined. A trajectory around the pentagon after a closedcircuit has to change the sublattice index.

The boundary conditions imposed at the edges of adisclination are shown in Fig. 26, identifying sites fromdifferent sublattices. In addition, the wave functions atthe K and K points are exchanged when moving aroundthe pentagon.

Far away from the defect, a slow rotation of the spino-rial wave function components can be described by agauge field that acts on the valley and sublattice indicesGonzález et al., 1992, 1993b. This gauge field is techni-cally non-Abelian, although a transformation can be de-fined that makes the resulting Dirac equation equivalentto one with an effective Abelian gauge field González etal., 1993b. The final continuum equation gives a reason-able description of the electronic spectrum of fullerenes

l+ +

−−

θ

α β

RR

z z

x

z z12

d

FIG. 25. Bending the surface of graphene by a radius R and itseffect on the pz orbitals.



with different sizes González et al., 1992, 1993b, andother structures that contain pentagons LeClair, 2000;Osipov et al., 2003; Kolesnikov and Osipov, 2004, 2006;Lammert and Crespi, 2004. It is easy to see that a hep-tagon leads to the opposite effective field.

An in-plane dislocation, that is, the inclusion of asemi-infinite row of sites, can be considered as inducedby a pentagon and a heptagon together. The non-Abelian field described above is canceled away from thecore. A closed path implies a shift by one or morelattice spacing. The wave functions at the K and Kpoints acquire phases, e±2i/3, under a translation by onelattice unit. Hence, the description of a dislocation in thecontinuum limit requires an Abelian vortex of charge±2 /3 at its core. Dislocations separated over distancesof the order of d lead to an effective flux through anarea of perimeter l of the order of Morpurgo andGuinea, 2006

d

kFl2 , 156

where kF is the Fermi vector of the electrons.In general, a local rotation of the honeycomb lattice

axes induces changes in the hopping which lead to mix-ing of the K and K wave functions, leading to a gaugefield like the one induced by a pentagon González et al.,2001. Graphene samples with disclinations and disloca-tions are feasible in particular substrates Coraux et al.,2008, and gauge fields related to the local curvature arethen expected to play a crucial role in such structures.The resulting electronic structure can be analyzed usingthe theory of quantum mechanics in curved space Bir-rell and Davies, 1982; Cortijo and Vozmediano, 2007a,2007b; de Juan et al., 2007.

C. Impurity states

Point defects, similar to impurities and vacancies,can nucleate electronic states in their vicinity. Hence,a concentration of ni impurities per carbon atom leadsto a change in the electronic density of the order of ni.The corresponding shift in the Fermi energy is F

vFni. In addition, impurities lead to a finite elastic

mean free path lelas ani−1/2 and to an elastic scattering

time "elas vFni−1. Hence, the regions with few impuri-ties can be considered low-density metals in the dirtylimit as "elas

−1 F.The Dirac equation allows for localized solutions that

satisfy many possible boundary conditions. It is knownthat small circular defects result in localized and semilo-calized states Dong et al., 1998, that is, states whosewave function decays as 1/r as a function of the distancefrom the center of the defect. A discrete version of thesestates can be realized in a nearest-neighbor tight-bindingmodel with unitary scatterers such as vacancies Pereiraet al., 2006. In the continuum, the Dirac equation 19for the wave function r= „1r ,2r… can be written,for E=0, as

w1w,w* = 0,

w*2w,w* = 0, 157

where w=x+ iy is a complex number. These equationsindicate that at zero energy the components of the wavefunction can only be either holomorphic or antiholomor-phic with respect to the variable w that is, 1w ,w*=1w* and 2w ,w*=2w. Since the boundary con-ditions require that the wave function vanishes at infin-ity, the only possible solutions have the form Kr „1/ x+ iyn ,0… or Kr „0,1 / x− iyn

…. The wavefunctions in the discrete lattice must be real, and at largedistances the actual solution found near a vacancy tendsto a superposition of two solutions formed from wavefunctions from the two valleys with equal weight, in away similar to the mixing at armchair edges Brey andFertig, 2006b.

The construction of the semilocalized state around avacancy in the honeycomb lattice can be extended toother discrete models, which leads to the Dirac equationin the continuum limit. One particular case is thenearest-neighbor square lattice with half flux perplaquette, or the nearest-neighbor square lattice withtwo flavors per site. The latter has been extensively stud-ied in relation to the effects of impurities on the elec-tronic structure of d-wave superconductors Balatsky etal., 2006, and numerical results are in agreement withthe existence of this solution. As the state is localized onone sublattice only, the solution can be generalized forthe case of two vacancies.

D. Localized states near edges, cracks, and voids

Localized states can be defined at edges where thenumber of atoms in the two sublattices is not compen-sated. The number of them depend on details of theedge. The graphene edges can be strongly deformed,due to the bonding of other atoms to carbon atoms atthe edges. These atoms should not induce states in thegraphene band. In general, a boundary inside thegraphene material will exist, as shown in Fig. 27, beyondwhich the sp2 hybridization is well defined. If this is the

FIG. 26. Color online Sketch of the boundary conditions as-sociated to a disclination pentagon in the honeycomb lattice.



case, the number of midgap states near the edge isroughly proportional to the difference in sites betweenthe two sublattices near this boundary.

Along a zigzag edge there is one localized state perthree lattice units. This implies that a precursor structurefor localized states at the Dirac energy can be found inribbons or constrictions of small lengths Muñoz-Rojaset al., 2006, which modifies the electronic structure andtransport properties.

Localized solutions can also be found near other de-fects that contain broken bonds or vacancies. Thesestates do not allow an analytical solution, although, asdiscussed above, the continuum Dirac equation is com-patible with many boundary conditions, and it shoulddescribe well localized states that vary slowly over dis-tances comparable to the lattice spacing. The existenceof these states can be investigated by analyzing the scal-ing of the spectrum near a defect as a function of thesize of the system L Vozmediano et al., 2005. A num-ber of small voids and elongated cracks show stateswhose energy scales as L−2, while the energy of ex-tended states scales as L−1. A state with energy scalingL−2 is compatible with continuum states for which themodulus of the wave function decays as r−2 as a functionof the distance from the defect.

E. Self-doping

The band-structure calculations discussed in the pre-vious sections show that the electronic structure of asingle graphene plane is not strictly symmetrical in en-ergy Reich et al., 2002. The absence of electron-holesymmetry shifts the energy of the states localized nearimpurities above or below the Fermi level, leading to atransfer of charge from or to the clean regions. Hence,the combination of localized defects and the lack of per-fect electron-hole symmetry around the Dirac pointsleads to the possibility of self-doping, in addition to theusual scattering processes.

Extended lattice defects, like edges, grain boundaries,or microcracks, are likely to induce a number of elec-tronic states proportional to their length L /a, where a isof the order of the lattice constant. Hence, a distributionof extended defects of length L at a distance equal to Litself gives rise to a concentration of L /a carriers per

carbon in regions of size of the order of L /a2. Theresulting system can be considered a metal with a lowdensity of carriers, ncarrier a /L per unit cell, and an elas-tic mean free path lelas L. Then, we obtain

F vF

aL,

1

"elas

vF

L, 158

and, therefore, "elas−1F when a /L1. Hence, theexistence of extended defects leads to the possibility ofself-doping but maintaining most of the sample in theclean limit. In this regime, coherent oscillations of trans-port properties are expected, although the observedelectronic properties may correspond to a shifted Fermienergy with respect to the nominally neutral defect-freesystem.

One can describe the effects that break electron-holesymmetry near the Dirac points in terms of a finite next-nearest-neighbor hopping between orbitals t in Eq.5. Consider, for instance, the electronic structure of aribbon of width L terminated by zigzag edges, which, asdiscussed, lead to surface states for t=0. The transla-tional symmetry along the axis of the ribbon allows us todefine bands in terms of a wave vector parallel to thisaxis. On the other hand, the localized surface bands, ex-tending from k = 2 /3 to k =−2 /3, acquire a dis-persion of the order of t. Hence, if the Fermi energyremains unchanged at the position of the Dirac pointsDirac=−3t, this band will be filled, and the ribbon willno longer be charge neutral. In order to restore chargeneutrality, the Fermi level needs to be shifted by anamount of the order of t. As a consequence, some ex-tended states near the Dirac points are filled, leading tothe phenomenon of self-doping. The local charge is afunction of distance to the edges, setting the Fermi en-ergy so that the ribbon is globally neutral. Note that thecharge transferred to the surface states is localizedmostly near the edges of the system.

The charge transfer is suppressed by electrostatic ef-fects, as large deviations from charge neutrality have anassociated energy cost Peres, Guinea, and Castro Neto,2006a. In order to study these charging effects, we addto the free-electron Hamiltonian 5 the Coulomb en-ergy of interaction between electrons,

HI = i,j

Ui,jninj, 159

where ni=ai,† ai,+bi,

† bi, is the number operator atsite Ri, and

Ui,j =e2

0Ri − Rj160

is the Coulomb interaction between electrons. We ex-pect, on physics grounds, that an electrostatic potentialbuilds up at the edges, shifting the position of the sur-face states, and reducing the charge transferred to or

FIG. 27. Color online Sketch of a rough graphene surface.The full line gives the boundary beyond which the lattice canbe considered undistorted. The number of midgap stateschanges depending on the difference in the number of re-moved sites for two sublattices.



from them. The potential at the edge induced by a con-stant doping per carbon atom is e2 /aW /a e2 /a isthe Coulomb energy per carbon, and W the width of theribbon W /a is the number of atoms involved. Thecharge transfer is stopped when the potential shifts thelocalized states to the Fermi energy, that is, when te2 /aW /a. The resulting self-doping is therefore

ta2

e2W, 161

which vanishes when W→.We treat the Hamiltonian 159 within the Hartree ap-

proximation that is, we replace HI by HM.F.=iVini,where Vi=jUi,jnj, and solve the problem self-consistently for ni. Numerical results for graphene rib-bons of length L=803a and different widths are shownin Fig. 28 t / t=0.2 and e2 /a=0.5t. The largest widthstudied is 0.1 m, and the total number of carbon at-oms in the ribbon is 105. Note that as W increases, theself-doping decreases indicating that, for a perfectgraphene plane W→, the self-doping effect disap-pears. For realistic parameters, we find that the amountof self-doping is 10−4−10−5 electrons per unit cell forsizes 0.1–1 m.

F. Vector potential and gauge field disorder

As discussed in Sec. IV, lattice distortions modify theDirac equation that describes the low-energy band struc-ture of graphene. We consider here deformations thatchange slowly on the lattice scale, so that they do not

mix the two inequivalent valleys. As shown earlier, per-turbations that hybridize the two sublattices lead toterms that change the Dirac Hamiltonian from vF ·k tovF ·k+ ·A. Hence, the vector A can be thought of asif induced by an effective gauge field A. In order topreserve time-reversal symmetry, this gauge field musthave opposite signs at the two Dirac cones AK=−AK.

A simple example is a distortion that changes the hop-ping between all bonds along a given axis of the lattice.We assume that the sites at the ends of those bondsdefine the unit cell, as shown in Fig. 29. If the distortionis constant, its only effect is to displace the Dirac pointsaway from the BZ corners. The two inequivalent pointsare displaced in opposite directions. This uniform distor-tion is the equivalent of a constant gauge field, whichdoes not change the electronic spectrum. The situationchanges if one considers a boundary that separates twodomains where the magnitude of the distortion is differ-ent. The shift of Dirac points leads to a deflection of theelectronic trajectories that cross the boundary, alsoshown in Fig. 29. The modulation of the gauge fieldleads to an effective magnetic field, which is of oppositesign for the two valleys.

We have shown in Sec. IV.B how topological latticedefects, such as disclinations and dislocations, can be de-scribed by an effective gauge field. Those defects canonly exist in graphene sheets that are intrinsicallycurved, and the gauge field only depends on the topol-ogy of the lattice. Changes in the nearest-neighbor hop-ping also lead to effective gauge fields. We next considertwo physical processes that induce effective gauge fields:i changes in the hopping induced by hybridization be-tween and bands, which arise in curved sheets, and

0 100 200 300 400 500Position

-0.04

-0.02

0

0.02

0.04

0.06E

lect

rost

atic

pote

ntia

l

0 10 200.4

0.45

0.5

0.5

0.5005

Cha

rge

dens

it y

0 200 400 600 800 1000 1200 1400Width

0

0.0001

0.0002

0.0003

0.0004

0.0005

Dop

ing

FIG. 28. Color online Displaced electronic charge close to agraphene zigzag edge. Top: self-consistent analysis of the dis-placed charge density in units of number of electrons per car-bon shown as a continuous line, and the corresponding elec-trostatic potential in units of t shown as a dashed line, for agraphene ribbon with periodic boundary conditions along thezigzag edge with a length of L=960a and with a circumfer-ence of size W=803a. Inset: The charge density near theedge. Due to the presence of the edge, there is a displacedcharge in the bulk bottom panel that is shown as a function ofwidth W. Note that the displaced charge vanishes in the bulklimit W→, in agreement with Eq. 161. Adapted fromPeres, Guinea, and Castro Neto, 2006a.

K

K’

K

K’

FIG. 29. Color online Gauge field induced by a simple elasticstrain. Top: the hopping along the horizontal bonds is assumedto be changed on the right-hand side of the graphene lattice,defining a straight boundary between the unperturbed and per-turbed regions dashed line. Bottom: the modified hoppingacts like a constant gauge field, which displaces the Diraccones in opposite directions at the K and K points of theBrillouin zone. The conservation of energy and momentumparallel to the boundary leads to a deflection of electrons bythe boundary.



ii changes in the hopping due to modulation in thebond length, which is associated with elastic strain. Thestrength of these fields depends on parameters that de-scribe the value of the - hybridization, and the depen-dence of hopping on the bond length.

A comparison of the relative strengths of the gaugefields induced by intrinsic curvature, - hybridizationextrinsic curvature, and elastic strains, arising from aripple of typical height and size, is given in Table II.

1. Gauge field induced by curvature

As discussed in Sec. IV.A, when the orbitals are notparallel, the hybridization between them depends ontheir relative orientation. The angle i determines therelative orientation of neighboring orbitals at some po-sition ri in the graphene sheet. The value of i dependson the local curvature of the layer. The relative angle ofrotation of two pz orbitals at position ri and rj can bewritten as cosi−j=Ni ·Nj, where Ni is the unit vectorperpendicular to the surface, defined in Eq. 125. If rj=ri+uij, we can write

Ni · Nj 1 + Ni · uij · Ni + 12Ni · uij · 2Ni ,

162

where we assume smoothly varying Nr. We use Eq.125 in terms of the height field hr Nrz−hr− h2z /2 to rewrite Eq. 162 as

Ni · Nj 1 − 12 uij · hri2. 163

Hence, bending of the graphene sheet leads to a modi-fication of the hopping amplitude between different sitesof the form

tij −tij0

2uij · hri2, 164

where tij0 is the bare hopping energy. A similar effect

leads to changes in the electronic states of carbon nano-tubes Kane and Mele, 1997. Using the results of Sec.IV, namely, Eq. 147, we can now show that a vector

potential is generated for nearest-neighbor hopping u=ab Eun-Ah Kim and Castro Neto, 2008,

Axh = −

3Eaba2

8x

2h2 − y2h2 ,

Ayh =

3Eaba2

4x

2h + y2hxhyh , 165

where the coupling constant Eab depends on microscopicdetails Eun-Ah Kim and Castro Neto, 2008. The flux ofeffective magnetic field through a ripple of lateral di-mension l and height h is given by

Eaba2h2

vFl3 , 166

where the radius of curvature is R−1hl−2. For a ripplewith l20 nm, h1 nm, taking Eab /vF10 Å−1, we find10−30.

2. Elastic strain

The elastic free energy for graphene can be written interms of the in-plane displacement ur= ux ,uy as

Fu =12 d2rB − G

i=1,2uii2

+ 2G i,j=1,2

uij2 ,

167

where B is the bulk modulus, G is the shear modulus,and

uij =12 ui

xj+

uj

xj 168

is the strain tensor x1=x and x2=y.There are many types of static deformation of the

honeycomb lattice that can affect the propagation ofDirac fermions. The simplest one is due to changes inthe area of the unit cell due to either dilation or contrac-tion. Changes in the unit-cell area lead to local changesin the density of electrons and, therefore, local changesin the chemical potential in the system. In this case, theireffect is similar to the one found in Eq. 148, and wehave

dpr = guxx + uyy , 169

and their effect is diagonal in the sublattice index.The nearest-neighbor hopping depends on the length

of the carbon bond. Hence, elastic strains that modifythe relative orientation of atoms also lead to an effectivegauge field, which acts on each K point separately, asfirst discussed in relation to carbon nanotubes Suzuuraand Ando, 2002b; Mañes, 2007. Consider two carbonatoms located in two different sublattices in the sameunit cell at Ri. The change in the local bond length canbe written as

TABLE II. Estimates of the effective magnetic length, andeffective magnetic fields generated by the deformations. Theintrinsic curvature entry also refers to the contribution fromtopological defects.

lB

B Th=1 nm, l=10 nm, a=0.1 nm

Intrinsic curvature l l

h 0.06

Extrinsic curvature l t

E

l3

ah20.006

Elastic strains l1

al

h26



ui =ab

a· uARi − uBRi + ab . 170

The local displacements of the atoms in the unit cell canbe related to ur by Ando, 2006a

ab · u = #−1uA − uB , 171

where # is a dimensionless quantity that depends on mi-croscopic details. Changes in the bond length lead tochanges in the hopping amplitude,

tij tij0 +

tij

aui, 172

and we can write

tabr ur

a, 173

where

=tab

lna. 174

Substituting Eq. 170 into Eq. 173 and the final resultinto Eq. 147, one finds Ando, 2006a

Axs = 3

4#uxx − uyy ,

Ays = 3

2#uxy. 175

We assume that the strains induced by a ripple of dimen-sion l and height h scale as uijh / l2. Then, using /vFa−11 Å−1, we find that the total flux through aripple is

h2

al. 176

For ripples such that h1 nm and l20 nm, this esti-mate gives 10−10, in reasonable agreement with theestimates by Morozov et al. 2006.

The strain tensor must satisfy some additional con-straints, as it is derived from a displacement vector field.These constraints are called Saint-Venant compatibilityconditions Landau and Lifshitz, 1959,

Wijkl =uij

xkxl+

ukl

xixj−

uil

xjxk−

ujk

xixl= 0. 177

An elastic deformation changes the distances in the crys-tal lattice and can be considered as a change in the met-ric,

gij = ij + uij. 178

The compatibility equations 177 are equivalent to thecondition that the curvature tensor derived from Eq.178 is zero. Hence, a purely elastic deformation cannotinduce intrinsic curvature in the sheet, which only arisesfrom topological defects. The effective fields associatedwith elastic strains can be large Morozov et al., 2006,leading to significant changes in the electronic wavefunctions. An analysis of the resulting state, and the pos-

sible instabilities that may occur, can be found in Guineaet al. 2008.

3. Random gauge fields

The preceding discussion suggests that the effectivefields associated with lattice defects can modify signifi-cantly the electronic properties. This is the case whenthe fields do not change appreciably on scales compa-rable to the effective magnetic length. The generalproblem of random gauge fields for Dirac fermions hasbeen extensively analyzed before the current interest ingraphene, as the topic is also relevant for the IQHELudwig et al., 1994 and d-wave superconductivityNersesyan et al., 1994. The one-electron nature of thistwo-dimensional problem makes it possible, at the Diracenergy, to map it onto models of interacting electrons inone dimension, where many exact results can be ob-tained Castillo et al., 1997. The low-energy density ofstates acquires an anomalous exponent 1−, where 0. The density of states is enhancednear the Dirac energy, reflecting the tendency of disor-der to close gaps. For sufficiently large values of therandom gauge field, a phase transition is also possibleChamon et al., 1996; Horovitz and Doussal, 2002.

Perturbation theory shows that random gauge fieldsare a marginal perturbation at the Dirac point, leadingto logarithmic divergences. These divergences tend tohave the opposite sign with respect to those induced bythe Coulomb interaction see Sec. V.B. As a result, arenormalization-group RG analysis of interacting elec-trons in a random gauge field suggests the possibility ofnontrivial phases Stauber et al., 2005; Aleiner and Efe-tov, 2006; Altland, 2006; Dell’Anna, 2006; Foster andLudwig, 2006a, 2006b; Nomura et al., 2007; Khvesh-chenko, 2008, where interactions and disorder canceleach other.

G. Coupling to magnetic impurities

Magnetic impurities in graphene can be introducedchemically by deposition and intercalation Calandraand Mauri, 2007; Uchoa et al., 2008, or self-generatedby the introduction of defects Kumazaki andHirashima, 2006, 2007. The energy dependence of thedensity of states in graphene leads to changes in theformation of a Kondo resonance between a magneticimpurity and the graphene electrons. The vanishing ofthe density of states at the Dirac energy implies that aKondo singlet in the ground state is not formed unlessthe exchange coupling exceeds a critical value, of theorder of the electron bandwidth, a problem alreadystudied in connection with magnetic impurities ind-wave superconductors Cassanello and Fradkin, 1996,1997; Polkovnikov et al., 2001; Polkovnikov, 2002; Fritzet al., 2006. For weak exchange couplings, the magneticimpurity remains unscreened. An external gate changesthe chemical potential, allowing for a tuning of theKondo resonance Sengupta and Baskaran, 2008. Thesituation changes significantly if the scalar potential in-



duced by the magnetic impurity is taken into account.This potential that can be comparable to the bandwidthallows the formation of midgap states and changes thephase shift associated with spin scattering Hentscheland Guinea, 2007. These phase shifts have a weak loga-rithmic dependence on the chemical potential, and aKondo resonance can exist, even close to the Dirac en-ergy.

The RKKY interaction between magnetic impuritiesis also modified in graphene. At finite fillings, the ab-sence of intravalley backscattering leads to a reductionof the Friedel oscillations, which decay as sin2kFr / r3Ando, 2006b; Cheianov and Fal’ko, 2006; Wunsch et al.,2006. This effect leads to an RKKY interaction at finitefillings, which oscillate and decay as r−3. When interval-ley scattering is included, the interaction reverts to theusual dependence on distance in two dimensions r−2

Cheianov and Fal’ko, 2006. At half filling extended de-fects lead to an RKKY interaction with an r−3 depen-dence Vozmediano et al., 2005; Dugaev et al., 2006.This behavior is changed when the impurity potential islocalized on atomic scales Brey et al., 2007; Saremi,2007, or for highly symmetrical couplings Saremi,2007.

H. Weak and strong localization

In sufficiently clean systems, where the Fermi wave-length is much shorter than the mean free path kFl1,electronic transport can be described in classical terms,assuming that electrons follow well-defined trajectories.At low temperatures, when electrons remain coherentover long distances, quantum effects lead to interferencecorrections to the classical expressions for the conduc-tivity, the weak-localization correction Bergman, 1984;Chakravarty and Schmid, 1986. These corrections areusually due to the positive interference between twopaths along closed loops, traversed in opposite direc-tions. As a result, the probability that the electron goesback to the origin is enhanced, so that quantum correc-tions decrease the conductivity. These interferences aresuppressed for paths longer than the dephasing length ldetermined by interactions between the electron and en-vironment. Interference effects can also be suppressedby magnetic fields that break down time-reversal sym-metry and add a random relative phase to the processdiscussed above. Hence, in most metals, the conductivityincreases when a small magnetic field is applied nega-tive magnetoresistance.

Graphene is special in this respect, due to the chiralityof its electrons. The motion along a closed path inducesa change in the relative weight of the two components ofthe wave function, leading to a new phase, which con-tributes to the interference processes. If the electrontraverses a path without being scattered from one valleyto the other, this Berry phase changes the sign of theamplitude of one path with respect to the time-reversedpath. As a consequence, the two paths interfere destruc-tively, leading to a suppression of backscattering Suz-uura and Ando, 2002a. Similar processes take place in

materials with strong spin-orbit coupling, as the spin di-rection changes along the path of the electron Berg-man, 1984; Chakravarty and Schmid, 1986. Hence, ifscattering between valleys in graphene can be neglected,one expects a positive magnetoresistance, i.e., weak an-tilocalization. In general, intravalley and intervalleyelastic scattering can be described in terms of two differ-ent scattering times "intra and "inter, so that if "intra"inter, one expects weak antilocalization processes,while if "inter"intra, ordinary weak localization will takeplace. Experimentally, localization effects are alwaysstrongly suppressed close to the Dirac point but can bepartially or, in rare cases, completely recovered at highcarrier concentrations, depending on a particular single-layer sample Morozov et al., 2006; Tikhonenko et al.,2008. Multilayer samples exhibit an additional positivemagnetoresistance in higher magnetic fields, which canbe attribued to classical changes in the current distribu-tion due to a vertical gradient of concentration Moro-zov et al., 2006 and antilocalization effects Wu et al.,2007.

The propagation of an electron in the absence of in-tervalley scattering can be affected by the effectivegauge fields induced by lattice defects and curvature.These fields can suppress the interference corrections tothe conductivity Morozov et al., 2006; Morpurgo andGuinea, 2006. In addition, the description in terms offree Dirac electrons is only valid near the neutralitypoint. The Fermi energy acquires a trigonal distortionaway from the Dirac point, and backward scatteringwithin each valley is no longer completely suppressedMcCann et al., 2006, leading to further suppression ofantilocalization effects at high dopings. Finally, the gra-dient of external potentials induces a small asymmetrybetween the two sublattices Morpurgo and Guinea,2006. This effect will also contribute to reduce antilocal-ization, without giving rise to localization effects.

The above analysis has to be modified for a graphenebilayer. Although the description of the electronic statesrequires a two-component spinor, the total phase arounda closed loop is 2, and backscattering is not suppressedKechedzhi et al., 2007. This result is consistent withexperimental observations, which show the existence ofweak localization effects in a bilayer Gorbachev et al.,2007.

When the Fermi energy is at the Dirac point, a replicaanalysis shows that the conductivity approaches a uni-versal value of the order of e2 /h Fradkin, 1986a, 1986b.This result is valid when intervalley scattering is ne-glected Ostrovsky et al., 2006, 2007; Ryu et al., 2007.Localization is induced when these terms are includedAleiner and Efetov, 2006; Altland, 2006, as also con-firmed by numerical calculations Louis et al., 2007. In-teraction effects tend to suppress the effects of disorder.The same result, namely, a conductance of the order ofe2 /h, is obtained for disordered graphene bilayers wherea self-consistent calculation leads to universal conductiv-ity at the neutrality point Nilsson, Castro Neto, Guinea,et al., 2006; Katsnelson, 2007c; Nilsson et al., 2008. In abiased graphene bilayer, the presence of impurities leads



to the appearance of impurity tails in the density ofstates due to the creation of midgap states, which aresensitive to the applied electric field that opens the gapbetween the conduction and valence bands Nilsson andCastro Neto, 2007.

We point out that most calculations of transport prop-erties assume self-averaging, that is, one can exchange aproblem with a lack of translational invariance by aneffective medium system with damping. This procedureonly works when the disorder is weak and the system isin the metallic phase. Close to the localized phase thisprocedure breaks down, the system divides itself intoregions of different chemical potential and one has tothink about transport in real space, usually described interms of percolation Cheianov, Fal’ko, Altshuler, et al.,2007; Shklovskii, 2007. Single electron transistor SETmeasurements of graphene show that this seems to bethe situation in graphene at half filling Martin et al.,2008.

Finally, we point out that graphene stacks suffer fromanother source of disorder, namely, c axis disorder,which is due to either impurities between layers or rota-tion of graphene planes relative to each other. In eithercase, the in-plane and out-of-plane transport is directlyaffected. This kind of disorder has been observed ex-perimentally by different techniques Bar et al., 2007;Hass, Varchon, Millán-Otoya, et al., 2008. In the case ofthe bilayer, the rotation of planes changes substantiallythe spectrum restoring the Dirac fermion descriptionLopes dos Santos et al., 2007. The transport propertiesin the out-of-plane direction are determined by the in-terlayer current operator jn,n+1= itcA,n,s

† cA,n+1,s

−cA,n+1,s† cA,n,s, where n is a layer index and A is a ge-

neric index that defines the sites coupled by the inter-layer hopping t. If we only consider hopping betweennearest-neighbor sites in consecutive layers, these sitesbelong to one of the two sublattices in each layer.

In a multilayer with Bernal stacking, these connectedsites are the ones where the density of states vanishes atzero energy, as discussed above. Hence, even in a cleansystem, the number of conducting channels in the direc-tion perpendicular to the layers vanishes at zero energyNilsson, Castro Neto, Guinea, and Peres, 2006; Nilssonet al., 2008. This situation is reminiscent of the in-planetransport properties of a single-layer graphene. Similarto the latter case, a self-consistent Born approximationfor a small concentration of impurities leads to a finiteconductivity, which becomes independent of the numberof impurities.

I. Transport near the Dirac point

In clean graphene, the number of channels availablefor electron transport decreases as the chemical poten-tial approaches the Dirac energy. As a result, the con-ductance through a clean graphene ribbon is at most4e2 /h, where the factor of 4 stands for the spin and val-ley degeneracy. In addition, only one out of every threepossible clean graphene ribbons has a conduction chan-

nel at the Dirac energy. The other two-thirds are semi-conducting, with a gap of the order of vF /W, where W isthe width. This result is a consequence of the additionalperiodicity introduced by the wave functions at the Kand K points of the Brillouin zone, irrespective of theboundary conditions.

A wide graphene ribbon allows for many channels,which can be approximately classified by the momentumperpendicular to the axis of the ribbon, ky. At the Diracenergy, transport through these channels is inhibited bythe existence of a gap ky

=vFky. Transport throughthese channels is suppressed by a factor of the order ofe−kyL, where L is the length of the ribbon. The numberof transverse channels increases as W /a, where W is thewidth of the ribbon and a is a length of the order of thelattice spacing. The allowed values of ky are ny /W,where ny is an integer. Hence, for a ribbon such thatWL, there are many channels that satisfy kyL1.Transport through these channels is not strongly inhib-ited, and their contribution dominates when the Fermienergy lies near the Dirac point. The conductance aris-ing from these channels is given by Katsnelson, 2006b;Tworzydlo et al., 2006

G e2

h

W

2 dkye−kyL e2

h

W

L. 179

The transmission at normal incidence ky=0 is 1, inagreement with the absence of backscattering ingraphene, for any barrier that does not induce interval-ley scattering Katsnelson et al., 2006. The transmissionof a given channel scales as Tky=1/cosh2kyL /2.

Equation 179 shows that the contribution from alltransverse channels leads to a conductance that scales,similar to a function of the length and width of the sys-tem, as the conductivity of a diffusive metal. Moreover,the value of the effective conductivity is of the order ofe2 /h. It can also be shown that the shot noise depends oncurrent in the same way as in a diffusive metal. A de-tailed analysis of possible boundary conditions at thecontacts and their influence on evanescent waves can befound in Robinson and Schomerus 2007 and Schom-erus 2007. The calculations leading to Eq. 179 can beextended to a graphene bilayer. The conductance is,again, a summation of terms arising from evanescentwaves between the two contacts, and it has the depen-dence on sample dimensions of a 2D conductivity of theorder of e2 /h Snyman and Beenakker, 2007, althoughthere is a prefactor twice as big as the one in single-layergraphene.

The calculation of the conductance of clean graphenein terms of transmission coefficients, using the Landauermethod, leads to an effective conductivity that is equalto the value obtained for bulk graphene using diagram-matic methods, the Kubo formula Peres, Guinea, andCastro Neto, 2006b, in the limit of zero impurity con-centration and zero doping. Moreover, this correspon-dence remains valid for the case of a bilayer without andwith trigonal warping effects Koshino and Ando, 2006;Cserti, Csordés, and Dévid, 2007.



Disorder at the Dirac energy changes the conductanceof graphene ribbons in two opposite directions Louis etal., 2007: i a sufficiently strong disorder, with short-range intervalley contributions, leads to a localized re-gime, where the conductance depends exponentially onthe ribbon length, and ii at the Dirac energy, disorderallows midgap states that can enhance the conductancemediated by evanescent waves discussed above. A fluc-tuating electrostatic potential also reduces the effectivegap for the transverse channels, further enhancing theconductance. The resonant tunneling regime mediatedby midgap states was suggested by analytical calcula-tions Titov, 2007. The enhancement of the conductanceby potential fluctuations can also be studied semianalyti-cally. In the absence of intervalley scattering, it leads toan effective conductivity that grows with ribbon lengthSan-Jose et al., 2007. In fact, analytical and numericalstudies Bardarson et al., 2007; Nomura et al., 2007; San-Jose et al., 2007; Lewenkopf et al., 2008, show that theconductivity obeys a universal scaling with the latticesize L,

L =2e2

hA lnL/ + B , 180

where is a length scale associated with a range of in-teractions and A and B are numbers of the order of unitA0.17 and B0.23 for a graphene lattice in the shapeof a square of size L Lewenkopf et al., 2008. Note,therefore, that the conductivity is always of the order ofe2 /h and has a weak dependence on size.

J. Boltzmann equation description of dc transport in dopedgraphene

It was shown experimentally that the dc conductivityof graphene depends linearly on the gate potential No-voselov et al., 2004; Novoselov, Geim, Morozov, et al.,2005; Novoselov, Jiang, Schedin, et al., 2005, except veryclose to the neutrality point see Fig. 30. Since the gatepotential depends linearly on the electronic density n,one has a conductivity n. As shown by Shon andAndo 1998, if the scatterers are short ranged, the dcconductivity should be independent of the electronicdensity, at odds with the experimental result. It has beenshown Ando, 2006b; Nomura and MacDonald, 2006,2007 that, by considering a scattering mechanism basedon screened charged impurities, it is possible to obtainfrom a Boltzmann equation approach a conductivityvarying linearly with the density, in agreement with theexperimental result Ando, 2006b; Novikov, 2007b;Peres, Lopes dos Santos, and Stauber, 2007; Trushin andSchliemann, 2007; Katsnelson and Geim, 2008.

The Boltzmann equation has the form Ziman, 1972

− vk · rfk − eE + vk H · kfk = − fk

t

scatt.

181

The solution of the Boltzmann equation in its generalform is difficult and one needs, therefore, to rely upon

some approximation. The first step in the usual approxi-mation scheme is to write the distribution as fk= f0k+gk, where f0k is the steady-state distribu-tion function and gk is assumed small. Inserting thisansatz in into Eq. 181 and keeping only terms that arelinear in the external fields, one obtains the linearizedBoltzmann equation Ziman, 1972, which reads

−f0k

kvk · −

k −

TrT + eE −

1

er

= − fk

t

scatt+ vk · rgk + evk H · kgk. 182

The second approximation has to do with the form ofthe scattering term. The simplest approach is to intro-duce a relaxation-time approximation,

− fk

t

scatt→

gk

"k, 183

where "k is the relaxation time, assumed to bemomentum-dependent. This momentum dependence isdetermined phenomenologically in such way that thedependence of the conductivity upon the electronic den-sity agrees with experimental data. The Boltzmannequation is certainly not valid at the Dirac point, butsince many experiments are performed at finite carrierdensity, controlled by an external gate voltage, we ex-pect the Boltzmann equation to give reliable results if anappropriate form for "k is used Adam et al., 2007.

FIG. 30. Color online Changes in conductivity of graphenewith varying gate voltage Vg and carrier concentration n. Here is proportional to n. Note that samples with higher mobility1 m2/V s normally show a sublinear dependence, presum-ably indicating the presence of different types of scatterers.Inset: Scanning-electron micrograph of one of experimentaldevices in false colors matching those seen in visible optics.The scale of the micrograph is given by the width of the Hallbar, which is 1 m. Adapted from Novoselov, Geim, Morozov,et al., 2005.



We compute the Boltzmann relaxation time "k for twodifferent scattering potentials: i a Dirac delta functionpotential and ii an unscreened Coulomb potential. Therelaxation time "k is defined as

1

"k= ni d kdk

22 Sk,k1 − cos , 184

where ni is impurity concentration per unit of area, andthe transition rate Sk ,k is given, in the Born approxi-mation, by

Sk,k = 2Hk,k21

vFk − k , 185

where vFk is the dispersion of Dirac fermions ingraphene and Hk,k is defined as

Hk,k = drk* rUSrkr , 186

with USr the scattering potential and kr is the elec-tronic spinor wave function of a clean graphene sheet. Ifthe potential is short ranged Shon and Ando, 1998 ofthe form US=v0r, the Boltzmann relaxation time isgiven by

"k =4vF

niv02

1

k. 187

On the other hand, if the potential is the Coulomb po-tential, given by USr=eQ /40r for charged impuri-ties of charge Q, the relaxation time is given by

"k =vF

u02 k , 188

where u02=niQ

2e2 /16022. As argued below, the phenom-

enology of Dirac fermions implies that the scattering ingraphene must be of the form 188.

Within the relaxation time approximation, the solu-tion of the linearized Boltzmann equation when an elec-tric field is applied to the sample is

gk = −f0k

ke"kvk · E , 189

and the electric current reads spin and valley indexesincluded

J =4

Ak

evkgk. 190

Since at low temperatures the following relation−f0k /k→−vFk holds, one can see that assum-ing Eq. 188 where k is measured relatively to the Diracpoint, the electronic conductivity turns out to be

xx = 2e2

h

2

u02 = 2

e2

h

vF2

u02 n , 191

where u0 is the strength of the scattering potential withdimensions of energy. The electronic conductivity de-pends linearly on the electron density, in agreement withthe experimental data. We stress that the Coulomb po-

tential is one possible mechanism of producing a scatter-ing rate of the form 188, but we do not exclude thatother mechanisms may exist see, for instance, Katsnel-son and Geim 2008.

K. Magnetotransport and universal conductivity

The description of the magnetotransport properties ofelectrons in a disordered honeycomb lattice is complexbecause of the interference effects associated with theHofstadter problem Gumbs and Fekete, 1997. We sim-plify our problem by describing electrons in the honey-comb lattice as Dirac fermions in the continuum ap-proximation, introduced in Sec. II.B. Furthermore, wefocus only on the problem of short-range scattering inthe unitary limit since in this regime many analyticalresults are obtained Kumazaki and Hirashima, 2006;Pereira et al., 2006; Peres, Guinea, and Castro Neto,2006a; Skrypnyk and Loktev, 2006, 2007; Mariani et al.,2007; Pereira, Lopes dos Santos, and Castro Neto, 2008.The problem of magnetotransport in the presence ofCoulomb impurities, as discussed, is still an open re-search problem. A similar approach was considered byAbrikosov in the quantum magnetoresistance study ofnonstoichiometric chalcogenides Abrikosov, 1998. Inthe case of graphene, the effective Hamiltonian describ-ing Dirac fermions in a magnetic field including disor-der can be written as H=H0+Hi, where H0 is given byEq. 5 and Hi is the impurity potential given by Peres,Guinea, and Castro Neto, 2006a

Hi = Vj=1

Ni

r − rjI . 192

The formulation of the problem in second quantizationrequires the solution of H0, which was done in Sec. II.I.The field operators, close to the K point, are defined asthe spin index is omitted for simplicity

r = k

eikx

L 0

0yck,−1

+ n,k,

eikx

2L ny − klB

2 n+1y − klB

2 ck,n,, 193

where ck,n, destroys an electron in band = ±1, withenergy level n and guiding center klB

2 ; ck,−1 destroys anelectron in the zero Landau level; the cyclotron fre-quency is given by Eq. 96. The sum over n=0,1 ,2 , . . . iscut off at n0 given by E1,n0=W, where W is of theorder of the electronic bandwidth. In this representa-tion, H0 becomes diagonal, leading to Green’s functionsof the form in the Matsubara representation

G0k,n, ;i =1

i − E,n, 194

where



E,n = cn 195

are the Landau levels for this problem = ±1 labels thetwo bands. Note that G0k ,n , ; i is effectively k in-dependent, and E ,−1=0 is the zero-energy Landaulevel. When expressed in the Landau basis, the scatter-ing Hamiltonian 192 connects Landau levels of nega-tive and positive energy.

1. The full self-consistent Born approximation (FSBA)

In order to describe the effect of impurity scatteringon the magnetoresistance of graphene, the Green’s func-tion for Landau levels in the presence of disorder needsto be computed. In the context of the 2D electron gas,an equivalent study was performed by Ohta and AndoOhta, 1971a, 1971b; Ando, 1974a, 1974b, 1974c, 1975;Ando and Uemura, 1974 using the averaging procedureover impurity positions of Duke 1968. Below, the aver-aging procedure over impurity positions is performed inthe standard way, namely, having determined theGreen’s function for a given impurity configurationr1 , . . . ,rNi

, the position averaged Green’s function is de-termined from

Gp,n, ;i ;r1, . . . ,rNi Gp,n, ;i

= L−2Nij=1

Ni drjGp,n, ;i ;r1, . . . ,rNi

.

196

In the presence of Landau levels, the average over im-purity positions involves the wave functions of the one-dimensional harmonic oscillator. After lengthy algebra,the Green’s function in the presence of vacancies, in theFSBA, can be written as

Gp,n, ; + 0+ = − En, − %1−1, 197

Gp,− 1; + 0+ = − %2−1, 198

where

%1 = − niZ−1, 199

%2 = − nigcGp,− 1; + 0+/2 + Z−1, 200

Z = gcGp,− 1; + 0+/2

+ gcn,

Gp,n, ; + 0+/2, 201

and gc=Ac /2lB2 is the degeneracy of a Landau level per

unit cell. One should note that the Green’s functions donot depend upon p explicitly. The self-consistent solu-tion of Eqs. 197–201 gives the density of states, theelectron self-energy, and the change of Landau level en-ergy position due to disorder.

The effect of disorder on the density of states of Diracfermions in a magnetic field is shown in Fig. 31. Forreference, we note that E1,1=0.14 eV for B=14 T,

and E1,1=0.1 eV for B=6 T. From Fig. 31 we seethat, for a given ni, the effect of broadening due to im-purities is less effective as the magnetic field increases. Itis also clear that the position of Landau levels is renor-malized relatively to the nondisordered case. The renor-malization of the Landau level position can be deter-mined from poles of Eqs. 197 and 198,

− E,n − Re % = 0. 202

Due to the importance of scattering at low energies, thesolution to Eq. 202 does not represent exact eigen-states of the system since the imaginary part of the self-energy is nonvanishing. However, these energies do de-termine the form of the density of states, as discussedbelow.

In Fig. 32, the graphical solution to Eq. 202 is givenfor two different energies E−1,n, with n=1,2. It isclear that the renormalization is important for the firstLandau level. This result is due to the increase in scat-tering at low energies, which is already present in thecase of zero magnetic field. The values of satisfyingEq. 202 show up in the density of states as the energyvalues where the oscillations due to the Landau levelquantization have a maximum. In Fig. 31, the position ofthe renormalized Landau levels is shown in the toppanel marked by two arrows, corresponding to thebare energies E−1,n, with n=1,2. The importance ofthis renormalization decreases with the decrease in thenumber of impurities. This is clear in Fig. 31, where a

0

0.002

0.004

0.006

0.008

DO

Spe

ru.

c.(1

/eV

)

ni = 0.001

ni = 0.005

ni = 0.0009

ωc = 0.14 eV, B = 12 T

-3 -2 -1 0 1 2 3ω / ωc

0

0.002

0.004

0.006

0.008

DO

Spe

ru.

c.(1

/eV

)

ni = 0.001

ni = 0.0008

ni = 0.0006

ωc = 0.1 eV, B = 6 T

FIG. 31. Color online Density of states of Dirac fermions ina magnetic field. Top: electronic density of states DOS as a function of /c c=0.14 eV in a magnetic field B=12 T for different impurity concentrations ni. Bottom: asa function of /c c=0.1 eV is the cyclotron frequency in amagnetic field B=6 T. The solid line shows the DOS in theabsence of disorder. The position of the Landau levels in theabsence of disorder is shown as vertical lines. The two arrowsin the top panel show the position of the renormalized Landaulevels see Fig. 32 given by the solution of Eq. 202. Adaptedfrom Peres, Guinea, and Castro Neto, 2006a.



visible shift toward low energies is evident when ni has asmall 10% change, from ni=10−3 to ni=910−4.

Studying of the magnetoresistance properties of thesystem requires calculation of the conductivity tensor.We compute the current-current correlation function,and from it the conductivity tensor is derived. The de-tails of the calculations are presented in Peres, Guinea,and Castro Neto 2006a. If, however, we neglect the realpart of the self-energy, assume for Im %i=! i=1,2 aconstant value, and consider that E1,1!, these re-sults reduce to those of Gorbar et al. 2002.

It is instructive to consider first the case ,T→0,which leads to xx0,0=0

0 =e2

h

4

Im %10/Im %20 − 1

1 + Im %10/c2

+n0 + 1

n0 + 1 + Im %10/c2 , 203

where we include a factor of 2 due to the valley degen-eracy. In the absence of a magnetic field c→0, theabove expression reduces to

0 =e2

h

4

1 −

Im %102

vF&2 + Im %102 , 204

where we have introduced the energy cutoff vF&. Eitherwhen Im %10 Im %20 and cIm %10 or n0

Im %10 /c, c=E0,1=2vF / lB2 or when &vF

Im %10, in the absence of an applied field, Eqs. 203and 204 reduce to

0 =4

e2

h, 205

which is the so-called universal conductivity of grapheneFradkin, 1986a, 1986b; Lee, 1993; Ludwig et al., 1994;Nersesyan et al., 1994; Ziegler, 1998; Yang and Nayak,2002; Katsnelson, 2006b; Peres, Guinea, and Castro

Neto, 2006a; Tworzydlo et al., 2006. This result was ob-tained previously by Ando and collaborators using thesecond-order self-consistent Born approximation Shonand Ando, 1998; Ando et al., 2002.

Because the dc magnetotransport properties ofgraphene are normally measured with the possibility oftuning its electronic density by a gate potential No-voselov et al., 2004, it is important to compute the con-ductivity kernel, since this has direct experimental rel-evance. In the case →0, we write the conductivityxx0,T as

xx0,T =e2

h

−

df

KB , 206

where the conductivity kernel KB is given in the Ap-pendix of Peres, Guinea, and Castro Neto 2006a. Themagnetic field dependence of kernel KB is shown inFig. 33. One can observe that the effect of disorder is thecreation of a region where KB remains constant be-fore it starts to increase in energy with superimposedoscillations coming from the Landau levels. The sameeffect, but with the absence of the oscillations, was iden-tified at the level of the self-consistent density of statesplotted in Fig. 31. Together with xx0,T, the Hall con-ductivity xy0,T allows the calculation of the resistivitytensor 109.

We now focus on the optical conductivity xxPeres, Guinea, and Castro Neto, 2006a; Gusynin et al.,2007. This quantity can be probed by reflectivity experi-ments in the subterahertz to midinfrared frequencyrange Bliokh, 2005. This quantity is shown in Fig. 34for different magnetic fields. It is clear that the first peakis controlled by E1,1−E1,−1, and we have checkedthat it does not obey any particular scaling form as afunction of /B. On the other hand, as the effect of

-6 -4 -2 0 2 4 60

0.2

0.4

0.6

0.8

-Im

Σ 1(ω)/

ωc

B = 12 TB = 6 T

ni=0.001

-6 -4 -2 0 2 4 6ω / ωc

0

0.2

0.4

0.6

0.8

-Im

Σ 2(ω)/

ωc

-6 -4 -2 0 2 4 6

-0.4

-0.2

0

0.2

0.4

Re

Σ 1(ω)/

ωc

ω−E(-1,1)ω−E(-1,2)

-6 -4 -2 0 2 4 6ω / ωc

-0.4

-0.2

0

0.2

0.4

Re

Σ 2(ω)/

ωcω

FIG. 32. Color online Imaginary right and real left partsof %1 top and %2 bottom, in units of c, as a functionof /c. The right panels also show the intercept of −E ,n with Re% as required by Eq. 202. Adapted fromPeres, Guinea, and Castro Neto, 2006a.

-0.4 -0.2 0 0.2 0.40

5

10

15

20

πK

B(ω

)h/e

2 B = 12 T

ni = 0.001

-0.4 -0.2 0 0.2 0.40

5

10

15

20

B = 10 T

-0.4 -0.2 0 0.2 0.4ω (eV)

0

5

10

15

20

πK

B(ω

)h/e

2 B = 8 T

-0.4 -0.2 0 0.2 0.4ω (eV)

0

5

10

15

20

B = 6 T

FIG. 33. Color online Conductivity kernel K in units ofe2 /h as a function of energy for different magnetic fieldsand ni=10−3. The horizontal lines mark the universal limit ofthe conductivity per cone 0=2e2 /h. The vertical lines showthe position of the Landau levels in the absence of disorder.Adapted from Peres, Guinea, and Castro Neto, 2006a.



scattering becomes less important, high-energy conduc-tivity oscillations start obeying the scaling /B, asshown in the lower right panel of Fig. 34.

V. MANY-BODY EFFECTS

A. Electron-phonon interactions

In Secs. IV.F.1 and IV.F.2, we discussed how static de-formations of the graphene sheet due to bending andstrain couple to the Dirac fermions via vector potentials.Just as bending has to do with the flexural modes of thegraphene sheet as discussed in Sec. III, strain fields arerelated to optical and acoustic modes Wirtz and Rubio,2004. Given the local displacements of the atoms ineach sublattice uA and uB, the electron-phonon couplinghas essentially the form discussed previously for staticfields.

Coupling to acoustic modes is the most straightfor-ward one, since it already appears in the elastic theory. Ifuac is the acoustic phonon displacement, then the rela-tion between this displacement and the atom displace-ment is given by Eq. 171, and its coupling to electronsis given by the vector potential 175 in the Dirac equa-tion 150.

For optical modes, the situation is slightly differentsince the optical mode displacement is Ando, 2006a,2007b

uop =12

uA − uB , 207

that is, the bond length deformation vector. To calculatethe coupling to electrons, we can proceed as previouslyand compute the change in the nearest-neighbor hop-ping energy due to the lattice distortion through Eqs.172, 173, 170, and 207. Once again the electron-phonon interaction becomes a problem of coupling elec-trons with a vector potential as in Eq. 150, where thecomponents of the vector potential are

Axop = −3

2

a2uyop,

Ayop = −3

2

a2uxop, 208

where =t / lna was defined in Eq. 174. Note that

we can write A op=−3/2 /a2 uop. A similar expres-

sion is valid close to the K’ point with A replaced by −A .Optical phonons are important in graphene research

because of Raman spectroscopy. The latter has playedan important role in the study of carbon nanotubesSaito et al., 1998 because of the 1D character of thesesystems, namely, the presence of van Hove singularitiesin the 1D spectrum leads to colossal enhancements ofthe Raman signal that can be easily detected, even for asingle isolated carbon nanotube. In graphene, the situa-tion is different since its 2D character leads to a muchsmoother density of states except for the van Hove sin-gularity at high energies of the order of the hoppingenergy t2.8 eV. Nevertheless, graphene is an opensurface and hence is readily accessible by Raman spec-troscopy. In fact, it has played an important role becauseit allows the identification of the number of planes Fer-rari et al., 2006; Gupta et al., 2006; Graf et al., 2007;Malard et al., 2007; Pisana et al., 2007; Yan et al., 2007and the study of the optical phonon modes in graphene,particularly the ones in the center of the BZ with mo-mentum q0. Similar studies have been performed ingraphite ribbons Cancado et al., 2004.

We consider the effect of Dirac fermions on the opti-cal modes. If one treats the vector potential, electron-phonon coupling, Eqs. 150 and 208 up to second-order perturbation theory, its main effect is thepolarization of the electron system by creating electron-hole pairs. In the QED language, the creation ofelectron-hole pairs is called pair electron-antielectronproduction Castro Neto, 2007. Pair production isequivalent to a renormalization of the phonon propaga-tor by a self-energy that is proportional to the polariza-tion function of the Dirac fermions.

The renormalized phonon frequency is given byAndo, 2006a, 2007b; Lazzeri and Mauri, 2006; CastroNeto and Guinea, 2007; Saha et al., 2007

0 1 2 3 4 5 6 7ω / ωc

2

2.5

3

πσ(

ω)h

/e2

B = 12 T

ni = 0.001

0 1 2 3 4 5 6 7ω / ωc

2

2.5

πσ(

ω)h

/e2

B = 6 T

0 1 2 3 4 5 6 7ω / ωc

2

2.5

3B = 10 T

0.22 0.24 0.26 0.28

ω / B1/2

(eV/ T1/2

)

22.12.22.32.42.52.62.7

B = 12 TB = 10 TB = 6 T

ab

c

FIG. 34. Color online Frequency-dependent conductivity percone in units of e2 /h at T=10 K and ni=10−3, as afunction of the energy in units of c for different values ofthe magnetic field B. The vertical arrows in the upper leftpanel, labeled a, b, and c, show the positions of the transitionsbetween different Landau levels: E1,1−E−1,0, E2,1−E−1,0, and E1,1−E1,−1, respectively. The horizontalcontinuous lines show the value of the universal conductivity.The lower right panel shows the conductivity for different val-ues of magnetic field as a function of /B. Adapted fromPeres, Guinea, and Castro Neto, 2006a.



'0q 0 −22

a20 q,0 , 209

where 0 is the bare phonon frequency, and theelectron-phonon polarization function is given by

q, = s,s=±1

d2k22

fEsk + q − fEsk

0 − Esk + q + Esk + i(,

210

where Esq is the Dirac fermion dispersion s= +1 forthe upper band and s=−1 for the lower band, and fEis the Fermi-Dirac distribution function. For Ramanspectroscopy, the response of interest is at q=0, whereonly the interband processes such that ss=−1 that is,processes between the lower and upper cones contrib-ute. The electron-phonon polarization function can becalculated using the linearized Dirac fermion dispersion7 and the low-energy density of states 15,

0,0 =63

vF2

0

vF&

dEEf− E − fE

1

0 + 2E + i(−

1

− 2E + i( , 211

where we have introduced the cutoff momentum &1/a so that the integral converges in the ultraviolet.At zero temperature T=0 we have fE=−E andwe assume electron doping 0, so that f−E=1 forthe case of hole doping, 0, obtained by electron-holesymmetry. The integration in Eq. 211 gives

0,0 =63

vF2 vF& −

+0

4ln0/2 +

0/2 − + i0/2 − ,

212

where the cutoff-dependent term is a contribution com-ing from the occupied states in the lower band andhence is independent of the chemical potential value.This contribution can be fully incorporated into the barevalue of 0 in Eq. 209. Hence the relative shift in thephonon frequency can be written as

0

0 −

4−

0+ ln0/2 +

0/2 − + i0/2 − ,

213

where

=363

2

8Ma20214

is the dimensionless electron-phonon coupling. Notethat Eq. 213 has a real and an imaginary part. The realpart represents the actual shift in frequency, while theimaginary part gives the damping of the phonon modedue to pair production see Fig. 35. There is a clear

change in behavior depending on whether is larger orsmaller than 0 /2. For 0 /2, there is a decrease inthe phonon frequency implying that the lattice is soften-ing, while for 0 /2, the lattice hardens. The interpre-tation for this effect is also given in Fig. 35. On the onehand, if the frequency of the phonon is larger than twicethe chemical potential, real electron-hole pairs are pro-duced, leading to stronger screening of the electron-ioninteraction, and hence to a softer phonon mode. At thesame time, the phonons become damped and decay. Onthe other hand, if the frequency of the phonon is smallerthan twice the chemical potential, the production ofelectron-hole pairs is halted by the Pauli principle andonly virtual excitations can be generated, leading to po-larization and lattice hardening. In this case, there is nodamping and the phonon is long lived. This result hasbeen observed experimentally by Raman spectroscopyPisana et al., 2007; Yan et al., 2007. Electron-phononcoupling has also been investigated theoretically for afinite magnetic field Ando, 2007a; Goerbig et al., 2007.In this case, resonant coupling occurs due to the largedegeneracy of the Landau levels, and different Ramantransitions are expected as compared with the zero-fieldcase. The coupling of electrons to flexural modes on afree-standing graphene sheet was discussed by Marianiand von Oppen 2008.

0 0.5 1 1.5 2ΜΩ0

-1.5

-1

-0.5

0

0.5

∆Ω0

Λ

Ω0

(b)

(a)

FIG. 35. Kohn anomaly in graphene. Top: the continuous lineis the relative phonon frequency shift as a function of /0,and the dashed line is the damping of the phonon due toelectron-hole pair creation. Bottom: a electron-hole processthat leads to phonon softening 02, and b electron-holeprocess that leads to phonon hardening 02.



B. Electron-electron interactions

Of all disciplines of condensed-matter physics, thestudy of electron-electron interactions is probably themost complex since it involves understanding the behav-ior of a macroscopic number of variables. Hence, theproblem of interacting systems is a field in constant mo-tion and we shall not try to give here a comprehensivesurvey of the problem for graphene. Instead, we focuson a small number of topics that are of current discus-sion in the literature.

Since graphene is a truly 2D system, it is informativeto compare it with the more standard 2DEG that hasbeen studied extensively in the past 25 years since thedevelopment of heterostructures and the discovery ofthe quantum Hall effect for a review, see Stone 1992.At the simplest level, metallic systems have two mainkinds of excitations: electron-hole pairs and collectivemodes such as plasmons.

Electron-hole pairs are incoherent excitations of theFermi sea and a direct result of Pauli’s exclusion prin-ciple: an electron inside the Fermi sea with momentum kis excited outside the Fermi sea to a new state with mo-mentum k+q, leaving a hole behind. The energy associ-ated with such an excitation is =k+q−k, and for statesclose to the Fermi surface kkF their energy scaleslinearly with the excitation momentum qvFq. In asystem with nonrelativistic dispersion such as normalmetals and semiconductors, the electron-hole continuumis made out of intraband transitions only and exists evenat zero energy since it is always possible to produceelectron-hole pairs with arbitrarily low energy close tothe Fermi surface, as shown in Fig. 36a. Besides that,the 2DEG can also sustain collective excitations such asplasmons that have dispersion plasmonq q, and existoutside the electron-hole continuum at sufficiently longwavelengths Shung, 1986a.

In systems with relativisticlike dispersion, such asgraphene, these excitations change considerably, espe-cially when the Fermi energy is at the Dirac point. Inthis case, the Fermi surface shrinks to a point and henceintraband excitations disappear and only interband tran-sitions between the lower and upper cones can exist seeFig. 36b. Therefore, neutral graphene has no electron-hole excitations at low energy, instead each electron-hole pair costs energy and hence the electron hole occu-pies the upper part of the energy versus momentumdiagram. In this case, plasmons are suppressed and nocoherent collective excitations can exist. If the chemicalpotential is moved away from the Dirac point, then in-traband excitations are restored and the electron-holecontinuum of graphene shares features of the 2DEGand undoped graphene. The full electron-hole con-tinuum of doped graphene is shown in Fig. 36c, and inthis case plasmon modes are allowed. As the chemicalpotential is moved away from the Dirac point, grapheneresembles more and more the 2DEG.

These features in the elementary excitations ofgraphene reflect its screening properties as well. In fact,the polarization and dielectric functions of undoped

graphene are different from the ones of the 2DEGLindhard function. In the random-phase approxima-tion RPA, the polarization function can be calculatedanalytically Shung, 1986a; González et al., 1993a, 1994,

)q, =q2

4vF2q2 − 2

, 215

and hence, for vFq, the polarization function isimaginary, indicating the damping of electron-hole pairs.Note that the static polarization function =0 vanisheslinearly with q, indicating the lack of screening in thesystem. This polarization function has also been calcu-lated in the presence of a finite chemical potentialShung, 1986a, 1986b; Ando, 2006b; Wunsch et al., 2006;Hwang and Das Sarma, 2007.

Undoped, clean graphene is a semimetal, with a van-ishing density of states at the Fermi level. As a result,the linear Thomas-Fermi screening length diverges, andthe long-range Coulomb interaction is not screened. Atfinite electron density n, the Thomas-Fermi screeninglength reads

TF 1

4

1

kF=

1

4

1n

, 216

where

=e2

0vF217

is the dimensionless coupling constant in the problemthe analog of Eq. 143 in the Coulomb impurity prob-lem. Going beyond the linear Thomas-Fermi regime, it

FIG. 36. Color online Electron-hole continuum and collec-tive modes of a a 2DEG, b undoped graphene, and cdoped graphene.



has been shown that the Coulomb law is modifiedKatsnelson, 2006a; Fogler, Novikov, and Shklovskii,2007; Zhang and Fogler, 2008.

The Dirac Hamiltonian in the presence of interactionscan be written as

H − ivF d2r†r · r

+e2

20 d2rd2r

1r − r

rr , 218

where

r = †rr 219

is the electronic density. One can observe that the Cou-lomb interaction, unlike in QED, is assumed to be in-stantaneous since vF /c1/300 and hence retardation ef-fects are very small. Moreover, photons propagate in 3Dspace whereas electrons are confined to the 2Dgraphene sheet. Hence, the Coulomb interaction breaksthe Lorentz invariance of the problem and makes themany-body situation different from the one in QEDBaym and Chin, 1976. Furthermore, the problem de-pends on two parameters: vF and e2 /0. Under a dimen-sional scaling, r→r , t→t ,→−1, both parametersremain invariant. In RG language, the Coulomb interac-tion is a marginal variable, whose strength relative to thekinetic energy does not change upon a change in scale. Ifthe units are chosen such that vF is dimensionless, thevalue of e2 /0 will also be rendered dimensionless. Thisis the case in theories considered renormalizable inquantum field theory.

The Fermi velocity in graphene is comparable to thatin half-filled metals. In solids with lattice constant a, thetotal kinetic energy per site 1/ma2, where m is the baremass of the electron, is of the same order of magnitudeas the electrostatic energy e2 /0a. The Fermi velocity forfillings of the order of unity is vF1/ma. Hence,e2 /0vF1. This estimate is also valid in graphene.Hence, unlike in QED, where QED=1/137, the cou-pling constant in graphene is 1.

Despite the fact that the coupling constant is of theorder of unity, a perturbative RG analysis can be ap-plied. RG techniques allow us to identify stable fixedpoints of the model, which may be attractive over abroader range than the one where a perturbative treat-ment can be rigorously justified. Alternatively, an RGscheme can be reformulated as the process of piecewiseintegration of high-energy excitations Shankar, 1994.This procedure leads to changes in the effective low-energy couplings. The scheme is valid if the energy ofthe renormalized modes is much larger than the scales ofinterest.

The Hartree-Fock correction due to Coulomb interac-tions between electrons given by Fig. 37 gives a loga-rithmic correction to the electron self-energy Gonzálezet al., 1994,

%HFk =

4k ln&

k , 220

where & is a momentum cutoff which sets the range ofvalidity of the Dirac equation. This result remains trueeven to higher order in perturbation theory Mish-chenko, 2007 and is also obtained in large N expansionsRosenstein et al., 1989, 1991; Son, 2007 where N is thenumber of flavors of Dirac fermions, with the onlymodification the prefactor in Eq. 220. This result im-plies that the Fermi velocity is renormalized towardhigher values. As a consequence, the density of statesnear the Dirac energy is reduced, in agreement with thegeneral trend of repulsive interactions to induce or in-crease gaps.

This result can be understood from the RG point ofview by studying the effect of reducing the cutoff from &to &−d& and its effect on the effective coupling. It canbe shown that obeys González et al., 1994

&

&= −

2

4. 221

Therefore, the Coulomb interaction becomes marginallyirrelevant. These features are confirmed by a full relativ-istic calculation, although the Fermi velocity cannot sur-pass the velocity of light González et al., 1994. Thisresult indicates that strongly correlated electronicphases, such as ferromagnetism Peres et al., 2005 andWigner crystals Dahal et al., 2006, are suppressed inclean graphene.

A calculation of higher-order self-energy terms leadsto a wave-function renormalization, and to a finite qua-siparticle lifetime, which grows linearly with quasiparti-cle energy González et al., 1994, 1996. The wave-function renormalization implies that the quasiparticleweight tends to zero as its energy is reduced. A strong-coupling expansion is also possible, assuming that thenumber of electronic flavors justifies an RPA expansion,keeping only electron-hole bubble diagrams Gonzálezet al., 1999. This analysis confirms that the Coulombinteraction is renormalized toward lower values.

The enhancement in the Fermi velocities leads to awidening of the electronic spectrum. This is consistentwith measurements of the gaps in narrow single-wallnanotubes, which show deviations from the scaling withR−1, where R is the radius, expected from the Diracequation Kane and Mele, 2004. The linear dependenceof the inverse quasiparticle lifetime with energy is con-sistent with photoemission experiments in graphite, forenergies larger with respect to the interlayer interactionsXu et al., 1996; Zhou, Gweon, and Lanzara, 2006; Zhou,Gweon, et al., 2006; Bostwick, Ohta, Seyller, et al., 2007;

k k+q

vq

FIG. 37. Color online Hartree-Fock self-energy diagram thatleads to a logarithmic renormalization of the Fermi velocity.



Sugawara et al., 2007. Note that in graphite, band-structure effects modify the lifetimes at low energiesSpataru et al., 2001. The vanishing of the quasiparticlepeak at low energies can lead to an energy-dependentrenormalization of the interlayer hopping Vozmedianoet al., 2002, 2003. Other thermodynamic properties ofundoped and doped graphene can also be calculatedBarlas et al., 2007; Vafek, 2007.

Nonperturbative calculations of the long-range inter-action effects in undoped graphene show that a transi-tion to a gapped phase is also possible, when the numberof electronic flavors is large Khveshchenko, 2001;Luk’yanchuk and Kopelevich, 2004; Khveshchenko andShively, 2006. The broken symmetry phase is similar tothe excitonic transition found in materials where it be-comes favorable to create electron-hole pairs that thenform bound excitons excitonic transition.

Undoped graphene cannot have well-defined plas-mons, as their energies fall within the electron-hole con-tinuum, and therefore have a significant Landau damp-ing. At finite temperatures, however, thermally excitedquasiparticles screen the Coulomb interaction, and anacoustic collective charge excitation can exist Vafek,2006.

Doped graphene shows a finite density of states at theFermi level, and the long-range Coulomb interaction isscreened. Accordingly, there are collective plasma inter-actions near q→0, which disperse as pq, since thesystem is 2D Shung, 1986a, 1986b; Campagnoli and To-satti, 1989. The fact that the electronic states are de-scribed by the massless Dirac equation implies that P n1/4, where n is the carrier density. The static dielectricconstant has a continuous derivative at 2kF, unlike in thecase of the 2D electron gas Ando, 2006b; Wunsch et al.,2006; Sarma et al., 2007. This fact is associated with thesuppressed backward scattering in graphene. The sim-plicity of the band structure of graphene allows analyti-cal calculation of the energy and momentum depen-dence of the dielectric function Wunsch et al., 2006;Sarma et al., 2007. The screening of the long-range Cou-lomb interaction implies that the low-energy quasiparti-cles show a quadratic dependence on energy with re-spect to the Fermi energy Hwang et al., 2007.

One way to probe the strength of the electron-electron interactions is via the electronic compressibility.Measurements of the compressibility using a single-electron transistor SET show little sign of interactionsin the system, being well fitted by the noninteractingresult that, contrary to the two-dimensional electron gas2DEG Eisenstein et al., 1994; Giuliani and Vignale,2005, is positively divergent Polini et al., 2007; Martinet al., 2008. Bilayer graphene, on the other hand, sharescharacteristics of the single layer and the 2DEG with anonmonotonic dependence of the compressibility on thecarrier density Kusminskiy et al., 2008. In fact, bilayergraphene very close to half filling has been predicted tobe unstable toward Wigner crystallization Dahal et al.,2007, just like the 2DEG. Furthermore, according toHartree-Fock calculations, clean bilayer graphene is un-stable toward ferromagnetism Nilsson et al., 2006a.

1. Screening in graphene stacks

The electron-electron interaction leads to the screen-ing of external potentials. In a doped stack, the chargetends to accumulate near the surfaces, and its distribu-tion is determined by the dielectric function of the stackin the out-of-plane direction. The same polarizability de-scribes the screening of an external field perpendicularto the layers, similar to the one induced by a gate inelectrically doped systems Novoselov et al., 2004. Theself-consistent distribution of charge in a biasedgraphene bilayer has been studied by McCann 2006.From the observed charge distribution and self-consistent calculations, an estimate of the band-structureparameters and their relation with the induced gap canbe obtained Castro, Novoselov, Morozov, et al., 2007.

In the absence of interlayer hopping, the polarizabilityof a set of stacks of 2D electron gases can be written asa sum of the screening by the individual layers. Usingthe accepted values for the effective masses and carrierdensities of graphene, this scheme gives a first approxi-mation to screening in graphite Visscher and Falicov,1971. The screening length in the out-of-plane directionis of about two graphene layers Morozov et al., 2005.This model is easily generalizable to a stack of semimet-als described by the 2D Dirac equation González et al.,2001. At half filling, the screening length in all direc-tions diverges, and the screening effects are weak.

Interlayer hopping modifies this picture significantly.The hopping leads to coherence Guinea, 2007. Theout-of-plane electronic dispersion is similar to that of aone-dimensional conductor. The out-of-plane polariz-ability of a multilayer contains intraband and interbandcontributions. The subbands in a system with Bernalstacking have a parabolic dispersion, when only thenearest-neighbor hopping terms are included. This bandstructure leads to an interband susceptibility describedby a sum of terms like those in Eq. 228, which divergesat half filling. In an infinite system, this divergence ismore pronounced at k= /c, that is, for a wave vectorequal to twice the distance between layers. This effectenhances Friedel-like oscillations in the charge distribu-tion in the out-of-plane direction, which can lead to thechanges in the sign of the charge in neighboring layersGuinea, 2007. Away from half filling, a graphene bi-layer behaves, from the point of view of screening, in away very similar to the 2DEG Wang and Chakraborty,2007b.

C. Short-range interactions

In this section, we discuss the effect of short-rangeCoulomb interactions on the physics of graphene. Thesimplest carbon system with a hexagonal shape is thebenzene molecule. The value of the Hubbard interactionamong electrons was, for this system, computed longago by Parr et al. 1950, yielding U=16.93 eV. For com-parison purposes, in polyacetylene the value for theHubbard interaction is U 10 eV and the hopping en-ergy is t2.5 eV Baeriswyl et al., 1986. These two ex-



amples show that the value of the on-site Coulomb in-teraction is fairly large for electrons. As a first guessfor graphene, one can take U to be of the same order asfor polyacethylene, with the hopping integral t 2.8 eV.Of course in pure graphene the electron-electron inter-action is not screened, since the density of states is zeroat the Dirac point, and one should work out the effect ofCoulomb interactions by considering the bare Coulombpotential. On the other hand, as shown before, defectsinduce a finite density of states at the Dirac point, whichcould lead to an effective screening of the long-rangeCoulomb interaction. We assume that the bare Coulombinteraction is screened in graphene and that Coulombinteractions are represented by the Hubbard interaction.This means that we must add to the Hamiltonian 5 aterm of the form

HU = URi

a↑†Ria↑Ria↓

†Ria↓Ri

+ b↑†Rib↑Rib↓

†Rib↓Ri . 222

The simplest question one can ask is whether this systemshows a tendency toward some kind of magnetic orderdriven by the interaction U. Within the simplestHartree-Fock approximation Peres et al., 2004, the in-stability line toward ferromagnetism is given by

UF =2

, 223

which is merely the Stoner criterion. Similar results areobtained in more sophisticated calculations Herbut,2006. At half filling the value for the density of states is0=0 and the critical value for UF is arbitrarily large.Therefore, we do not expect a ferromagnetic groundstate at the neutrality point of one electron per carbonatom. For other electronic densities, becomes finiteproducing a finite value for UF. We note that the inclu-sion of t does not change these findings, since the den-sity of states remains zero at the neutrality point.

The critical interaction strength toward an antiferro-magnetic ground state is given by Peres et al., 2004

UAF =2

1/N k,0

1/E+k, 224

where E+k is given in Eq. 6. This result gives a finiteUAF at the neutrality point Sorella and Tosatti, 1992;Martelo et al., 1997,

UAF0 = 2.23t . 225

Quantum Monte Carlo calculations Sorella and Tosatti,1992; Paiva et al., 2005, however, raise its value to

UAF0 5t . 226

Taking for graphene the same value for U as in poly-acetylene and t=2.8 eV, one obtains U / t 3.6, whichputs the system far from the transition toward an anti-ferromagnet ground state. Yet another possibility is thatthe system may be in a sort of a quantum spin liquid

Lee and Lee, 2005 as originally proposed by Pauling1972 in 1956 since mean-field calculations give a criti-cal value for U to be of the order of U / t 1.7. Whetherthis type of ground state really exists and whether quan-tum fluctuations push this value of U toward larger val-ues is not known.

1. Bilayer graphene: Exchange

The exchange interaction can be large in an unbiasedgraphene bilayer with a small concentration of carriers.It was shown that the exchange contribution to the elec-tronic energy of a single graphene layer does not lead toa ferromagnetic instability Peres et al., 2005. The rea-son for this is a significant contribution from the inter-band exchange, which is a term usually neglected indoped semiconductors. This contribution depends onthe overlap of the conduction and valence wave func-tions, and it is modified in a bilayer. The interband ex-change energy is reduced in a bilayer Nilsson et al.,2006b, and a positive contribution that depends loga-rithmically on the bandwidth in graphene is absent in itsbilayer. As a result, the exchange energy becomes nega-tive, and scales as n3/2, where n is the carrier density,similar to the 2DEG. The quadratic dispersion at lowenergies implies that the kinetic energy scales as n2,again as in the 2DEG. This expansion leads to

E = Ekin + Eexc vF

2n2

8t

−e2n3/2

270

. 227

Writing n↑= n+s /2, n↓= n−s /2, where s is the magne-tization, Eq. 227 predicts a second-order transition to aferromagnetic state for n=4e4t2 /813vF

40. Higher-ordercorrections to Eq. 227 lead to a first-order transition atslightly higher densities Nilsson et al., 2006b. For a ra-tio 1 /00.1, this analysis implies that a graphene bi-layer should be ferromagnetic for carrier densities suchthat n41010 cm−2.

A bilayer is also the unit cell of Bernal graphite, andthe exchange instability can also be studied in an infinitesystem. Taking into account nearest-neighbor interlayerhopping only, bulk graphite should also show an ex-change instability at low doping. In fact, there is experi-mental evidence for a ferromagnetic instability instrongly disordered graphite Esquinazi et al., 2002,2003; Kopelevich and Esquinazi, 2007.

The analysis described above can be extended to thebiased bilayer, where a gap separates the conduction andvalence bands Stauber et al., 2007. The analysis of thiscase is somewhat different, as the Fermi surface at lowdoping is a ring, and the exchange interaction canchange its bounds. The presence of a gap further re-duces the mixing of the valence and conduction bands,leading to an enhancement of the exchange instability.At all doping levels, where the Fermi surface is ringshaped, the biased bilayer is unstable toward ferromag-netism.



2. Bilayer graphene: Short-range interactions

The band structure of a graphene bilayer, at half fill-ing, leads to logarithmic divergences in different re-sponse functions at q=0. The two parabolic bands thatare tangent at k=0 lead to a susceptibility given by that

q , q &

d2k1

− vF2/tk2

ln &

t/vF2 ,

228

where &t2 /vF2 is a high momentum cutoff. These

logarithmic divergences are similar to the ones thatshow up when the Fermi surface of a 2D metal is near asaddle point in the dispersion relation González et al.,1996. A full treatment of these divergences requires aRG approach Shankar, 1994. Within a simpler mean-field treatment, however, it is easy to note that the diver-gence of the bilayer susceptibility gives rise to an insta-bility toward an antiferromagnetic phase, where thecarbon atoms that are not connected to the neighboringlayers acquire a finite magnetization, while the magneti-zation of atoms with neighbors in the contiguous layersremains zero. A scheme of the expected ordered state isshown in Fig. 38.

D. Interactions in high magnetic fields

The formation of Landau levels enhances the effect ofinteractions due to the quenching of the kinetic energy.This effect is most pronounced at low fillings, when onlythe lowest levels are occupied. New phases may appearat low temperatures. We consider here phases differentfrom the fractional quantum Hall effect, which has notbeen observed in graphene so far. The existence of newphases can be inferred from the splitting of the valley orspin degeneracy of the Landau levels, which can be ob-served in spectroscopy measurements Sadowski et al.,2006; Henriksen, Tung, Jiang, et al., 2007, or in the ap-pearance of new quantum Hall plateaus Zhang et al.,2006; Abanin, Novoselov, Zeitler, et al., 2007; Giesbers etal., 2007; Goswami et al., 2007; Jiang, Zhang, Stormer, etal., 2007.

Interactions can lead to new phases when their effectovercomes that of disorder. An analysis of the competi-tion between disorder and interactions has been foundby Nomura and MacDonald 2007. The energy splittingof the different broken symmetry phases, in a clean sys-tem, is determined by lattice effects, so that it is reducedby factors of order a / lB, where a is a length of the orderof the lattice spacing and lB is the magnetic length Ali-

cea and Fisher, 2006; Goerbig et al., 2006, 2007; Wang etal., 2008. The combination of disorder and a magneticfield may also lift the degeneracy between the two val-leys, favoring valley-polarized phases Abanin, Lee, andLevitov, 2007.

In addition to phases with enhanced ferromagnetismor with broken valley symmetry, interactions at highmagnetic fields can lead to excitonic instabilities Gusy-nin et al., 2006 and Wigner crystal phases Zhang andJoglekar, 2007. When only the n=0 state is occupied,the Landau levels have all their weight in a given sub-lattice. Then, the breaking of valley degeneracy can beassociated with a charge-density wave, which opens agap Fuchs and Lederer, 2007. It is interesting to notethat in these phases new collective excitations are pos-sible Doretto and Morais Smith, 2007.

Interactions modify the edge states in the quantumHall regime. A novel phase can appear when the n=0 isthe last filled level. The Zeeman splitting shifts the elec-tronlike and holelike chiral states, which disperse in op-posite directions near the boundary of the sample. Theresulting level crossing between an electronlike levelwith spin antiparallel to the field, and a holelike levelwith spin parallel to the field, may lead to Luttinger-liquid features in the edge states Fertig and Brey, 2006;Abanin, Novoselov, Zeitler, et al., 2007.

VI. CONCLUSIONS

Graphene is a unique system in many ways. It is truly2D, has unusual electronic excitations described in termsof Dirac fermions that move in a curved space, is aninteresting mix of a semiconductor zero density ofstates and a metal gaplessness, and has properties ofsoft matter. The electrons in graphene seem to be almostinsensitive to disorder and electron-electron interactionsand have very long mean free paths. Hence, graphene’sproperties are different from what is found in usual met-als and semiconductors. Graphene has also a robust butflexible structure with unusual phonon modes that donot exist in ordinary 3D solids. In some sense, graphenebrings together issues in quantum gravity and particlephysics, and also from soft and hard condensed matter.Interestingly enough, these properties can be easilymodified with the application of electric and magneticfields, addition of layers, control of its geometry, andchemical doping. Moreover, graphene can be directlyand relatively easily probed by various scanning probetechniques from mesoscopic down to atomic scales, be-cause it is not buried inside a 3D structure. This makesgraphene one of the most versatile systems incondensed-matter research.

Besides the unusual basic properties, graphene hasthe potential for a large number of applications Geimand Novoselov, 2007, from chemical sensors Chen, Lin,Rooks, et al., 2007; Schedin et al., 2007 to transistorsNilsson et al., 2006b; Oostinga et al., 2007. Graphenecan be chemically and/or structurally modified in orderto change its functionality and henceforth its potentialapplications. Moreover, graphene can be easily obtained

FIG. 38. Color online Sketch of the expected magnetizationfor a graphene bilayer at half filling.



from graphite, a material that is abundant on the Earth’ssurface. This particular characteristic makes grapheneone of the most readily available materials for basic re-search since it frees economically challenged researchinstitutions in developing countries from the depen-dence of expensive sample-growing techniques.

Many of graphene’s properties are currently subject ofintense research and debate. Understanding the natureof the disorder and how it affects the transport proper-ties a problem of fundamental importance for applica-tions, the effect of phonons on electronic transport, thenature of electron-electron interactions, and how theymodify its physical properties are research areas that arestill in their infancy. In this review, we have only touchedthe surface.

Whereas many papers have been written on mono-layer graphene in the past few years, only a small frac-tion actually deal with multilayers. The majority of thetheoretical and experimental efforts have concentratedon the single layer, perhaps because of its simplicity andthe natural attraction that a one atom thick material,which can be produced by simple methods in almost anylaboratory, creates. Nevertheless, few-layer graphene isequally interesting and unusual with a technological po-tential, perhaps larger than the single layer. Indeed, thetheoretical understanding and experimental explorationof multilayers is far behind the single layer. This is afertile and open field of research for the future.

Finally, we have focused entirely on pure carbongraphene where the band structure is dominated by theDirac description. Nevertheless, chemical modificationof graphene can lead to entirely new physics. Dependingon the nature of chemical dopants and how they areintroduced into the graphene lattice adsorption, substi-tution, or intercalation, there can be many results.Small concentrations of adsorbed alkali metal can beused to change the chemical potential while adsorbedtransition elements can lead to strong hybridization ef-fects that affect the electronic structure. In fact, the in-troducion of d- and f-electron atoms in the graphenelattice may produce a significant enhancement of theelectron-electron interactions. Hence, it is easy to envi-sion a plethora of many-body effects that can be inducedby doping and have to be studied in the context of Diracelectrons: Kondo effect, ferromagnetism, antiferromag-netism, and charge- and spin-density waves. The studyof chemically induced many-body effects in graphenewould add a new chapter to the short but fascinatinghistory of this material. Only time will tell, but the po-tential for more amazement is lurking on the horizon.

ACKNOWLEDGMENTS

We have benefited immensely from discussions withmany colleagues and friends in the last few years, but wewould like to especially thank Boris Altshuler, Eva An-drei, Alexander Balatsky, Carlo Beenakker, Sankar DasSarma, Walt de Heer, Millie Dresselhaus, VladimirFalko, Andrea Ferrari, Herb Fertig, Eduardo Fradkin,Ernie Hill, Mihail Katsnelson, Eun-Ah Kim, Philip Kim,

Valery Kotov, Alessandra Lanzara, Leonid Levitov, Al-lan MacDonald, Sergey Morozov, Johan Nilsson, VitorPereira, Philip Phillips, Ramamurti Shankar, João Lopesdos Santos, Shan-Wen Tsai, Bruno Uchoa, and MariaVozmediano. N.M.R.P. acknowledges financial supportfrom POCI 2010 via project PTDC/FIS/64404/2006. F.G.was supported by MEC Spain Grant No. FIS2005-05478-C02-01 and EU Contract No. 12881 NEST.A.H.C.N was supported through NSF Grant No. DMR-0343790. K.S.N. and A.K.G. were supported by EPSRCUK and the Royal Society.

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