The electronic properties of graphene

REVIEWS OF MODERN PHYSICS, VOLUME 81, JANUARY–MARCH 2009 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 共Published 14 January 2009兲 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 controlled by 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 Hall effect. The electronic properties of graphene stacks are discussed and vary with stacking order and number of layers. Edge 共surface兲 states in graphene depend on the edge termination 共zigzag or armchair兲 and affect the physical properties of nanoribbons. Different types of disorder modify the Dirac equation leading to unusual spectroscopic and transport properties. The effects of electron-electron and electron-phonon interactions in single layer and multilayer graphene are also presented. DOI: 10.1103/RevModPhys.81.109 PACS number共s兲: 81.05.Uw, 73.20.⫺r, 03.65.Pm, 82.45.Mp CONTENTS I. Introduction II. Elementary Electronic Properties of Graphene A. Single layer: Tight-binding approach 1. Cyclotron mass 2. Density of states B. Dirac fermions 1. Chiral tunneling and Klein paradox 2. Confinement and Zitterbewegung C. Bilayer graphene: Tight-binding approach D. Epitaxial graphene E. Graphene stacks 1. Electronic structure of bulk graphite F. Surface states in graphene G. Surface states in graphene stacks H. The spectrum of graphene nanoribbons 1. Zigzag nanoribbons 2. Armchair nanoribbons I. Dirac fermions in a magnetic field J. The anomalous integer quantum Hall effect K. Tight-binding model in a magnetic field L. Landau levels in graphene stacks M. Diamagnetism N. Spin-orbit coupling III. Flexural Phonons, Elasticity, and Crumpling IV. Disorder in Graphene A. Ripples 0034-6861/2009/81共1兲/109共54兲 110 112 112 113 114 114 115 117 118 119 120 121 122 124 124 125 126 126 128 128 130 130 131 132 134 135 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 共FSBA兲 146 V. Many-Body Effects 148 A. Electron-phonon interactions B. Electron-electron interactions 1. Screening in graphene stacks C. Short-range interactions 2. Bilayer graphene: Short-range interactions D. Interactions in high magnetic fields 109 150 152 152 1. Bilayer graphene: Exchange VI. Conclusions 148 153 154 154 154 Acknowledgments 155 References 155 ©2009 The American Physical Society 110 Castro Neto et al.: The electronic properties of graphene FIG. 1. 共Color online兲 Graphene 共top left兲 is a honeycomb lattice of carbon atoms. Graphite 共top right兲 can be viewed as a stack of graphene layers. Carbon nanotubes are rolled-up cylinders of graphene 共bottom left兲. Fullerenes 共C60兲 are molecules consisting of wrapped graphene by the introduction of pentagons on the hexagonal lattice. From Castro Neto et al., 2006a. I. INTRODUCTION Carbon is the materia prima for life and the basis of all organic chemistry. Because of the flexibility of its bonding, carbon-based systems show an unlimited number of different structures with an equally large variety of physical properties. These physical properties are, in great part, the result of the dimensionality of these structures. Among systems with only carbon atoms, graphene—a two-dimensional 共2D兲 allotrope of carbon—plays an important role since it is the basis for the understanding of the electronic properties in other allotropes. Graphene is made out of carbon atoms arranged on a honeycomb structure made out of hexagons 共see Fig. 1兲, and can be thought of as composed of benzene rings stripped out from their hydrogen atoms 共Pauling, 1972兲. Fullerenes 共Andreoni, 2000兲 are molecules where carbon atoms are arranged spherically, and hence, from the physical point of view, are zerodimensional objects with discrete energy states. Fullerenes can be obtained from graphene with the introduction of pentagons 共that create positive curvature defects兲, and hence, fullerenes can be thought as wrapped-up graphene. Carbon nanotubes 共Saito et al., 1998; Charlier et al., 2007兲 are obtained by rolling graphene along a given direction and reconnecting the carbon bonds. Hence carbon nanotubes have only hexagons and can be thought of as one-dimensional 共1D兲 objects. Graphite, a three dimensional 共3D兲 allotrope of carbon, became widely known after the invention of the pencil in 1564 共Petroski, 1989兲, and its usefulness as an instrument for writing comes from the fact that graphite is made out of stacks of graphene layers that are weakly coupled by van der Waals forces. Hence, when one presses a pencil against a sheet of paper, one is actually producing graphene stacks and, somewhere among them, there could be individual graphene layers. Although graphene is the mother for all these different allotropes and has been presumably produced every time someone writes with a pencil, it was only isolated 440 years after its invention 共Novoselov et al., 2004兲. The reason is that, first, no one actually expected graphene to exist in the free state and, second, even with the benRev. Mod. Phys., Vol. 81, No. 1, January–March 2009 efit of hindsight, no experimental tools existed to search for one-atom-thick flakes among the pencil debris covering macroscopic areas 共Geim and MacDonald, 2007兲. Graphene was eventually spotted due to the subtle optical effect it creates on top of a chosen SiO2 substrate 共Novoselov et al., 2004兲 that allows its observation with an ordinary optical microscope 共Abergel et al., 2007; Blake et al., 2007; Casiraghi et al., 2007兲. Hence, graphene is relatively straightforward to make, but not so easy to find. The structural flexibility of graphene is reflected in its electronic properties. The sp2 hybridization between one s orbital and two p orbitals leads to a trigonal planar structure with a formation of a ␴ bond between carbon atoms that are separated by 1.42 Å. The ␴ band is responsible for the robustness of the lattice structure in all allotropes. Due to the Pauli principle, these bands have a filled shell and, hence, form a deep valence band. The unaffected p orbital, which is perpendicular to the planar structure, can bind covalently with neighboring carbon atoms, leading to the formation of a ␲ band. Since each p orbital has one extra electron, the ␲ band is half filled. Half-filled bands in transition elements have played an important role in the physics of strongly correlated systems since, due to their strong tight-binding character, the Coulomb energies are large, leading to strong collective effects, magnetism, and insulating behavior due to correlation gaps or Mottness 共Phillips, 2006兲. In fact, Linus Pauling proposed in the 1950s that, on the basis of the electronic properties of benzene, graphene should be a resonant valence bond 共RVB兲 structure 共Pauling, 1972兲. RVB states have become popular in the literature of transition-metal oxides, and particularly in studies of cuprate-oxide superconductors 共Maple, 1998兲. This point of view should be contrasted with contemporaneous band-structure studies of graphene 共Wallace, 1947兲 that found it to be a semimetal with unusual linearly dispersing electronic excitations called Dirac electrons. While most current experimental data in graphene support the band structure point of view, the role of electron-electron interactions in graphene is a subject of intense research. It was P. R. Wallace in 1946 who first wrote on the band structure of graphene and showed the unusual semimetallic behavior in this material 共Wallace, 1947兲. At that time, the thought of a purely 2D structure was not reality and Wallace’s studies of graphene served him as a starting point to study graphite, an important material for nuclear reactors in the post–World War II era. During the following years, the study of graphite culminated with the Slonczewski-Weiss-McClure 共SWM兲 band structure of graphite, which provided a description of the electronic properties in this material 共McClure, 1957; Slonczewski and Weiss, 1958兲 and was successful in describing the experimental data 共Boyle and Nozières 1958; McClure, 1958; Spry and Scherer, 1960; Soule et al., 1964; Williamson et al., 1965; Dillon et al., 1977兲. From 1957 to 1968, the assignment of the electron and hole states within the SWM model were opposite to Castro Neto et al.: The electronic properties of graphene what is accepted today. In 1968, Schroeder et al. 共Schroeder et al., 1968兲 established the currently accepted location of electron and hole pockets 共McClure, 1971兲. The SWM model has been revisited in recent years because of its inability to describe the van der Waals–like interactions between graphene planes, a problem that requires the understanding of many-body effects that go beyond the band-structure description 共Rydberg et al., 2003兲. These issues, however, do not arise in the context of a single graphene crystal but they show up when graphene layers are stacked on top of each other, as in the case, for instance, of the bilayer graphene. Stacking can change the electronic properties considerably and the layering structure can be used in order to control the electronic properties. One of the most interesting aspects of the graphene problem is that its low-energy excitations are massless, chiral, Dirac fermions. In neutral graphene, the chemical potential crosses exactly the Dirac point. This particular dispersion, that is only valid at low energies, mimics the physics of quantum electrodynamics 共QED兲 for massless fermions except for the fact that in graphene the Dirac fermions move with a speed vF, which is 300 times smaller than the speed of light c. Hence, many of the unusual properties of QED can show up in graphene but at much smaller speeds 共Castro Neto et al., 2006a; Katsnelson et al., 2006; Katsnelson and Novoselov, 2007兲. Dirac fermions behave in unusual ways when compared to ordinary electrons if subjected to magnetic fields, leading to new physical phenomena 共Gusynin and Sharapov, 2005; Peres, Guinea, and Castro Neto, 2006a兲 such as the anomalous integer quantum Hall effect 共IQHE兲 measured experimentally 共Novoselov, Geim, Morozov, et al., 2005a; Zhang et al., 2005兲. Besides being qualitatively different from the IQHE observed in Si and GaAlAs 共heterostructures兲 devices 共Stone, 1992兲, the IQHE in graphene can be observed at room temperature because of the large cyclotron energies for “relativistic” electrons 共Novoselov et al., 2007兲. In fact, the anomalous IQHE is the trademark of Dirac fermion behavior. Another interesting feature of Dirac fermions is their insensitivity to external electrostatic potentials due to the so-called Klein paradox, that is, the fact that Dirac fermions can be transmitted with probability 1 through a classically forbidden region 共Calogeracos and Dombey, 1999; Itzykson and Zuber, 2006兲. In fact, Dirac fermions behave in an unusual way in the presence of confining potentials, leading to the phenomenon of Zitterbewegung, or jittery motion of the wave function 共Itzykson and Zuber, 2006兲. In graphene, these electrostatic potentials can be easily generated by disorder. Since disorder is unavoidable in any material, there has been a great deal of interest in trying to understand how disorder affects the physics of electrons in graphene and its transport properties. In fact, under certain conditions, Dirac fermions are immune to localization effects observed in ordinary electrons 共Lee and Ramakrishnan, 1985兲 and it has been established experimentally that electrons can propagate without scattering over large Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 111 distances of the order of micrometers in graphene 共Novoselov et al., 2004兲. The sources of disorder in graphene are many and can vary from ordinary effects commonly found in semiconductors, such as ionized impurities in the Si substrate, to adatoms and various molecules adsorbed in the graphene surface, to more unusual defects such as ripples associated with the soft structure of graphene 共Meyer, Geim, Katsnelson, Novoselov, Booth, et al., 2007a兲. In fact, graphene is unique in the sense that it shares properties of soft membranes 共Nelson et al., 2004兲 and at the same time it behaves in a metallic way, so that the Dirac fermions propagate on a locally curved space. Here analogies with problems of quantum gravity become apparent 共Fauser et al., 2007兲. The softness of graphene is related with the fact that it has outof-plane vibrational modes 共phonons兲 that cannot be found in 3D solids. These flexural modes, responsible for the bending properties of graphene, also account for the lack of long range structural order in soft membranes leading to the phenomenon of crumpling 共Nelson et al., 2004兲. Nevertheless, the presence of a substrate or scaffolds that hold graphene in place can stabilize a certain degree of order in graphene but leaves behind the so-called ripples 共which can be viewed as frozen flexural modes兲. It was realized early on that graphene should also present unusual mesoscopic effects 共Peres, Castro Neto, and Guinea, 2006a; Katsnelson, 2007a兲. These effects have their origin in the boundary conditions required for the wave functions in mesoscopic samples with various types of edges graphene can have 共Nakada et al., 1996; Wakabayashi et al., 1999; Peres, Guinea, and Castro Neto, 2006a; Akhmerov and Beenakker, 2008兲. The most studied edges, zigzag and armchair, have drastically different electronic properties. Zigzag edges can sustain edge 共surface兲 states and resonances that are not present in the armchair case. Moreover, when coupled to conducting leads, the boundary conditions for a graphene ribbon strongly affect its conductance, and the chiral Dirac nature of fermions in graphene can be used for applications where one can control the valley flavor of the electrons besides its charge, the so-called valleytronics 共Rycerz et al., 2007兲. Furthermore, when superconducting contacts are attached to graphene, they lead to the development of supercurrent flow and Andreev processes characteristic of the superconducting proximity effect 共Heersche et al., 2007兲. The fact that Cooper pairs can propagate so well in graphene attests to the robust electronic coherence in this material. In fact, quantum interference phenomena such as weak localization, universal conductance fluctuations 共Morozov et al., 2006兲, and the Aharonov-Bohm effect in graphene rings have already been observed experimentally 共Recher et al., 2007; Russo, 2007兲. The ballistic electronic propagation in graphene can be used for field-effect devices such as p-n 共Cheianov and Fal’ko, 2006; Cheianov, Fal’ko, and Altshuler, 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, 2008兲 and p-n-p 共Ossipov et al., 2007兲 junctions, and as “neu- 112 Castro Neto et al.: The electronic properties of graphene trino” billiards 共Berry and Modragon, 1987; Miao et al., 2007兲. It has also been suggested that Coulomb interactions are considerably enhanced in smaller geometries, such as graphene quantum dots 共Milton Pereira et al., 2007兲, leading to unusual Coulomb blockade effects 共Geim and Novoselov, 2007兲 and perhaps to magnetic phenomena such as the Kondo effect. The transport properties of graphene allow for their use in a plethora of applications ranging from single molecule detection 共Schedin et al., 2007; Wehling et al., 2008兲 to spin injection 共Cho et al., 2007; Hill et al., 2007; Ohishi et al., 2007; Tombros et al., 2007兲. Because of its unusual structural and electronic flexibility, graphene can be tailored chemically and/or structurally in many different ways: deposition of metal atoms 共Calandra and Mauri, 2007; Uchoa et al., 2008兲 or molecules 共Schedin et al., 2007; Leenaerts et al., 2008; Wehling et al., 2008兲 on top; intercalation 关as done in graphite intercalated compounds 共Dresselhaus et al., 1983; Tanuma and Kamimura, 1985; Dresselhaus and Dresselhaus, 2002兲兴; incorporation of nitrogen and/or boron in its structure 共Martins et al., 2007; Peres, Klironomos, Tsai, et al., 2007兲 关in analogy with what has been done in nanotubes 共Stephan et al., 1994兲兴; and using different substrates that modify the electronic structure 共Calizo et al., 2007; Giovannetti et al., 2007; Varchon et al., 2007; Zhou et al., 2007; Das et al., 2008; Faugeras et al., 2008兲. The control of graphene properties can be extended in new directions allowing for the creation of graphene-based systems with magnetic and superconducting properties 共Uchoa and Castro Neto, 2007兲 that are unique in their 2D properties. Although the graphene field is still in its infancy, the scientific and technological possibilities of this new material seem to be unlimited. The understanding and control of this material’s properties can open doors for a new frontier in electronics. As the current status of the experiment and potential applications have recently been reviewed 共Geim and Novoselov, 2007兲, in this paper we concentrate on the theory and more technical aspects of electronic properties with this exciting new material. II. ELEMENTARY ELECTRONIC PROPERTIES OF GRAPHENE A. Single layer: Tight-binding approach Graphene is made out of carbon atoms arranged in hexagonal structure, as shown in Fig. 2. The structure can be seen as a triangular lattice with a basis of two atoms per unit cell. The lattice vectors can be written as a a a1 = 共3, 冑3兲, a2 = 共3,− 冑3兲, 共1兲 2 2 where a ⬇ 1.42 Å is the carbon-carbon distance. The reciprocal-lattice vectors are given by b1 = 2␲ 共1, 冑3兲, 3a b2 = 2␲ 共1,− 冑3兲. 3a Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 共2兲 A δ3 δ1 a1 δ2 ky B b1 K Γ M K’ kx a2 b2 FIG. 2. 共Color online兲 Honeycomb lattice and its Brillouin zone. Left: lattice structure of graphene, made out of two interpenetrating triangular lattices 共a1 and a2 are the lattice unit vectors, and ␦i, i = 1 , 2 , 3 are the nearest-neighbor vectors兲. Right: corresponding Brillouin zone. The Dirac cones are located at the K and K⬘ points. Of particular importance for the physics of graphene are the two points K and K⬘ at the corners of the graphene Brillouin zone 共BZ兲. These are named Dirac points for reasons that will become clear later. Their positions in momentum space are given by K= 冉 冊 2␲ 2␲ , , 3a 3冑3a K⬘ = 冉 冊 2␲ 2␲ . ,− 3a 3冑3a 共3兲 The three nearest-neighbor vectors in real space are given by a 2 a 2 ␦1 = 共1, 冑3兲 ␦2 = 共1,− 冑3兲 ␦3 = − a共1,0兲 共4兲 while the six second-nearest neighbors are located at ␦1⬘ = ± a1, ␦2⬘ = ± a2, ␦3⬘ = ± 共a2 − a1兲. The tight-binding Hamiltonian for electrons in graphene considering that electrons can hop to both nearest- and next-nearest-neighbor atoms has the form 共we use units such that ប = 1兲 H=−t 共a␴† ,ib␴,j + H.c.兲 兺 具i,j典,␴ − t⬘ 共a␴† ,ia␴,j + b␴† ,ib␴,j + H.c.兲, 兺 具具i,j典典,␴ 共5兲 where ai,␴ 共ai,† ␴兲 annihilates 共creates兲 an electron with spin ␴ 共␴ = ↑ , ↓ 兲 on site Ri on sublattice A 共an equivalent definition is used for sublattice B兲, t共⬇2.8 eV兲 is the nearest-neighbor hopping energy 共hopping between different sublattices兲, and t⬘ is the next nearest-neighbor hopping energy1 共hopping in the same sublattice兲. The energy bands derived from this Hamiltonian have the form 共Wallace, 1947兲 E±共k兲 = ± t冑3 + f共k兲 − t⬘f共k兲, 1 The value of t⬘ is not well known but ab initio calculations 共Reich et al., 2002兲 find 0.02t ⱗ t⬘ ⱗ 0.2t depending on the tightbinding parametrization. These calculations also include the effect of a third-nearest-neighbors hopping, which has a value of around 0.07 eV. A tight-binding fit to cyclotron resonance experiments 共Deacon et al., 2007兲 finds t⬘ ⬇ 0.1 eV. 113 Castro Neto et al.: The electronic properties of graphene FIG. 3. 共Color online兲 Electronic dispersion in the honeycomb lattice. Left: energy spectrum 共in units of t兲 for finite values of t and t⬘, with t = 2.7 eV and t⬘ = −0.2t. Right: zoom in of the energy bands close to one of the Dirac points. 冉冑 冊 冉 冊 f共k兲 = 2 cos共冑3kya兲 + 4 cos 3 3 kya cos kxa , 2 2 共6兲 where the plus sign applies to the upper 共␲*兲 and the minus sign the lower 共␲兲 band. It is clear from Eq. 共6兲 that the spectrum is symmetric around zero energy if t⬘ = 0. For finite values of t⬘, the electron-hole symmetry is broken and the ␲ and ␲* bands become asymmetric. In Fig. 3, we show the full band structure of graphene with both t and t⬘. In the same figure, we also show a zoom in of the band structure close to one of the Dirac points 共at the K or K⬘ point in the BZ兲. This dispersion can be obtained by expanding the full band structure, Eq. 共6兲, close to the K 共or K⬘兲 vector, Eq. 共3兲, as k = K + q, with 兩q 兩 Ⰶ 兩K兩 共Wallace, 1947兲, E±共q兲 ⬇ ± vF兩q兩 + O关共q/K兲2兴, 共7兲 where q is the momentum measured relatively to the Dirac points and vF is the Fermi velocity, given by vF = 3ta / 2, with a value vF ⯝ 1 ⫻ 106 m / s. This result was first obtained by Wallace 共1947兲. The most striking difference between this result and the usual case, ⑀共q兲 = q2 / 共2m兲, where m is the electron mass, is that the Fermi velocity in Eq. 共7兲 does not depend on the energy or momentum: in the usual case we have v = k / m = 冑2E / m and hence the velocity changes substantially with energy. The expansion of the spectrum around the Dirac point including t⬘ up to second order in q / K is given by 冉 冊 9t⬘a2 3ta2 ± sin共3␪q兲 兩q兩2 , E±共q兲 ⯝ 3t⬘ ± vF兩q兩 − 4 8 where 冉 冊 q ␪q = arctan x qy 共8兲 1. Cyclotron mass The energy dispersion 共7兲 resembles the energy of ultrarelativistic particles; these particles are quantum mechanically described by the massless Dirac equation 共see Sec. II.B for more on this analogy兲. An immediate consequence of this massless Dirac-like dispersion is a cyclotron mass that depends on the electronic density as its square root 共Novoselov, Geim, Morozov, et al., 2005; Zhang et al., 2005兲. The cyclotron mass is defined, within the semiclassical approximation 共Ashcroft and Mermin, 1976兲, as m* = 共9兲 冋 册 1 ⳵A共E兲 2␲ ⳵E , 共10兲 E=EF with A共E兲 the area in k space enclosed by the orbit and given by A共E兲 = ␲q共E兲2 = ␲ E2 vF2 . 共11兲 Using Eq. 共11兲 in Eq. 共10兲, one obtains m* = is the angle in momentum space. Hence, the presence of t⬘ shifts in energy the position of the Dirac point and breaks electron-hole symmetry. Note that up to order 共q / K兲2 the dispersion depends on the direction in momentum space and has a threefold symmetry. This is the so-called trigonal warping of the electronic spectrum 共Ando et al., 1998, Dresselhaus and Dresselhaus, 2002兲. Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 FIG. 4. 共Color online兲 Cyclotron mass of charge carriers in graphene as a function of their concentration n. Positive and negative n correspond to electrons and holes, respectively. Symbols are the experimental data extracted from the temperature dependence of the SdH oscillations; solid curves are the best fit by Eq. 共13兲. m0 is the free-electron mass. Adapted from Novoselov, Geim, Morozov, et al., 2005. EF vF2 = kF . vF 共12兲 The electronic density n is related to the Fermi momentum kF as kF2 / ␲ = n 共with contributions from the two Dirac points K and K⬘ and spin included兲, which leads to m* = 冑␲ 冑n. 共13兲 vF Fitting Eq. 共13兲 to the experimental data 共see Fig. 4兲 provides an estimation for the Fermi velocity and the 114 Castro Neto et al.: The electronic properties of graphene 0.2 5 4 ρ(ε) hopping parameter as vF ⬇ 106 ms−1 and t ⬇ 3 eV, respectively. Experimental observation of the 冑n dependence on the cyclotron mass provides evidence for the existence of massless Dirac quasiparticles in graphene 共Novoselov, Geim, Morozov, et al., 2005; Zhang et al., 2005; Deacon et al., 2007; Jiang, Henriksen, Tung, et al., 2007兲—the usual parabolic 共Schrödinger兲 dispersion implies a constant cyclotron mass. t’=0.2t 0.15 3 0.1 2 0.05 1 0 -4 -2 0 2 0 0 0.2 0.4 0.6 0.8 1 0.4 1 ρ(ε) 2. Density of states The density of states per unit cell, derived from Eq. 共5兲, is given in Fig. 5 for both t⬘ = 0 and t⬘ ⫽ 0, showing in both cases semimetallic behavior 共Wallace, 1947; Bena and Kivelson, 2005兲. For t⬘ = 0, it is possible to derive an analytical expression for the density of states per unit cell, which has the form 共Hobson and Nierenberg, 1953兲 ␳共E兲 = Z0 = Z1 = 冦 冦 冉 冑 冊 4 兩E兩 1 ␲ F , ␲2 t2 冑Z0 2 冉 冏 冏冊 冏冏 冏冏 冉 冏 冏冊 1+ E t 2 − Z1 , Z0 0 − 3t 艋 E 艋 − t ∨ t 艋 E 艋 3t, 4 E , t −t艋E艋t 关共E/t兲2 − 1兴2 , − 3t 艋 E 艋 − t ∨ t 艋 E 艋 3t, − 4 2Ac 兩E兩 , ␲ vF2 共15兲 where Ac is the unit cell area given by Ac = 3冑3a2 / 2. It is worth noting that the density of states for graphene is different from the density of states of carbon nanotubes 共Saito et al., 1992a, 1992b兲. The latter shows 1 / 冑E singularities due to the 1D nature of their electronic spectrum, which occurs due to the quantization of the momentum in the direction perpendicular to the tube axis. From this perspective, graphene nanoribbons, which also have momentum quantization perpendicular to the ribbon length, have properties similar to carbon nanotubes. Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 -2 0 ε /t 2 0 -0.8 -0.4 0 ε /t 0.4 0.8 FIG. 5. Density of states per unit cell as a function of energy 共in units of t兲 computed from the energy dispersion 共5兲, t⬘ = 0.2t 共top兲 and t⬘ = 0 共bottom兲. Also shown is a zoom-in of the density of states close to the neutrality point of one electron per site. For the case t⬘ = 0, the electron-hole nature of the spectrum is apparent and the density of states close to the neutrality point can be approximated by ␳共⑀兲 ⬀ 兩⑀兩. 关共E/t兲2 − 1兴2 , −t艋E艋t 4 where F共␲ / 2 , x兲 is the complete elliptic integral of the first kind. Close to the Dirac point, the dispersion is approximated by Eq. 共7兲 and the density of states per unit cell is given by 共with a degeneracy of 4 included兲 ␳共E兲 = 0.1 0.2 E , t 2 0.2 0.6 0.4 4 E 1+ t 0.3 t’=0 0.8 冧 冧 共14兲 B. Dirac fermions We consider the Hamiltonian 共5兲 with t⬘ = 0 and the Fourier transform of the electron operators, an = 1 e 冑N c 兺 k −ik·Rn a共k兲, 共16兲 where Nc is the number of unit cells. Using this transformation, we write the field an as a sum of two terms, coming from expanding the Fourier sum around K⬘ and K. This produces an approximation for the representation 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 , 共17兲 115 Castro Neto et al.: The electronic properties of graphene where the index i = 1 共i = 2兲 refers to the K 共K⬘兲 point. These new fields, ai,n and bi,n, are assumed to vary slowly over the unit cell. The procedure for deriving a theory that is valid close to the Dirac point consists in using this representation in the tight- H⯝−t 冕 dxdy⌿̂†1共r兲 + ⌿̂†2共r兲 = − ivF 冕 冋冉 冋冉 0 − 3a共1 + i冑3兲/4 0 − 3a共1 − i冑3兲/4 3a共1 − i冑3兲/4 3a共1 + i冑3兲/4 0 0 冊 冉 ⳵x + ⳵x + 3a共− i − 冑3兲/4 0 − 3a共i − 冑3兲/4 − 3a共− i − 冑3兲/4 0 3a共i − 冑3兲/4 0 冊册 冊册 ⳵y ⌿̂1共r兲 ⳵y ⌿̂2共r兲 dxdy关⌿̂†1共r兲␴ · ⵜ⌿̂1共r兲 + ⌿̂†2共r兲␴* · ⵜ⌿̂2共r兲兴, − ivF␴ · ⵜ␺共r兲 = E␺共r兲. 共19兲 The wave function, in momentum space, for the momentum around K has the form 1 冑2 冉 冊 e−i␪k/2 共20兲 ±ei␪k/2 for HK = vF␴ · k, where the ⫾ signs correspond to the eigenenergies E = ± vFk, that is, for the ␲* and ␲ bands, respectively, and ␪k is given by Eq. 共9兲. The wave function for the momentum around K⬘ has the form ␺±,K⬘共k兲 = 冊 冉 0 with Pauli matrices ␴ = 共␴x , ␴y兲, ␴* = 共␴x , −␴y兲, and ⌿̂†i = 共a†i , b†i 兲 共i = 1 , 2兲. It is clear that the effective Hamiltonian 共18兲 is made of two copies of the massless Diraclike Hamiltonian, one holding for p around K and the other for p around K⬘. Note that, in first quantized language, the two-component electron wave function ␺共r兲, close to the K point, obeys the 2D Dirac equation, ␺±,K共k兲 = binding Hamiltonian and expanding the operators up to a linear order in ␦. In the derivation, one uses the fact that 兺␦e±iK·␦ = 兺␦e±iK⬘·␦ = 0. After some straightforward algebra, we arrive at 共Semenoff, 1984兲 冑 冉 1 ei␪k/2 −i␪ /2 2 ±e k 冊 共21兲 for HK⬘ = vF␴* · k. Note that the wave functions at K and K⬘ are related by time-reversal symmetry: if we set the origin of coordinates in momentum space in the M point of the BZ 共see Fig. 2兲, time reversal becomes equivalent to a reflection along the kx axis, that is, 共kx , ky兲 → 共kx , −ky兲. Also note that if the phase ␪ is rotated by 2␲, the wave function changes sign indicating a phase of ␲ 共in the literature this is commonly called a Berry’s phase兲. This change of phase by ␲ under rotation is characteristic of spinors. In fact, the wave function is a twocomponent spinor. A relevant quantity used to characterize the eigenfunctions is their helicity defined as the projection of the momentum operator along the 共pseudo兲spin direction. The quantum-mechanical operator for the helicity has the form Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 共18兲 1 p ĥ = ␴ · . 2 兩p兩 共22兲 It is clear from the definition of ĥ that the states ␺K共r兲 and ␺K⬘共r兲 are also eigenstates of ĥ, ĥ␺K共r兲 = ± 21 ␺K共r兲, 共23兲 and an equivalent equation for ␺K⬘共r兲 with inverted sign. Therefore, electrons 共holes兲 have a positive 共negative兲 helicity. Equation 共23兲 implies that ␴ has its two eigenvalues either in the direction of 共⇑兲 or against 共⇓兲 the momentum p. This property says that the states of the system close to the Dirac point have well defined chirality or helicity. Note that chirality is not defined in regard to the real spin of the electron 共that has not yet appeared in the problem兲 but to a pseudospin variable associated with the two components of the wave function. The helicity values are good quantum numbers as long as the Hamiltonian 共18兲 is valid. Therefore, the existence of helicity quantum numbers holds only as an asymptotic property, which is well defined close to the Dirac points K and K⬘. Either at larger energies or due to the presence of a finite t⬘, the helicity stops being a good quantum number. 1. Chiral tunneling and Klein paradox In this section, we address the scattering of chiral electrons in two dimensions by a square barrier 共Katsnelson et al., 2006; Katsnelson, 2007b兲. The one-dimensional scattering of chiral electrons was discussed earlier in the context on nanotubes 共Ando et al., 1998; McEuen et al., 1999兲. We start by noting that by a gauge transformation the wave function 共20兲 can be written as 116 Castro Neto et al.: The electronic properties of graphene Energy ␺I共r兲 = D 1 冑2 冉 冊 1 se i␾ ei共kxx+kyy兲 + r 冑2 冉 1 se i共␲−␾兲 冊 ei共−kxx+kyy兲 , 共25兲 V0 E y with ␾ = arctan共ky / kx兲, kx = kF cos ␾, ky = kF sin ␾, and kF the Fermi momentum. In region II, we have x D I III II θ φ ␺II共r兲 = φ ␺K共k兲 = 冑2 冉 冊 ±ei␪k cos2 ␾ . 1 − cos2共Dqx兲sin2 ␾ 共30兲 In Fig. 7, we show the angular dependence of T共␾兲 for two different values of the potential V0; it is clear that there are several directions for which the transmission is Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 2 s ⬘e 1 i共␲−␪兲 冊 ei共−qxx+kyy兲 , 共27兲 t 冑2 冉 冊 1 sei␾ ei共kxx+kyy兲 , 共28兲 with s = sgn共E兲 and s⬘ = sgn共E − V0兲. The coefficients r, a, b, and t are determined from the continuity of the wave function, which implies that the wave function has to obey the conditions ␺I共x = 0 , y兲 = ␺II共x = 0 , y兲 and ␺II共x = D , y兲 = ␺III共x = D , y兲. Unlike the Schrödinger equation, we only need to match the wave function but not its derivative. The transmission through the barrier is obtained from T共␾兲 = tt* and has the form cos2 ␪ cos2 ␾ . 关cos共Dqx兲cos ␾ cos ␪兴2 + sin2共Dqx兲共1 − ss⬘ sin ␾ sin ␪兲2 This expression does not take into account a contribution from evanescent waves in region II, which is usually negligible, unless the chemical potential in region II is at the Dirac energy 共see Sec. IV.A兲. Note that T共␾兲 = T共−␾兲, and for values of Dqx satisfying the relation Dqx = n␲, with n an integer, the barrier becomes completely transparent since T共␾兲 = 1, independent of the value of ␾. Also, for normal incidence 共␾ → 0 and ␪ → 0兲 and any value of Dqx, one obtains T共0兲 = 1, and the barrier is again totally transparent. This result is a manifestation of the Klein paradox 共Calogeracos and Dombey, 1999; Itzykson and Zuber, 2006兲 and does not occur for nonrelativistic electrons. In this latter case and for normal incidence, the transmission is always smaller than 1. In the limit 兩V0 兩 Ⰷ 兩E兩, Eq. 共29兲 has the following asymptotic form: T共␾兲 ⯝ 冑 冉 b and finally in region III we have a transmitted wave only, 共24兲 We further assume that the scattering does not mix the momenta around K and K⬘ points. In Fig. 6, we depict the scattering process due to the square barrier of width D. The wave function in the different regions can be written in terms of incident and reflected waves. In region I, we have T共␾兲 = ei共qxx+kyy兲 + qx = 冑共V0 − E兲2/共vF2 兲 − k2y , ␺III共r兲 = . 2 s ⬘e i␪ with ␪ = arctan共ky / qx兲 and FIG. 6. 共Color online兲 Klein tunneling in graphene. Top: schematic of the scattering of Dirac electrons by a square potential. Bottom: definition of the angles ␾ and ␪ used in the scattering formalism in regions I, II, and III. 1 1 共26兲 x 1 冊 冑 冉 a 共29兲 1. Similar calculations were done for a graphene bilayer 共Katsnelson et al., 2006兲 with the absence of tunneling in the forward 共ky = 0兲 direction its most distinctive behavior. The simplest example of a potential barrier is a square potential discussed previously. When intervalley scattering and the lack of symmetry between sublattices are neglected, a potential barrier shows no reflection for electrons incident in the normal direction 共Katsnelson et al., 2006兲. Even when the barrier separates regions where the Fermi surface is electronlike on one side and holelike on the other, a normally incident electron continues propagating as a hole with 100% efficiency. This phenomenon is another manifestation of the chirality of the Dirac electrons within each valley, which prevents backscattering in general. The transmission and reflection probabilities of electrons at different angles depend on the potential profile along the barrier. A slowly varying barrier is more efficient in reflecting electrons at nonzero incident angles 共Cheianov and Fal’ko, 2006兲. Electrons moving through a barrier separating p- and n-doped graphene, a p-n junction, are transmitted as 117 Castro Neto et al.: The electronic properties of graphene 1 T(φ) 0.8 0.6 V0 = 200 meV 0.4 V0 = 285 meV 0.2 0 -90 -75 -60 -45 -30 -15 -75 -60 -45 -30 -15 0 15 30 45 60 75 90 0 15 angle φ 30 45 60 75 90 1 T(φ) 0.8 0.6 0.4 0.2 0 -90 FIG. 7. 共Color online兲 Angular behavior of T共␾兲 for two different values of V0: V0 = 200 meV, dashed line; V0 = 285 meV, solid line. The remaining parameters are D = 110 nm 共top兲, D = 50 nm 共bottom兲 E = 80 meV, kF = 2␲ / ␭, and ␭ = 50 nm. holes. The relation between the velocity and the momentum for a hole is the inverse of that for an electron. This implies that, if the momentum parallel to the barrier is conserved, the velocity of the quasiparticle is inverted. When the incident electrons emerge from a source, the transmitting holes are focused into an image of the source. This behavior is the same as that of photons moving in a medium with negative reflection index 共Cheianov, Fal’ko, and Altshuler, 2007兲. Similar effects can occur in graphene quantum dots, where the inner and outer regions contain electrons and holes, respectively 共Cserti, Palyi, and Peterfalvi, 2007兲. Note that the fact that barriers do not impede the transmission of normally incident electrons does not preclude the existence of sharp resonances, due to the confinement of electrons with a finite parallel momentum. This leads to the possibility of fabricating quantum dots with potential barriers 共Silvestrov and Efetov, 2007兲. Finally, at half filling, due to disorder graphene can be divided in electron and hole charge puddles 共Katsnelson et al., 2006; Martin et al., 2008兲. Transport is determined by the transmission across the p-n junctions between these puddles 共Cheianov, Fal’ko, Altshuler, et al., 2007; Shklovskii, 2007兲. There is much progress in the measurement of transport properties of graphene ribbons with additional top gates that play the role of tunable potential barriers 共Han et al., 2007; Huard et al., 2007; Lemme et al., 2007; Özyilmaz et al., 2007; Williams et al., 2007兲. A magnetic field and potential fluctuations break both the inversion symmetry of the lattice and time-reversal symmetry. The combination of these effects also breaks the symmetry between the two valleys. The transmission coefficient becomes valley dependent, and, in general, electrons from different valleys propagate along different paths. This opens the possibility of manipulating the valley index 共Tworzydlo et al., 2007兲 共valleytronics兲 in a way similar to the control of the spin in mesoscopic devices 共spintronics兲. For large magnetic fields, a p-n junction separates regions with different quantized Hall conRev. Mod. Phys., Vol. 81, No. 1, January–March 2009 ductivities. At the junction, chiral currents can flow at both edges 共Abanin and Levitov, 2007兲, inducing backscattering between the Hall currents at the edges of the sample. The scattering of electrons near the Dirac point by graphene-superconductor junctions differs from Andreev scattering process in normal metals 共Titov and Beenakker, 2006兲. When the distance between the Fermi energy and the Dirac energy is smaller than the superconducting gap, the superconducting interaction hybridizes quasiparticles from one band with quasiholes in the other. As in the case of scattering at a p-n junction, the trajectories of the incoming electron and reflected hole 共note that hole here is meant as in the BCS theory of superconductivity兲 are different from those in similar processes in metals with only one type of carrier 共Bhattacharjee and Sengupta, 2006; Maiti and Sengupta, 2007兲. 2. Confinement and Zitterbewegung Zitterbewegung, or jittery motion of the wave function of the Dirac problem, occurs when one tries to confine the Dirac electrons 共Itzykson and Zuber, 2006兲. Localization of a wave packet leads, due to the Heisenberg principle, to uncertainty in the momentum. For a Dirac particle with zero rest mass, uncertainty in the momentum translates into uncertainty in the energy of the particle as well 共this should be contrasted with the nonrelativistic case, where the position-momentum uncertainty relation is independent of the energy-time uncertainty relation兲. Thus, for an ultrarelativistic particle, a particlelike state can have holelike states in its time evolution. Consider, for instance, if one tries to construct a wave packet at some time t = 0, and assume, for simplicity, that this packet has a Gaussian shape of width w with momentum close to K, ␺0共r兲 = e−r 2/2w2 冑␲ w eiK·r␾ , 共31兲 where ␾ is spinor composed of positive energy states 关associated with ␺+,K of Eq. 共20兲兴. The eigenfunction of the Dirac equation can be written in terms of the solution 共20兲 as ␺共r,t兲 = 冕 d 2k 兺 ␣a,k␺a,K共k兲e−ia共k·r+vFkt兲 , 共2␲兲2 a=±1 共32兲 where ␣±,k are Fourier coefficients. We can rewrite Eq. 共31兲 in terms of Eq. 共32兲 by inverse Fourier transform and find that ␣±,k = 冑␲we−k 2w2/2 † ␺±,K 共k兲␾ . 共33兲 Note that the relative weight of positive energy states with respect to negative energy states 兩␣+ / ␣−兩, given by Eq. 共20兲 is 1, that is, there are as many positive energy states as negative energy states in a wave packet. Hence, these will cause the wave function to be delocalized at any time t ⫽ 0. Thus, a wave packet of electronlike states has holelike components, a result that puzzled many re- 118 Castro Neto et al.: The electronic properties of graphene EeV 1 0.8 0.6 0.4 0.2 0.2 Π  2
 k ya 3 FIG. 8. 共Color online兲 Energy spectrum 共in units of t兲 for a graphene ribbon 600a wide, as a function of the momentum k along the ribbon 共in units of 1 / 冑3a兲, in the presence of confining potential with V0 = 1 eV, ␭ = 180a. searchers in the early days of QED 共Itzykson and Zuber, 2006兲. Consider the tight-binding description 共Peres, Castro Neto, and Guinea, 2006b; Chen, Apalkov, and Chakraborty, 2007兲 of Sec. II.A when a potential Vi on site Ri is added to the problem, H e = 兺 V in i , FIG. 9. 共Color online兲 Lattice structure of bilayer graphene, its respective electronic hopping energies, and Brillouin zone. 共a兲 Lattice structure of the bilayer with the various hopping parameters according to the SWM model. The A sublattices are indicated by darker spheres. 共b兲 Brillouin zone. Adapted from Malard et al., 2007. The tight-binding Hamiltonian for this problem can be written as † Ht.b. = − ␥0 兺 共am,i, ␴bm,j,␴ + H.c.兲 具i,j典 m,␴ † − ␥1 兺 共a1,j, ␴a2,j,␴ + H.c.兲, 共34兲 j,␴ i where ni is the local electronic density. For simplicity, we assume that the confining potential is 1D, that is, that Vi vanishes in the bulk but becomes large at the edge of the sample. We assume a potential that decays exponentially away from the edges into the bulk with a penetration depth ␭. In Fig. 8, we show the electronic spectrum for a graphene ribbon of width L = 600a, in the presence of a confining potential, V共x兲 = V0关e−共x−L/2兲/␭ + e−共L/2−x兲/␭兴, 共35兲 where x is the direction of confinement and V0 is the strength of the potential. One can see that in the presence of the confining potential the electron-hole symmetry is broken and, for V0 ⬎ 0, the hole part of the spectrum is distorted. In particular, for k close to the Dirac point, we see that the hole dispersion is given by En,␴=−1共k兲 ⬇ −␥nk2 − ␨nk4, where n is a positive integer, and ␥n ⬍ 0 共␥n ⬎ 0兲 for n ⬍ N* 共n ⬎ N*兲. Hence, at n = N* the hole effective mass diverges 共␥N* = 0兲 and, by tuning the chemical potential ␮ via a back gate to the hole region of the spectrum 共␮ ⬍ 0兲 one should be able to observe an anomaly in the Shubnikov–de Haas 共SdH兲 oscillations. This is how Zitterbewegung could manifest itself in magnetotransport. C. Bilayer graphene: Tight-binding approach The tight-binding model developed for graphite can be easily extended to stacks with a finite number of graphene layers. The simplest generalization is a bilayer 共McCann and Fal’ko, 2006兲. A bilayer is interesting because the IQHE shows anomalies, although different from those observed in a single layer 共Novoselov et al., 2006兲, and also a gap can open between the conduction and valence band 共McCann and Fal’ko, 2006兲. The bilayer structure, with the AB stacking of 3D graphite, is shown in Fig. 9. Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 † † − ␥4 兺 共a1,j, ␴b2,j,␴ + a2,j,␴b1,j,␴ + H.c.兲 j,␴ † − ␥3 兺 共b1,j, ␴b2,j,␴ + H.c.兲, 共36兲 j,␴ where am,i,␴ 共bm,i␴兲 annihilates an electron with spin ␴, on sublattice A 共B兲, in plane m = 1 , 2, at site Ri. Here we use the graphite nomenclature for the hopping parameters: ␥0 = t is the in-plane hopping energy and ␥1 关␥1 = t⬜ ⬇ 0.4 eV in graphite 共Brandt et al., 1988; Dresselhaus and Dresselhaus, 2002兲兴 is the hopping energy between atom A1 and atom A2 共see Fig. 9兲, ␥4 关␥4 ⬇ 0.04 eV in graphite 共Brandt et al., 1988; Dresselhaus and Dresselhaus, 2002兲兴 is the hopping energy between atom A1 共A2兲 and atom B2 共B1兲, and ␥3 关␥3 ⬇ 0.3 eV in graphite 共Brandt et al., 1988; Dresselhaus and Dresselhaus, 2002兲兴 connects B1 and B2. In the continuum limit, by expanding the momentum close to the K point in the BZ, the Hamiltonian reads H = 兺 ⌿k† · HK · ⌿k , 共37兲 k where 共ignoring ␥4 for the time being兲 HK ⬅ 冢 −V v Fk 0 3␥3ak* v Fk * −V ␥1 ␥1 0 0 V v Fk 3␥3ak 0 v Fk * V 冣 , 共38兲 and k = kx + iky is a complex number; we have added V, which is here half the shift in electrochemical potential between the two layers 共this term will appear if a potential bias is applied between the layers兲, and ⌿k† = „b†1共k兲,a†1共k兲,a†2共k兲,b†2共k兲… is a four-component spinor. 共39兲 119 Castro Neto et al.: The electronic properties of graphene FIG. 10. 共Color online兲 Band structure of bilayer graphene of V = 0 and ␥3 = 0. If V = 0 and ␥3 , vFk Ⰶ ␥1, one can eliminate the highenergy states perturbatively and write an effective Hamiltonian, HK ⬅ 冢 0 vF2 共k*兲2 ␥1 + 3␥3ak vF2 k2 ␥1 + 3␥3ak* 0 冣 . 共40兲 The hopping ␥4 leads to a k-dependent coupling between the sublattices or a small renormalization of ␥1. The same role is played by the inequivalence between sublattices within a layer. For ␥3 = 0, Eq. 共40兲 gives two parabolic bands, ⑀k,± ⬇ ± vF2 k2 / t⬜, which touch at ⑀ = 0 共as shown in Fig. 10兲. The spectrum is electron-hole symmetric. There are two additional bands that start at ±t⬜. Within this approximation, the bilayer is metallic, with a constant density of states. The term ␥3 changes qualitatively the spectrum at low energies since it introduces a trigonal distortion, or warping, of the bands 关note that this trigonal distortion, unlike the one introduced by large momentum in Eq. 共8兲, occurs at low energies兴. The electron-hole symmetry is preserved but, instead of two bands touching at k = 0, we obtain three sets of Dirac-like linear bands. One Dirac point is at ⑀ = 0 and k = 0, while the three other Dirac points, also at ⑀ = 0, lie at three equivalent points with a finite momentum. The stability of points where bands touch can be understood using topological arguments 共Mañes et al., 2007兲. The winding number of a closed curve in the plane around a given point is an integer representing the total number of times that the curve travels counterclockwise around the point so that the wave function remains unaltered. The winding number of the point where the two parabolic bands come together for ␥3 = 0 has winding number +2. The trigonal warping term ␥3 splits it into a Dirac point at k = 0 and winding number −1, and three Dirac points at k ⫽ 0 and winding numbers +1. An in-plane magnetic field, or a small rotation of one layer with respect to the other, splits the ␥3 = 0 degeneracy into two Dirac points with winding number +1. The term V in Eq. 共38兲 breaks the equivalence of the two layers, or, alternatively, inversion symmetry. In this case, the dispersion relation becomes Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 FIG. 11. 共Color online兲 Band structure of bilayer graphene for V ⫽ 0 and ␥3 = 0. 2 2 4 /2 ± 冑4V2vF2 k2 + t2vF2 k2 + t⬜ = V2 + vF2 k2 + t⬜ ⑀±,k /4, 共41兲 giving rise to the dispersion shown in Fig. 11, and to the opening of a gap close to, but not directly at, the K point. For small momenta, and V Ⰶ t, the energy of the conduction band can be expanded, 2 V. ⑀k ⬇ V − 2VvF2 k2/t⬜ + vF4 k4/2t⬜ 共42兲 The dispersion for the valence band can be obtained by replacing ⑀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 hence can be measured experimentally 共McCann, 2006; McCann and Fal’ko, 2006; Castro, Novoselov, Morozov, et al., 2007兲. The ability to open a gap makes bilayer graphene interesting for technological applications. D. Epitaxial graphene It has been known for a long time that monolayers of graphene could be grown epitaxially on metal surfaces using catalytic decomposition of hydrocarbons or carbon oxide 共Shelton et al., 1974; Eizenberg and Blakely, 1979; Campagnoli and Tosatti, 1989; Oshima and Nagashima, 1997; Sinitsyna and Yaminsky, 2006兲. When such surfaces are heated, oxygen or hydrogen desorbs, and the carbon atoms form a graphene monolayer. The resulting graphene structures could reach sizes up to a micrometer with few defects, and they were characterized by different surface-science techniques and local scanning probes 共Himpsel et al., 1982兲. For example, graphene grown on ruthenium has zigzag edges and also ripples associated with a 共10⫻ 10兲 reconstruction 共Vázquez de Parga et al., 2008兲. 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 surface 共Bommel 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兲. The number of layers can be controlled by limiting time or temperature of the heating treatment. The quality and the number of layers in the samples depend on the SiC face used for their growth 共de Heer et al., 2007兲 共the carbon-terminated surface produces few layers but with a low mobility, whereas the silicon-terminated surface produces several layers but with higher mobility兲. Epitaxially grown multilayers exhibit SdH oscillations with 120 Castro Neto et al.: The electronic properties of graphene a Berry phase shift of ␲ 共Berger et al., 2006兲, which is the same as the phase shift for Dirac fermions observed in a single layer as well as for some subbands present in multilayer graphene and graphite 共Luk’yanchuk and Kopelevich, 2004兲. The carbon layer directly on top of the substrate is expected to be strongly bonded to it, and it shows no ␲ bands 共Varchon et al., 2007兲. The next layer shows a 共6冑3 ⫻ 6冑3兲 reconstruction due to the substrate, and has graphene properties. An alternate route to produce few layers of graphene is based on synthesis from nanodiamonds 共Affoune et al., 2001兲. Angle-resolved photoemission experiments 共ARPES兲 show that epitaxial graphene grown on SiC has linearly dispersing quasiparticles 共Dirac fermions兲 共Zhou, Gweon, et al., 2006; Bostwick, Ohta, Seyller, et al., 2007; Ohta et al., 2007兲, in agreement with the theoretical expectation. Nevertheless, these experiments show that the electronic properties can change locally in space, indicating a certain degree of inhomogeneity due to the growth method 共Zhou et al., 2007兲. Similar inhomogeneities due to disorder in the c-axis orientation of graphene planes are observed in graphite 共Zhou, Gweon, and Lanzara, 2006兲. Moreover, graphene grown this way is heavily doped due to the charge transfer from the substrate to the graphene layer 共with the chemical potential well above the Dirac point兲 and therefore all samples have strong metallic character with large electronic mobilities 共Berger et al., 2006; de Heer et al., 2007兲. There is also evidence for strong interaction between a substrate and the graphene layer leading to the appearance of gaps at the Dirac point 共Zhou et al., 2007兲. Indeed, gaps can be generated by the breaking of the sublattice symmetry and, as in the case of other carbon-based systems such as polyacethylene 共Su et al., 1979, 1980兲, it can lead to solitonlike excitations 共Jackiw and Rebbi 1976; Hou et al., 2007兲. Multilayer graphene grown on SiC have also been studied with ARPES 共Bostwick, Ohta, McChesney, et al., 2007; Ohta et al., 2007兲 and the results seem to agree quite well with band-structure calculations 共Mattausch and Pankratov, 2007兲. Spectroscopy measurements also show the transitions associated with Landau levels 共Sadowski et al., 2006兲 and weak-localization effects at low magnetic fields, also expected for Dirac fermions 共Wu et al., 2007兲. Local probes reveal a rich structure of terraces 共Mallet et al., 2007兲 and interference patterns due to defects at or below the graphene layers 共Rutter et al., 2007兲. E. Graphene stacks In stacks with more than one graphene layer, two consecutive layers are normally oriented in such a way that the atoms in one of the two sublattices An of the honeycomb structure of one layer are directly above one-half of the atoms in the neighboring layer, sublattice An±1. The second set of atoms in one layer sits on top of the 共empty兲 center of a hexagon in the other layer. The shortest distance between carbon atoms in different layers is dAnAn±1 = c = 3.4 Å. The next distance is Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 FIG. 12. 共Color online兲 Sketch of the three inequivalent orientations of graphene layers with respect to each other. dAnBn±1 = 冑c2 + a2. This is the most common arrangement of nearest-neighbor layers observed in nature, although a stacking order in which all atoms in one layer occupy positions directly above the atoms in the neighboring layers 共hexagonal stacking兲 has been considered theoretically 共Charlier et al., 1991兲 and appears in graphite intercalated compounds 共Dresselhaus and Dresselhaus, 2002兲. The relative position of two neighboring layers allows for two different orientations of the third layer. If we label the positions of the first two atoms as 1 and 2, the third layer can be of type 1, leading to the sequence 121, or it can fill a third position different from 1 and 2 共see Fig. 12兲, labeled 3. There are no more inequivalent positions where a new layer can be placed, so that thicker stacks can be described in terms of these three orientations. In the most common version of bulk graphite, the stacking order is 1212… 共Bernal stacking兲. Regions with the stacking 123123… 共rhombohedral stacking兲 have also been observed in different types of graphite 共Bacon, 1950; Gasparoux, 1967兲. Finally, samples with no discernible stacking order 共turbostratic graphite兲 are also commonly reported. Beyond two layers, the stack ordering can be arbitrarily complex. Simple analytical expressions for the electronic bands can be obtained for perfect Bernal 共1212…兲 and rhombohedral 共123123…兲 stacking 共Guinea et al., 2006; Partoens and Peeters, 2006兲. Even if we consider one interlayer hopping t⬜ = ␥1, the two stacking orders show different band structures near ⑀ = 0. A Bernal stack with N layers, N even, has N / 2 electronlike and N / 2 holelike parabolic subbands touching at ⑀ = 0. When N is odd, an additional subband with linear 共Dirac兲 dispersion emerges. Rhombohedral systems have only two subbands that touch at ⑀ = 0. These subbands disperse as kN, and become surface states localized at the top and bottom layers when N → ⬁. In this limit, the remaining 2N − 2 subbands of a rhombohedral stack become Diraclike, with the same Fermi velocity as a single graphene layer. The subband structure of a trilayer with the Bernal stacking includes two touching parabolic bands, and one with Dirac dispersion, combining the features of bilayer and monolayer graphene. The low-energy bands have different weights on the two sublattices of each graphene layer. The states at a Castro Neto et al.: The electronic properties of graphene site directly coupled to the neighboring planes are pushed to energies ⑀ ⬇ ± t⬜. The bands near ⑀ = 0 are localized mostly at the sites without neighbors in the next layers. For the Bernal stacking, this feature implies that the density of states at ⑀ = 0 at sites without nearest neighbors in the contiguous layers is finite, while it vanishes linearly at the other sites. In stacks with rhombohedral stacking, all sites have one neighbor in another plane, and the density of states vanishes at ⑀ = 0 共Guinea et al., 2006兲. This result is consistent with the well known fact that only one of the two sublattices at a graphite surface can be resolved by scanning tunneling microscopy 共STM兲 共Tománek et al., 1987兲. As in the case of a bilayer, an inhomogeneous charge distribution can change the electrostatic potential in the different layers. For more than two layers, this breaking of the equivalence between layers can take place even in the absence of an applied electric field. It is interesting to note that a gap can open in a stack with Bernal ordering and four layers if the electronic charge at the two surface layers is different from that at the two inner ones. Systems with a higher number of layers do not show a gap, even in the presence of charge inhomogeneity. Four representative examples are shown in Fig. 13. The band structure analyzed here will be modified by the inclusion of the trigonal warping term, ␥3. Experimental studies of graphene stacks have showed that, with an increasing number of layers, the system becomes increasingly metallic 共concentration of charge carriers at zero energy gradually increases兲, and there appear several types of electronlike and holelike carries 共Novoselov et al., 2004; Morozov et al., 2005兲. An inhomogeneous charge distribution between layers becomes very important in this case, leading to 2D electron and hole systems that occupy only a few graphene layers near the surface, and can completely dominate transport properties of graphene stacks 共Morozov et al., 2005兲. The degeneracies of the bands at ⑀ = 0 can be studied using topological arguments 共Mañes et al., 2007兲. Multilayers with an even number of layers and Bernal stacking have inversion symmetry, leading to degeneracies with winding number +2, as in the case of a bilayer. The trigonal lattice symmetry implies that these points can lead, at most, to four Dirac points. In stacks with an odd number of layers, these degeneracies can be completely removed. The winding number of the degeneracies found in stacks with N layers and orthorhombic ordering is ±N. The inclusion of trigonal warping terms will lead to the existence of many weaker degeneracies near ⑀ = 0. Furthermore, it is well known that in graphite, the planes can be rotated relative to each other giving rise to Moiré patterns that are observed in STM of graphite surfaces 共Rong and Kuiper, 1993兲. The graphene layers can be rotated relative to each other due to the weak coupling between planes that allows for the presence of many different orientational states that are quasidegenerate in energy. For certain angles, the graphene layers become commensurate with each other leading to a lowering of the electronic energy. Such a phenomenon is Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 121 (a) (b) (c) (d) FIG. 13. 共Color online兲 Electronic bands of graphene multilayers. 共a兲 Biased bilayer. 共b兲 Trilayer with Bernal stacking. 共c兲 Trilayer with orthorhombic stacking. 共d兲 Stack with four layers where the top and bottom layers are shifted in energy with respect to the two middle layers by +0.1 eV. quite similar to the commensurate-incommensurate transitions observed in certain charge-density-wave systems or adsorption of gases on graphite 共Bak, 1982兲. This kind of electronic structure dependence on the relative rotation angle between graphene layers leads to what is called superlubricity in graphite 共Dienwiebel et al., 2004兲, namely, the vanishing of the friction between layers as a function of the rotation angle. In the case of bilayer graphene, a rotation by a small commensurate angle leads to the effective decoupling between layers and recovery of the linear Dirac spectrum of the single layer, albeit with a modification on the value of the Fermi velocity 共Lopes dos Santos et al., 2007兲. 1. Electronic structure of bulk graphite The tight-binding description of graphene described earlier can be extended to systems with an infinite number of layers. The coupling between layers leads to hopping terms between ␲ orbitals in different layers, leading to the so-called Slonczewski-Weiss-McClure model 共Slonczewski and Weiss, 1958兲. This model describes the band structure of bulk graphite with the Bernal stacking 122 Castro Neto et al.: The electronic properties of graphene TABLE I. Band-structure parameters of graphite 共Dresselhaus and Dresselhaus, 2002兲. ␥0 ␥1 ␥2 ␥3 ␥4 ␥5 ⌬ 3.16 eV 0.39 eV −0.020 eV 0.315 eV 0.044 eV 0.038 eV −0.008 eV order in terms of seven parameters: ␥0, ␥1, ␥2, ␥3, ␥4, ␥5, and ⌬. The parameter ␥0 describes the hopping within each layer, and it has been considered previously. The coupling between orbitals in atoms that are nearest neighbors in successive layers is ␥1, which we called t⬜ earlier. The parameters ␥3 and ␥4 describe the hopping between orbitals at next nearest neighbors in successive layers and were discussed in the case of the bilayer. The couplings between orbitals at next-nearest-neighbor layers are ␥2 and ␥5. Finally, ⌬ is an on-site energy that reflects the inequivalence between the two sublattices in each graphene layer once the presence of neighboring layers is taken into account. The values of these parameters, and their dependence with pressure, or, equivalently, the interatomic distances, have been extensively studied 共McClure, 1957, Nozières, 1958; Dresselhaus and Mavroides, 1964; Soule et al., 1964; Dillon et al., 1977; Brandt et al., 1988兲. A representative set of values is shown in Table I. The unit cell of graphite with Bernal stacking includes two layers, and two atoms within each layer. The tightbinding Hamiltonian described previously can be represented as a 4 ⫻ 4 matrix. In the continuum limit, the two inequivalent corners of the BZ can be treated separately, and the in-plane terms can be described by the Dirac equation. The next terms of importance for the lowenergy electronic spectrum are the nearest-neighbor couplings ␥1 and ␥3. The influence of the parameter ␥4 on the low-energy bands is much smaller, as discussed below. Finally, the fine details of the spectrum of bulk graphite are determined by ⌬, which breaks the electron-hole symmetry of the bands preserved by ␥0, ␥1, and ␥3. It is usually assumed to be much smaller than the other terms. We label the two atoms from the unit cell in one layer as 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 written as HK 11 = 2␥2 cos共2␲kz/c兲, HK 12 = vF共kx + iky兲, HK 13 = 3 ␥ 4a 共1 + eikzc兲共kx + iky兲, 2 Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 HK 14 = 3 ␥ 3a 共1 + eikzc兲共kx − iky兲, 2 HK 22 = ⌬ + 2␥5 cos共2␲kz/c兲, ikzc HK 兲, 23 = ␥1共1 + e HK 24 = 3 ␥ 4a 共1 + eikzc兲共kx + iky兲, 2 HK 33 = ⌬ + 2␥5 cos共2␲kz/c兲, HK 34 = vF共kx + iky兲, HK 44 = 2␥2 cos共2␲kz/c兲, 共43兲 where c is the lattice constant in the out-of-plane direction, equal to twice the interlayer spacing. The matrix elements of HK⬘ can be obtained by replacing kx by −kx 共other conventions for the unit cell and the orientation of the lattice lead to different phases兲. Recent ARPES experiments 共Ohta et al., 2006; Zhou, Gweon, and Lanzara, 2006; Zhou, Gweon, et al., 2006; Bostwick, Ohta, Seyller, et al., 2007兲 performed in epitaxially grown graphene stacks 共Berger et al., 2004兲 confirm the main features of this model, formulated on the basis of Fermi surface measurements 共McClure, 1957; Soule et al., 1964兲. The electronic spectrum of the model can also be calculated in a magnetic field 共de Gennes, 1964; Nakao, 1976兲, and the results are also consistent with STM on graphite surfaces 共Kobayashi et al., 2005; Matsui et al., 2005; Niimi et al., 2006; Li and Andrei, 2007兲, epitaxially grown graphene stacks 共Mallet et al., 2007兲, and optical measurements in the infrared range 共Li et al., 2006兲. F. Surface states in graphene So far we have discussed the basic bulk properties of graphene. Nevertheless, graphene has very interesting surface 共edge兲 states that do not occur in other systems. A semi-infinite graphene sheet with a zigzag edge has a band of zero-energy states localized at the surface 共Fujita et al., 1996; Nakada et al., 1996; Wakabayashi et al., 1999兲. In Sec. II.H, we discuss the existence of edge states using the Dirac equation. Here we discuss the same problem using the tight-binding Hamiltonian. To see why these edge states exist, we consider the ribbon geometry with zigzag edges shown in Fig. 14. The ribbon width is such that it has N unit cells in the transverse cross section 共y direction兲. We assume that the ribbon has infinite length in the longitudinal direction 共x direction兲. We rewrite Eq. 共5兲, with t⬘ = 0, in terms of the integer indices m and n, introduced in Fig. 14, and labeling the unit cells, Castro Neto et al.: The electronic properties of graphene n−1 A a1 n 0 = ␤共k,0兲, 共54兲 0 = ␣共k,N − 1兲. 共55兲 Equations 共52兲 and 共53兲 are easily solved, giving B a2 123 ␣共k,n兲 = 关− 2 cos共ka/2兲兴nei共ka/2兲n␣共k,0兲, n+1 共56兲 y x n+2 m−1 m m+ 1 ⫻e−i共ka/2兲共N−1−n兲␤共k,N − 1兲. m+ 2 FIG. 14. 共Color online兲 Ribbon geometry with zigzag edges. H=−t 兺 关a␴† 共m,n兲b␴共m,n兲 + a␴† 共m,n兲b␴共m − 1,n兲 m,n,␴ + a␴† 共m,n兲b␴共m,n − 1兲 + H.c.兴. 共44兲 Given that the ribbon is infinite in the a1 direction, one can introduce a Fourier decomposition of the operators leading to H=−t 冕 dk 兺 关a† 共k,n兲b␴共k,n兲 + eikaa␴† 共k,n兲b␴共k,n兲 2␲ n,␴ ␴ + a␴† 共k,n兲b␴共k,n − 1兲 + H.c.兴, 共45兲 where c␴† 共k , n兲兩0典 = 兩c , ␴ , k , n典 and c = a , b. The oneparticle Hamiltonian can be written as H1p = − t 冕 dk 兺 关共1 + eika兲兩a,k,n, ␴典具b,k,n, ␴兩 n,␴ + 兩a,k,n, ␴典具b,k,n − 1, ␴兩 + H.c.兴. 共46兲 The solution of the Schrödinger equation H1p兩␮ , k , ␴典 = E␮,k兩␮ , k , ␴典 can be generally expressed as 兩␮,k, ␴典 = 兺 关␣共k,n兲兩a,k,n, ␴典 + ␤共k,n兲兩b,k,n, ␴典兴, n 共47兲 where the coefficients ␣ and ␤ satisfy the following equations: E␮,k␣共k,n兲 = − t关共1 + eika兲␤共k,n兲 + ␤共k,n − 1兲兴, 共48兲 E␮,k␤共k,n兲 = − t关共1 + e−ika兲␣共k,n兲 + ␣共k,n + 1兲兴. 共49兲 As the ribbon has a finite width, we have to be careful with the boundary conditions. Since the ribbon only exists between n = 0 and n = N − 1 at the boundary, Eqs. 共48兲 and 共49兲 read E␮,k␣共k,0兲 = − t共1 + eika兲␤共k,0兲, 共50兲 E␮,k␤共k,N − 1兲 = − t共1 + e−ika兲␣共k,N − 1兲. 共51兲 The surface 共edge兲 states are solutions of Eqs. 共48兲–共51兲 with E␮,k = 0, 0 = 共1 + eika兲␤共k,n兲 + ␤共k,n − 1兲, 共52兲 0 = 共1 + e−ika兲␣共k,n兲 + ␣共k,n + 1兲, 共53兲 Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 ␤共k,n兲 = 关− 2 cos共ka/2兲兴N−1−n 共57兲 Consider, for simplicity, a semi-infinite system with a single edge. We must require the convergence condition 兩−2 cos共ka / 2兲兩 ⬍ 1 in Eq. 共57兲 because otherwise the wave function would diverge in the semi-infinite graphene sheet. Therefore, the semi-infinite system has edge states for ka in the region 2␲ / 3 ⬍ ka ⬍ 4␲ / 3, which corresponds to 1 / 3 of the possible momenta. Note that the amplitudes of the edge states are given by 兩␣共k,n兲兩 = 兩␤共k,n兲兩 = 冑 冑 2 −n/␭共k兲 , e ␭共k兲 共58兲 2 −共N−1−n兲/␭共k兲 e , ␭共k兲 共59兲 where the penetration length is given by ␭共k兲 = − 1/ln兩2 cos共ka/2兲兩. 共60兲 It is easily seen that the penetration length diverges when ka approaches the limits of the region 兴2␲ / 3 , 4␲ / 3关. Although the boundary conditions defined by Eqs. 共54兲 and 共55兲 are satisfied for solutions 共56兲 and 共57兲 in the semi-infinite system, they are not in the ribbon geometry. In fact, Eqs. 共58兲 and 共59兲 are eigenstates only in the semi-infinite system. In the graphene ribbon the two edge states, which come from both sides of the edge, will overlap with each other. The bonding and antibonding states formed by the two edge states will then be the ribbon eigenstates 共Wakabayashi et al., 1999兲 共note that at zero energy there are no other states with which the edge states could hybridize兲. As bonding and antibonding states result in a gap in energy, the zero-energy flat bands of edge states will become slightly dispersive, depending on the ribbon width N. The overlap between the two edge states is larger as ka approaches 2␲ / 3 and 4␲ / 3. This means that deviations from zero-energy flatness will be stronger near these points. Edge states in graphene nanoribbons, as in carbon nanotubes, are predicted to be Luttinger liquids, that is, interacting one-dimensional electron systems 共Castro Neto et al., 2006b兲. Hence, clean nanoribbons must have 1D square root singularities in their density of states 共Nakada et al., 1996兲 that can be probed by Raman spectroscopy. Disorder may smooth out these singularities, however. In the presence of a magnetic field, when the bulk states are gapped, the edge states are responsible for the transport of spin and charge 共Abanin et al., 2006; Abanin, Lee, and Levitov, 2007; Abanin and Levitov, 2007; Abanin, Novoselov, Zeitler, et al., 2007兲. 124 Castro Neto et al.: The electronic properties of graphene A B B2 A1 a2 n a1 y B1 x n+1 armchair edge A2 y x m- 1 m m+ 1 m+ 2 FIG. 15. Sketch of a zigzag termination of a graphene bilayer. As discussed by Castro, Peres, Lopes dos Santos, et al. 共2008兲, there is a band of surface states completely localized in the bottom layer, and another surface band which alternates between the two. H. The spectrum of graphene nanoribbons The spectrum of graphene nanoribbons depends on the nature of their edges: zigzag or armchair 共Brey and Fertig, 2006a, 2006b; Nakada et al., 1996兲. In Fig. 16, we show a honeycomb lattice having zigzag edges along the x direction and armchair edges along the y direction. If we choose the ribbon to be infinite in the x direction, we produce a graphene nanoribbon with zigzag edges; conversely, choosing the ribbon to be macroscopically large along the y but finite in the x direction, we produce a graphene nanoribbon with armchair edges. In Fig. 17, we show 14 energy levels, calculated in the tight-binding approximation, closest to zero energy for a nanoribbon with zigzag and armchair edges and of width N = 200 unit cells. We show that both are metallic, and that the zigzag ribbon presents a band of zero-energy modes that is absent in the armchair case. This band at Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 FIG. 16. 共Color online兲 A piece of a honeycomb lattice displaying both zigzag and armchair edges. zero energy is the surface states living near the edge of the graphene ribbon. More detailed ab initio calculations of graphene nanoribbon spectra show that interaction effects can lead to electronic gaps 共Son et al., 2006b兲 and magnetic states close to the graphene edges, independent of their nature 共Son et al., 2006a; Yang, Cohen, and Louie, 2007; Yang, Park, Son, et al., 2007兲. From the experimental point of view, however, graphene nanoribbons currently have a high degree of roughness at the edges. Such edge disorder can change significantly the properties of edge states 共Areshkin and White, 2007; Gunlycke et al., 2007兲, leading to Anderson localization and anomalies in the quantum Hall effect 共Castro Neto et al., 2006b; Martin and Blanter, 2007兲 as well as Coulomb blockade effects 共Sols et al., 2007兲. Such effects have already been observed in lithographically engineered graphene nanoribbons 共Han et al., 2007; energy / t Single-layer graphene can be considered a zero gap semiconductor, which leads to the possibility of midgap states, at ⑀ = 0, as discussed in the previous section. The most studied such states are those localized near a graphene zigzag edge 共Fujita et al., 1996; Wakayabashi and Sigrist, 2000兲. It can be shown analytically 共Castro, Peres, Lopes dos Santos, et al., 2008兲 that a bilayer zigzag edge, like that shown in Fig. 15, analyzed within the nearest-neighbor tight-binding approximation described before, has two bands of localized states, one completely localized in the top layer and indistinguishable from similar states in single-layer graphene, and another band that 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 onethird of the BZ of a stripe bounded by zigzag edges. They become dispersive in a biased bilayer. As graphite can 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 one surface band per layer. zigzag edge 1 0.2 0 0 -1 energy / t G. Surface states in graphene stacks a 0 1 2 3 4 5 6 7 -0.2 1 0.2 0 0 -1 0 1 2 3 4 5 momentum ka 6 7 0 -0.2 1.5 0.1 2 0.2 3 2.5 3.5 momentum ka 0.3 0.4 4 4.5 FIG. 17. Electronic dispersion for graphene nanoribbons. Left: energy spectrum, as calculated from the tight-binding equations, for a nanoribbon with armchair 共top兲 and zigzag 共bottom兲 edges. The width of the nanoribbon is N = 200 unit cells. Only 14 eigenstates are depicted. Right: zoom of the lowenergy states shown on the right. 125 Castro Neto et al.: The electronic properties of graphene Özyilmaz et al., 2007兲. Furthermore, the problem of edge passivation by hydrogen or other elements is not currently understood experimentally. Passivation can be modeled in the tight-binding approach by modifications of the hopping energies 共Novikov, 2007兲 or via additional phases in the boundary conditions 共Kane and Mele, 1997兲. Theoretical modeling of edge passivation indicate that those have a strong effect on the electronic properties at the edge of graphene nanoribbons 共Barone et al., 2006; Hod et al., 2007兲. In what follows, we derive the spectrum for both zigzag and armchair edges directly from the Dirac equation. This was originally done both with and without a magnetic field 共Nakada et al., 1996; Brey and Fertig, 2006a, 2006b兲. In the geometry of Fig. 16, the unit-cell vectors are a1 = a0共1 , 0兲 and a2 = a0共1 / 2 , 冑3 / 2兲, which generate the unit vectors of the BZ given by b1 冑 冑 冑 = 4␲ / 共a0 3兲共 3 / 2 , −1 / 2兲 and b2 = 4␲ / 共a0 3兲共0 , 1兲. From these two vectors, we find two inequivalent Dirac points given by K = 共4␲ / 3a0 , 0兲 = 共K , 0兲 and K⬘ = 共−4␲ / 3a0 , 0兲 = 共−K , 0兲, with a0 = 冑3a. The Dirac Hamiltonian around the Dirac point K reads in momentum space 冉 0 px − ipy px + ipy 0 冊 and around the K⬘ as H K⬘ = v F 冉 0 px + ipy px − ipy 0 共61兲 , 冊 . ⬘ 共r兲, ⌿A共r兲 = eiK·r␺A共r兲 + eiK⬘·r␺A 共62兲 共63兲 and for sublattice B is given by ⬘ 共r兲, ⌿B共r兲 = eiK·r␺B共r兲 + eiK⬘·r␺B 共68兲 The boundary conditions 共67兲 and 共68兲 are satisfied for any x by the choice ⬘ 共L兲 = ␾B共0兲 = ␾B⬘ 共0兲 = 0. ␾A共L兲 = ␾A 共69兲 We need now to find out the form of the envelope functions. The eigenfunction around the point K has the form 冉 0 kx − ⳵y kx + ⳵y 0 冊冉 冊 冉 冊 ␾A共y兲 ␾A共y兲 = ⑀˜ , ␾B共y兲 ␾B共y兲 共70兲 共kx − ⳵y兲␾B = ⑀˜ ␾A , 共kx + ⳵y兲␾A = ⑀˜ ␾B . 共71兲 Applying the operator kx + ⳵y to the first of Eqs. 共71兲 leads to 共− ⳵2y + k2x兲␾B = ⑀˜ 2␾B , 共72兲 with ␾A given by 1 ␾A = 共kx − ⳵y兲␾B . ⑀˜ 共73兲 冉 冊 ␾A共y兲 , ␾B共y兲 leading to Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 共74兲 leading to an eigenenergy ⑀˜ 2 = k2x − z2. The boundary conditions for a zigzag edge require that ␾A共y = L兲 = 0 and ␾B共y = 0兲 = 0, leading to ␾B共y = 0兲 = 0 ⇔ A + B = 0, 共75兲 共64兲 共65兲 and a similar equation for the spinor of Hamiltonian 共62兲. For zigzag edges, the boundary conditions at the edge of the ribbon 共located at y = 0 and y = L, where L is the ribbon width兲 are ⌿B共y = 0兲 = 0, ␾B = Aezy + Be−zy , ␾A共y = L兲 = 0 ⇔ 共kx − z兲AezL + 共kx + z兲Be−zL = 0, where ␺A and ␺B are the components of the spinor wave ⬘ and ␺B⬘ have idenfunction of Hamiltonian 共61兲 and ␺A tical meaning but relative to Eq. 共62兲. Assume that the edges of the nanoribbons are parallel to the x axis. In this case, the translational symmetry guarantees that the spinor wave function can be written as ⌿A共y = L兲 = 0, ⬘ 共0兲. 0 = eiKxeikxx␾B共0兲 + e−iKxeikxx␾B The solution of Eq. 共72兲 has the form The wave function, in real space, for the sublattice A is given by ␺共r兲 = eikxx 共67兲 with ⑀˜ = ⑀ / vF and ⑀ the energy eigenvalue. The eigenproblem can be written as two linear differential equations of the form 1. Zigzag nanoribbons HK = vF ⬘ 共L兲, 0 = eiKxeikxx␾A共L兲 + e−iKxeikxx␾A 共66兲 which leads to an eigenvalue equation of the form e−2zL = kx − z . kx + z 共76兲 Equation 共76兲 has real solutions for z, whenever kx is positive; these solutions correspond to surface waves 共edge states兲 existing near the edge of the graphene ribbon. In Sec. II.F, we discussed these states from the point of view of the tight-binding model. In addition to real solutions for z, Eq. 共76兲 also supports complex ones, of the form z = ikn, leading to kx = kn . tan共knL兲 共77兲 The solutions of Eq. 共77兲 correspond to confined modes in the graphene ribbon. 126 Castro Neto et al.: The electronic properties of graphene If we apply the same procedure to the Dirac equation around the Dirac point K⬘, we obtain a different eigenvalue equation given by kx + z e−2zL = . kx − z 共78兲 This equation supports real solutions for z if kx is negative. Therefore, we have edge states for negative values kx, with momentum around K⬘. As in the case of K, the system also supports confined modes, given by kn . kx = − tan共knL兲 determined from Eq. 共73兲. The solutions of Eq. 共72兲 have the form ␾B = Aeiknx + Be−iknx , 共88兲 ␾B⬘ = Ceiknx + De−iknx . 共89兲 Applying the boundary conditions 共86兲 and 共87兲, one obtains 0 = A + B + C + D, 共90兲 0 = Aei共kn+K兲L + De−i共kn+K兲L + Be−i共kn−K兲L + Cei共kn−K兲L . 共79兲 One should note that the eigenvalue equations for K⬘ are obtained from those for K by inversion, kx → −kx. We finally note that the edge states for zigzag nanoribbons are dispersionless 共localized in real space兲 when t⬘ = 0. When electron-hole symmetry is broken 共t⬘ ⫽ 0兲, these states become dispersive with a Fermi velocity ve ⬇ t⬘a 共Castro Neto et al., 2006b兲. 共91兲 The boundary conditions are satisfied with the choice A = − D, B = C = 0, 共92兲 which leads to sin关共kn + K兲L兴 = 0. Therefore, the allowed values of kn are given by kn = n␲ 4␲ , − L 3a0 共93兲 and the eigenenergies are given by 2. Armchair nanoribbons We now consider an armchair nanoribbon with armchair edges along the y direction. The boundary conditions at the edges of the ribbon 共located at x = 0 and x = L, where L is the width of the ribbon兲, ⌿A共x = 0兲 = ⌿B共x = 0兲 = ⌿A共x = L兲 = ⌿B共x = L兲 = 0. 共80兲 Translational symmetry guarantees that the spinor wave function of Hamiltonian 共61兲 can be written as ␺共r兲 = eikyy 冉 冊 ␾A共x兲 , ␾B共x兲 共81兲 and a similar equation for the spinor of the Hamiltonian 共62兲. The boundary conditions have the form ⬘ 共0兲, 0 = eikyy␾A共0兲 + eikyy␾A 共82兲 ⬘ 共0兲, 0 = eikyy␾B共0兲 + eikyy␾B 共83兲 ⬘ 共L兲, 0 = eiKLeikyy␾A共L兲 + e−iKLeikyy␾A 共84兲 ⬘ 共L兲, 0 = eiKLeikyy␾B共L兲 + e−iKLeikyy␾B 共85兲 and are satisfied for any y if ␾␮共0兲 + ␾␮⬘ 共0兲 = 0 共86兲 eiKL␾␮共L兲 + e−iKL␾␮⬘ 共L兲 = 0, 共87兲 and with ␮ = A , B. It is clear that these boundary conditions mix states from the two Dirac points. Now we must find the form of the envelope functions obeying the boundary conditions 共86兲 and 共87兲. As before, the functions ␾B ⬘ obey the second-order differential equation 共72兲 and ␾B ⬘ are 共with y replaced by x兲, and the functions ␾A and ␾A Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 ⑀˜ 2 = k2y + kn2 . 共94兲 No surface states exist in this case. I. Dirac fermions in a magnetic field We now consider the problem of a uniform magnetic field B applied perpendicular to the graphene plane.2 We use the Landau gauge: A = B共−y , 0兲. Note that the magnetic field introduces a new length scale in the problem, ᐉB = 冑 c , eB 共95兲 which is the magnetic length. The only other scale in the problem is the Fermi-Dirac velocity. Dimensional analysis shows that the only quantity with dimensions of energy we can make is vF / ᐉB. In fact, this determines the cyclotron frequency of the Dirac fermions, ␻c = 冑2 vF ᐉB 共96兲 共the 冑2 factor comes from the quantization of the problem, see below兲. Equations 共95兲 and 共96兲 show that the cyclotron energy scales like 冑B, in contrast with the nonrelativistic problem where the cyclotron energy is linear in B. This implies that the energy scale associated with the Dirac fermions is rather different from the one found in the ordinary 2D electron gas. For instance, for fields of the order of B ⬇ 10 T, the cyclotron energy in the 2D electron gas is of the order of 10 K. In contrast, 2 The problem of transverse magnetic and electric fields can also be solved exactly. See Lukose et al. 共2007兲 and Peres and Castro 共2007兲. 127 Castro Neto et al.: The electronic properties of graphene for the Dirac fermion problem and the same fields, the cyclotron energy is of the order of 1000 K, that is, two orders of magnitude larger. This has strong implications for the observation of the quantum Hall effect at room temperature 共Novoselov et al., 2007兲. Furthermore, for B = 10 T the Zeeman energy is relatively small, g␮BB ⬇ 5 K, and can be disregarded. We now consider the Dirac equation in more detail. Using the minimal coupling in Eq. 共19兲 共i.e., replacing −iⵜ by −i ⵜ + eA / c兲, we find ជ · 共− i ⵜ + eA/c兲兴␺共r兲 = E␺共r兲, v F关 ␴ 共97兲 in the Landau gauge the generic solution for the wave function has the form ␺共x , y兲 = eikx␾共y兲, and the Dirac equation reads vF 冋 册 ⳵y − k + Bey/c ␾共y兲 = E␾共y兲, 0 − ⳵y − k + Bey/c 0 共98兲 which can be rewritten as ␻c 冋 册 0 O O † 0 ␾共␰兲 = E␾共␰兲, 共99兲 or equivalently 共O␴+ + O†␴−兲␾ = 共2E/␻c兲␾ , 共100兲 where ␴± = ␴x ± i␴y, and we have defined the dimensionless length scale ␰= y − ᐉ Bk ᐉB 共101兲 ␾0共␰兲 = ␺0共␰兲 丢 兩 ⇓ 典, 1 冑2 共⳵␰ + ␰兲, where 兩⇓典 indicates the state localized on sublattice A, and 兩⇑典 indicates the state localized on sublattice B. Furthermore, O ␺ 0共 ␰ 兲 = 0 ␾N,±共␰兲 = ␺N−1共␰兲 丢 兩 ⇑ 典 ± ␺N共␰兲 丢 兩 ⇓ 典 = O† = 1 冑2 共− ⳵␰ + ␰兲, 共103兲 ជ is of dimenand since the Hilbert space generated by ␴ sion 2, and the spectrum generated by O† is bounded from below, we just need to ensure that O␾0 = 0, ␴−␾0 = 0, 共104兲 in order for Eq. 共103兲 to be fulfilled. The obvious zeromode solution is Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 共106兲 冉 冊 ␺N−1共␰兲 , ± ␺ N共 ␰ 兲 共102兲 which obey canonical commutation relations 关O , O†兴 = 1. The number operator is simply N = O†O. First, we note that Eq. 共100兲 allows for a solution with zero energy, 共O␴+ + O†␴−兲␾0 = 0, 共105兲 is the ground states of the 1D harmonic oscillator. All the solutions can now be constructed from the zero mode, and 1D harmonic-oscillator operators O= FIG. 18. 共Color online兲 SdH oscillations observed in longitudinal resistivity ␳xx of graphene as a function of the charge carrier concentration n. Each peak corresponds to the population of one Landau level. Note that the sequence is not interrupted when passing through the Dirac point, between electrons and holes. The period of oscillations is ⌬n = 4B / ⌽0, where B is the applied field and ⌽0 is the flux quantum 共Novoselov, Geim, Morozov, et al., 2005兲. 共107兲 and their energy is given by 共McClure, 1956兲 E±共N兲 = ± ␻c冑N, 共108兲 where N = 0 , 1 , 2 , . . . is a positive integer and ␺N共␰兲 is the solution of the 1D harmonic oscillator 关explicitly, ␺N共␰兲 = 2−N/2共N!兲−1/2 exp兵−␰2 / 2其HN共␰兲, where HN共␰兲 is a Hermite polynomial兴. The Landau levels at the opposite Dirac point K⬘ have exactly the same spectrum and hence each Landau level is doubly degenerate. Of particular importance for the Dirac problem discussed here is the existence of a zero-energy state N = 0, which is responsible for the anomalies observed in the quantum Hall effect. This particular Landau level structure has been observed by many different experimental probes, from Shubnikov–de Haas oscillations in single layer graphene 共see Fig. 18兲 共Novoselov, Geim, Morozov, et al., 2005; Zhang et al., 2005兲, to infrared spectroscopy 128 Castro Neto et al.: The electronic properties of graphene FIG. 19. 共Color online兲 Geometry of Laughlin’s thought experiment with a graphene ribbon: a magnetic field B is applied normal to the surface of the ribbon; a current I circles the loop, generating a Hall voltage VH and a magnetic flux ⌽. 共Jiang, Henriksen, Tung, et al., 2007兲, and to scanning tunneling spectroscopy 共Li and Andrei, 2007兲 共STS兲 on a graphite surface. J. The anomalous integer quantum Hall effect In the presence of disorder, Landau levels get broadened and mobility edges appear 共Laughlin, 1981兲. Note that there will be a Landau level at zero energy that separates states with hole character 共␮ ⬍ 0兲 from states with electron character 共␮ ⬎ 0兲. The components of the resistivity and conductivity tensors are given by ␳xx = ␳xy = ␴xx 2 , + ␴xy 2 ␴xx ␴xy 2 , + ␴xy 2 ␴xx 共109兲 where ␴xx 共␳xx兲 is the longitudinal component and ␴xy 共␳xy兲 is the Hall component of the conductivity 共resistivity兲. When the chemical potential is inside a region of localized 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 delocalized states, when the chemical potential is crossing a Landau level, we have ␴xx ⫽ 0 and ␴xy varies continuously 共Sheng et al., 2006, 2007兲. The value of ␴xy in the region of localized states can be obtained from Laughlin’s gauge invariance argument 共Laughlin, 1981兲: one imagines making a graphene ribbon such as shown in Fig. 19 with a magnetic field B normal through its surface and a current I circling its loop. Due to the Lorentz force, the magnetic field produces a Hall voltage VH perpendicular to the field and current. The circulating current generates a magnetic flux ⌽ that threads the loop. The current is given by I=c ␦E , ␦⌽ 共110兲 where E is the total energy of the system. The localized states do not respond to changes in ⌽, only the delocalized ones. When the flux is changed by a flux quantum ␦⌽ = ⌽0 = hc / e, the extended states remain the same by Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 gauge invariance. If the chemical potential is in the region of localized states, all the extended states below the chemical potential will be filled both before and after the change of flux by ⌽0. However, during the change of flux, an integer number of states enter the cylinder at one edge and leave at the opposite edge. The question is: How many occupied states are transferred between edges? We consider a naive and, as shown further, incorrect calculation in order to show the importance of the zero mode in this problem. Each Landau level contributes with one state times its degeneracy g. In the case of graphene, we have g = 4 since there are two spin states and two Dirac cones. Hence, we expect that when the flux changes by one flux quantum, the change in energy would be ␦Einc = ± 4NeVH, where N is an integer. The plus sign applies to electron states 共charge +e兲 and the minus sign to hole states 共charge −e兲. Hence, we conclude that Iinc = ± 4共e2 / h兲VH and hence ␴xy,inc = I / VH = ± 4Ne2 / h, which is the naive expectation. The problem with this result is that when the chemical potential is exactly at half filling, that is, at the Dirac point, it would predict a Hall plateau at N = 0 with ␴xy,inc = 0, which is not possible since there is an N = 0 Landau level, with extended states at this energy. The solution for this paradox is rather simple: because of the presence of the zero mode that is shared by the two Dirac points, there are exactly 2 ⫻ 共2N + 1兲 occupied states that are transferred from one edge to another. Hence, the change in energy is ␦E = ± 2共2N + 1兲eVH for a change of flux of ␦⌽ = hc / e. Therefore, the Hall conductivity is 共Schakel, 1991; Gusynin and Sharapov, 2005; Herbut, 2007; Peres, Guinea, and Castro Neto, 2006a, 2006b兲 ␴xy = e2 c ␦E I = ± 2共2N + 1兲 , = VH VH ␦⌽ h 共111兲 without any Hall plateau at N = 0. This result has been observed experimentally 共Novoselov, Geim, Morozov, et al., 2005; Zhang et al., 2005兲 as shown in Fig. 20. K. Tight-binding model in a magnetic field In the tight-binding approximation, the hopping integrals are replaced by a Peierls substitution, R⬘ R⬘ eie兰R A·drtR,R⬘ = ei共2␲/⌽0兲兰R A·drtR,R⬘ , 共112兲 where tR,R⬘ represents the hopping integral between the sites R and R⬘, with no field present. The tight-binding Hamiltonian for a single graphene layer, in a constant magnetic field perpendicular to the plane, is conveniently written as H=−t 兺 关ei␲共⌽/⌽0兲n关共1+z兲/2兴a␴† 共m,n兲b␴共m,n兲 m,n,␴ + e−i␲共⌽/⌽0兲na␴† 共m,n兲b␴„m − 1,n − 共1 − z兲/2… + ei␲共⌽/⌽0兲n关共z−1兲/2兴a␴† 共m,n兲b␴共m,n − z兲 + H.c.兴, 共113兲 129 Castro Neto et al.: The electronic properties of graphene energy / t zigzag: N=200, φ/φ0=1/701 armchair: N=200, φ/φ0=1/701 1 1 0.5 0.5 0 0 -0.5 -0.5 -1 0 1 2 3 4 5 -1 7 6 0.4 0.4 0.2 0.2 0 1 2 3 4 5 7 6 energy / t µ µ 0 0 -0.2 -0.2 -0.4 -0.4 2 3 4 5 momentum ka 0.5 6 1 1.5 momentum ka 2 FIG. 21. 共Color online兲 Fourteen energy levels of tight-binding electrons in graphene in the presence of a magnetic flux ⌽ = ⌽0 / 701, for a finite stripe with N = 200 unit cells. The bottom panels are zoom-in images of the top ones. The dashed line represents the chemical potential ␮. FIG. 20. 共Color online兲 Quantum Hall effect in graphene as a function of charge-carrier concentration. The peak at n = 0 shows that in high magnetic fields there appears a Landau level at zero energy where no states exist in zero field. The field draws electronic states for this level from both conduction and valence bands. The dashed lines indicate plateaus in ␴xy described by Eq. 共111兲. Adapted from Novoselov, Geim, Morozov, et al., 2005. holding for a graphene stripe with a zigzag 共z = 1兲 and armchair 共z = −1兲 edges oriented along the x direction. Fourier transforming along the x direction gives H=−t 兺 关ei␲共⌽/⌽0兲n关共1+z兲/2兴a␴† 共k,n兲b␴共k,n兲 + e−i␲共⌽/⌽0兲neikaa␴† 共k,n兲b␴„k,n − 共1 − z兲/2… + 冋 冉 − z兲 + H.c.兴. 册 冊 冋 共114兲 冉 E␮,k␤共k,n兲 = − t e−ika/22 cos ␲ 册 冊 ⌽ ka n− ␣共k,n兲 ⌽0 2 + ␣共k,n + 1兲 , 共115兲 where the coefficients ␣共k , n兲 and ␤共k , n兲 show up in Hamiltonian’s eigenfunction 兩␺共k兲典 written in terms of lattice-position-state states as Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 Equations 共114兲 and 共115兲 hold in the bulk. Considering that the zigzag ribbon has N unit cells along its width, from n = 0 to n = N − 1, the boundary conditions at the edges are obtained from Eqs. 共114兲 and 共115兲, and read 冉 冊 E␮,k␣共k,0兲 = − teika/22 cos ka ␤共k,0兲, 2 ⫻␣共k,N − 1兲. ⌽ ka n− ␤共k,n兲 ⌽0 2 + ␤共k,n − 1兲 , 共116兲 E␮,k␤共k,N − 1兲 = − 2te−ika/2 cos ␲ We now consider the case of zigzag edges. The eigenproblem can be rewritten in terms of Harper’s equations 共Harper, 1955兲, and for zigzag edges we obtain 共Rammal, 1985兲 E␮,k␣共k,n兲 = − t eika/22 cos ␲ n,␴ 冋 k,n,␴ ei␲共⌽/⌽0兲n关共z−1兲/2兴a␴† 共k,n兲b␴共k,n 兩␺共k兲典 = 兺 关␣共k,n兲兩a;k,n, ␴典 + ␤共k,n兲兩b;k,n, ␴典兴. 共117兲 ⌽ ka 共N − 1兲 − ⌽0 2 册 共118兲 Similar equations hold for a graphene ribbon with armchair edges. In Fig. 21, we show 14 energy levels, around zero energy, for both the zigzag and armchair cases. The formation of the Landau levels is signaled by the presence of flat energy bands, following the bulk energy spectrum. From Fig. 21, it is straightforward to obtain the value of the Hall conductivity in the quantum Hall effect regime. We assume that the chemical potential is in between two Landau levels at positive energies, shown by the dashed line in Fig. 21. The Landau level structure shows two zero-energy modes; one of them is electronlike 共holelike兲, since close to the edge of the sample its energy is shifted upwards 共downwards兲. The other Landau levels are doubly degenerate. The determination of the values for the Hall conductivity is done by counting how many energy levels 共of electronlike nature兲 are below the chemical potential. This counting produces the value 2N + 1, with N = 0 , 1 , 2 , . . . 共for the case of Fig. 21 one has 130 Castro Neto et al.: The electronic properties of graphene of the bands, as shown in Fig. 22. Note that, in a trilayer with Bernal stacking, two sets of levels have a 冑N dependence, while the energies of other two depend linearly on N. In an infinite rhombohedral stack, the Landau levels remain discrete and quasi-2D 共Guinea et al., 2006兲. Note that, even in an infinite stack with the Bernal structure, the Landau level closest to E = 0 forms a band that does not overlap with the other Landau levels, leading to the possibility of a 3D integer quantum Hall effect 共Luk’yanchuk and Kopelevich, 2004; Kopelevich et al., 2006; Bernevig et al., 2007兲. The optical transitions between Landau levels can also be calculated. The selection rules are the same as for a graphene single layer, and only transitions between subbands with Landau level indices M and N such that 兩N兩 = 兩M ± 1兩 are allowed. The resulting transitions, with their respective spectral strengths, are shown in Fig. 23. The transitions are grouped into subbands, which give rise to a continuum when the number of layers tends to infinity. In Bernal stacks with an odd number of layers, the transitions associated with Dirac subbands with linear dispersion have the largest spectral strength, and they give a significant contribution to the total absorption even if the number of layers is large, NL ⱗ 30– 40 共Sadowski et al., 2006兲. (a) (b) (c) M. Diamagnetism (d) FIG. 22. 共Color online兲 Landau levels of graphene stacks shown in Fig. 13. The applied magnetic field is 1 T. N = 2兲. From this counting, the Hall conductivity is given, including an extra factor of 2 accounting for the spin degree of freedom, by ␴xy = ± 2 冉 冊 1 e2 e2 N+ . 共2N + 1兲 = ± 4 2 h h 共119兲 Equation 共119兲 leads to the anomalous integer quantum Hall effect discussed previously, which is the hallmark of single-layer graphene. L. Landau levels in graphene stacks The dependence of the Landau level energies on the Landau index N roughly follows the dispersion relation Back in 1952, Mrozowski 共Mrozowski, 1952兲 studied diamagnetism of polycrystalline graphite and other condensed-matter molecular-ring systems. It was concluded that in such ring systems diamagnetism has two contributions: 共i兲 diamagnetism from the filled bands of electrons, and 共ii兲 Landau diamagnetism of free electrons and holes. For graphite the second source of diamagnetism is by far the largest of the two. McClure 共1956兲 computed diamagnetism of a 2D honeycomb lattice using the theory introduced by Wallace 共1947兲, a calculation he later generalized to threedimensional graphite 共McClure, 1960兲. For the honeycomb plane, the magnetic susceptibility 共in units of emu/g兲 is ␹=− 冉 冊 0.0014 2 ␮ ␥0 sech2 , T 2kBT 共120兲 where ␮ is the Fermi energy, T is the temperature, and kB is the Boltzmann constant. For graphite, the magnetic susceptibility is anisotropic and the difference between FIG. 23. 共Color online兲 Relative spectral strength of the low energy optical transitions between Landau levels in graphene stacks with Bernal ordering and an odd number of layers. The applied magnetic field is 1 T. Left: 3 layers. Middle: 11 layers. Right: 51 layers. The large circles are the transitions in a single layer. Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 Castro Neto et al.: The electronic properties of graphene the susceptibility parallel to the principal axis and that perpendicular to the principal axis is −21.5 ⫻ 10−6 emu/ g. The susceptibility perpendicular to the principal axis is equal to about the free-atom susceptibility, −0.5⫻ 10−6 emu/ g. In the 2D model, the susceptibility turns out to be large due to the presence of fast moving electrons, with a velocity of the order of vF ⯝ 106 m / s, which in turn is a consequence of the large value of the hopping parameter ␥0. In fact, the susceptibility turns out to be proportional to the square of ␥0. Later Sharma et al. extended the calculation of McClure for graphite by including the effect of trigonal warping and showed that the low-temperature diamagnetism increases 共Sharma et al., 1974兲. Safran and DiSalvo 共1979兲, interested in the susceptibility of graphite intercalation compounds, recalculated, in a tight-binding model, the susceptibility perpendicular to a graphite plane using Fukuyama’s theory 共Fukuyama, 1971兲, which includes interband transitions. The results agree with those by McClure 共1956兲. Later, Safran computed the susceptibility of a graphene bilayer showing that this system is diamagnetic at small values of the Fermi energy, but there appears a paramagnetic peak when the Fermi energy is of the order of the interlayer coupling 共Safran, 1984兲. The magnetic susceptibility of other carbon-based materials, such as carbon nanotubes and C60 molecular solids, was measured 共Heremans et al., 1994兲 showing a diamagnetic response at finite magnetic fields different from that of graphite. Studying the magnetic response of nanographite ribbons with both zigzag and armchair edges was done by Wakabayashi et al. using a numerical differentiation of the free energy 共Wakabayashi et al., 1999兲. From these two systems, the zigzag edge state is of particular interest since the system supports edge states even in the presence of a magnetic field, leading to very high density of states near the edge of the ribbon. The high-temperature response of these nanoribbons was found to be diamagnetic, whereas the lowtemperature susceptibility is paramagnetic. The Dirac-like nature of the electronic quasiparticles in graphene led Ghosal et al. 共2007兲 to consider in general the problem of the diamagnetism of nodal fermions, and Nakamura to study the orbital magnetism of Dirac fermions in weak magnetic fields 共Nakamura, 2007兲. Koshino and Ando considered the diamagnetism of disordered graphene in the self-consistent Born approximation, with a disorder potential of the form U共r兲 = 1ui␦共r − R兲 共Koshino and Ando, 2007兲. At the neutrality point and zero temperature, the susceptibility of disordered graphene is given by ␹共0兲 = − gvgs 2 2 2W e ␥0 , 3␲2 ⌫0 共121兲 where gs = gv = 2 is the spin and valley degeneracies, W is a dimensionless parameter for the disorder strength, defined as W = niu2i / 4␲␥20, ni is the impurity concentration, and ⌫0 = ⑀c exp关−1 / 共2W兲兴, with ⑀c a parameter defining a cutoff function used in the theory. At finite Fermi energy Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 131 ⑀F and zero temperature, the magnetic susceptibility is given by ␹ 共 ⑀ F兲 = − g vg s 2 2 W e ␥0 . 3␲ 兩 ⑀ F兩 共122兲 In the clean limit, the susceptibility is given by 共McClure, 1956; Safran and DiSalvo, 1979; Koshino and Ando, 2007兲 ␹ 共 ⑀ F兲 = − g vg s 2 2 e ␥0␦共⑀F兲. 6␲ 共123兲 N. Spin-orbit coupling Spin-orbit coupling describes a process in which an electron changes simultaneously its spin and angular momentum or, in general, moves from one orbital wave function to another. The mixing of the spin and the orbital motion is a relativistic effect, which can be derived from Dirac’s model of the electron. The mixing is large in heavy ions, where the average velocity of the electrons is higher. Carbon is a light atom, and the spin orbit interaction is expected to be weak. The symmetries of the spin-orbit interaction, however, allow the formation of a gap at the Dirac points in clean graphene. The spin-orbit interaction leads to a spindependent shift of the orbitals, which is of a different sign for the two sublattices, acting as an effective mass within each Dirac point 共Dresselhaus and Dresselhaus, 1965; Kane and Mele, 2005; Wang and Chakraborty, 2007a兲. The appearance of this gap leads to a nontrivial spin Hall conductance, similar to other models that study the parity anomaly in relativistic field theory in 2 + 1 dimensions 共Haldane, 1988兲. When the inversion symmetry of the honeycomb lattice is broken, either because the graphene layer is curved or because an external electric field is applied 共Rashba interaction兲, additional terms, which modulate the nearest-neighbor hopping, are induced 共Ando, 2000兲. The intrinsic and extrinsic spin-orbit interactions can be written as 共Dresselhaus and Dresselhaus, 1965; Kane and Mele, 2005兲 HSO;int ⬅ ⌬so HSO;ext ⬅ ␭R 冕 冕 d2r⌿̂†共r兲ŝz␴ˆ z␶ˆ z⌿̂共r兲, d2r⌿̂†共r兲共− ŝx␴ˆ y + ŝy␴ˆ x␶ˆ z兲⌿̂共r兲, 共124兲 where ␴ˆ and ␶ˆ are Pauli matrices that describe the sublattice and valley degrees of freedom and ŝ are Pauli matrices acting on actual spin space. ⌬so is the spin-orbit coupling and ␭R is the Rashba coupling. The magnitude of the spin-orbit coupling in graphene can be inferred from the known spin-orbit coupling in the carbon atom. This coupling allows for transitions between the pz, px, and py orbitals. An electric field also induces transitions between the pz and s orbitals. These intra-atomic processes mix the ␲ and ␴ bands in 132 Castro Neto et al.: The electronic properties of graphene graphene. Using second-order perturbation theory, one obtains a coupling between the low-energy states in the ␲ band. Tight-binding 共Huertas-Hernando et al., 2006; Zarea and Sandler, 2007兲 and band-structure calculations 共Min et al., 2006; Yao et al., 2007兲 give estimates for the intrinsic and extrinsic spin-orbit interactions in the range 0.01− 0.2 K, and hence much smaller than the other energy scales in the problem 共kinetic, interaction, and disorder兲. III. FLEXURAL PHONONS, ELASTICITY, AND CRUMPLING Graphite, in the Bernal stacking configuration, is a layered crystalline solid with four atoms per unit cell. Its basic structure is essentially a repetition of the bilayer structure discussed earlier. The coupling between layers, as discussed, is weak and, therefore, graphene has always been the starting point for the discussion of phonons in graphite 共Wirtz and Rubio, 2004兲. Graphene has two atoms per unit cell, and if we consider graphene as strictly 2D it should have two acoustic modes 关with dispersion ␻ac共k兲 ⬀ k as k → 0兴 and two optical modes 关with dispersion ␻op共k兲 ⬀ const, as k → 0兴 solely due to the in-plane translation and stretching of the graphene lattice. Nevertheless, graphene exists in the 3D space and hence the atoms can oscillate out-of-plane leading to two out-of-plane phonons 共one acoustic and another optical兲 called flexural modes. The acoustic flexural mode has dispersion ␻flex共k兲 ⬀ k2 as k → 0, which represents the translation of the whole graphene plane 共essentially a one-atom-thick membrane兲 in the perpendicular direction 共free-particle motion兲. The optical flexural mode represents the out-of-phase, out-of-plane oscillation of the neighboring atoms. In first approximation, if we neglect the coupling between graphene planes, graphite has essentially the same phonon modes, although they are degenerate. The coupling between planes has two main effects: 共i兲 it lifts the degeneracy of the phonon modes, and 共ii兲 it leads to a strong suppression of the energy of the flexural modes. Theoretically, flexural modes should become ordinary acoustic and optical modes in a fully covalent 3D solid, but in practice the flexural modes survive due to the fact that the planes are coupled by weak van der Waals–like forces. These modes have been measured experimentally in graphite 共Wirtz and Rubio, 2004兲. Graphene can also be obtained as a suspended membrane, that is, without a substrate, supported only by a scaffold or bridging micrometerscale gaps 共Bunch et al., 2007; Meyer, Geim, Katsnelson, Novoselov, Booth, et al., 2007; Meyer, Geim, Katsnelson, Novoselov, Obergfell, et al., 2007兲. Figure 24 shows a suspended graphene sheet and an atomic resolution image of its crystal lattice. Because the flexural modes disperse like k2, they dominate the behavior of structural fluctuations in graphene at low energies 共low temperatures兲 and long wavelengths. It is instructive to understand how these modes appear from the point of view of elasticity theory 共Chaikin and Lubensky, 1995; Nelson et al., 2004兲. ConRev. Mod. Phys., Vol. 81, No. 1, January–March 2009 (a) (b) FIG. 24. 共Color online兲 Suspended graphene sheet. 共a兲 Brightfield transmission-electron-microscope image of a graphene membrane. Its central part 共homogeneous and featureless region兲 is monolayer graphene. Adapted from Meyer, Geim, Katsnelson, Novoselov, Booth, et al., 2007. 共b兲 Despite only one atom thick, graphene remains a perfect crystal at this atomic resolution. The image is obtained in a scanning transmission electron microscope. The visible periodicity is given by the lattice of benzene rings. Adapted from Booth et al., 2008. sider, for instance, a graphene sheet in 3D and parametrize the position of the sheet relative of a fixed coordinate frame in terms of the in-plane vector r and the height variable h共r兲 so that a position in the graphene is given by the vector R = „r , h共r兲…. The unit vector normal to the surface is given by N= z − ⵜh 共125兲 冑1 + 共ⵜh兲2 , where ⵜ = 共⳵x , ⳵y兲 is the 2D gradient operator and z is the unit vector in the third direction. In a flat graphene configuration, all the normal vectors are aligned and therefore ⵜ · N = 0. Deviations from the flat configuration require misalignment of the normal vectors and cost elastic energy. This elastic energy can be written as E0 = ␬ 2 冕 d2r共ⵜ · N兲2 ⬇ ␬ 2 冕 d2r共ⵜ2h兲2 , 共126兲 where ␬ is the bending stiffness of graphene, and the expression in terms of h共r兲 is valid for smooth distortions of the graphene sheet 关共ⵜh兲2 Ⰶ 1兴. The energy 共126兲 is valid in the absence of a surface tension or a substrate that breaks the rotational and translational symmetry of the problem, respectively. In the presence of tension, 133 Castro Neto et al.: The electronic properties of graphene there is an energy cost for solid rotations of the graphene sheet 共ⵜh ⫽ 0兲 and hence a new term has to be added to the elastic energy, ␥ ET = 2 冕 2 共127兲 2 d r共ⵜh兲 , where ␥ is the interfacial stiffness. A substrate, described by a height variable s共r兲, pins the graphene sheet through van der Waals and other electrostatic potentials so that the graphene configuration tries to follow the substrate h共r兲 ⬃ s共r兲. Deviations from this configuration cost extra elastic energy that can be approximated by a harmonic potential 共Swain and Andelman, 1999兲, ES = v 2 冕 共128兲 d2r关s共r兲 − h共r兲兴2 , where v characterizes the strength of the interaction potential between substrate and graphene. First, consider the free floating graphene problem 共126兲 that we can rewrite in momentum space as E0 = ␬ 兺 k4h−khk . 2 k 共129兲 We now canonically quantize the problem by introducing a momentum operator Pk that has the following commutator with hk: 共130兲 关hk,Pk⬘兴 = i␦k,k⬘ , and we write the Hamiltonian as H=兺 k 再 4 冎 P−kPk ␬k h−khk , + 2 2␴ 共131兲 冉冊 ␬ ␴ 1/2 k2 , 共132兲 which is the long-wavelength dispersion of flexural modes. In the presence of tension, it is easy to see that the dispersion is modified to ␻共k兲 = k 冑 ␬ 2 ␥ k + , ␴ ␴ 共133兲 indicating that the dispersion of flexural modes becomes linear in k, as k → 0, under tension. That is what happens in graphite, where the interaction between layers breaks the rotational symmetry of the graphene layers. Equation 共132兲 also allows us to relate the bending energy of graphene with the Young modulus Y of graphite. The fundamental resonance frequency of a macroscopic graphite sample of thickness t is given by 共Bunch et al., 2007兲 Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 冉冊 Y ␳ 1/2 共134兲 tk2 , where ␳ = ␴ / t is the 3D mass density. Assuming that Eq. 共134兲 works down to the single plane level, that is, when t is the distance between planes, we find 共135兲 ␬ = Yt3 , which provides a simple relationship between the bending stiffness and the Young modulus. Given that Y ⬇ 1012 N / m and t ⬇ 3.4 Å we find ␬ ⬇ 1 eV. This result is in good agreement with ab initio calculations of the bending rigidity 共Lenosky et al., 1992; Tu and Ou-Yang, 2002兲 and experiments in graphene resonators 共Bunch et al., 2007兲. The problem of structural order of a “free-floating” graphene sheet can be fully understood from the existence of the flexural modes. Consider, for instance, the number of flexural modes per unit of area at a certain temperature T, Nph = 冕 1 d 2k 2 nk = 共2␲兲 2␲ 冕 ⬁ dk 0 k e ␤冑␬/␴k2 −1 , 共136兲 where nk is the Bose-Einstein occupation number 共␤ = 1 / kBT兲. For T ⫽ 0, the above integral is logarithmically divergent in the infrared 共k → 0兲 indicating a divergent number of phonons in the thermodynamic limit. For a system with finite size L, the smallest possible wave vector is of the order of kmin ⬃ 2␲ / L. Using kmin as a lower cutoff in the integral 共136兲, we find Nph = where ␴ is graphene’s 2D mass density. From the Heisenberg equations of motion for the operators, it is trivial to find that hk oscillates harmonically with a frequency given by ␻flex共k兲 = ␯共k兲 = 冉 where LT = 冊 1 ␲ 2 2 , 2 ln LT 1 − e−LT/L 2␲ 冉冊 ␬ 冑kBT ␴ 共137兲 1/4 共138兲 is the thermal wavelength of flexural modes. Note that when L Ⰷ LT, the number of flexural phonons in Eq. 共137兲 diverges logarithmically with the size of the system, Nph ⬇ 2␲ LT2 冉 冊 ln L , LT 共139兲 indicating that the system cannot be structurally ordered at any finite temperature. This is nothing but the crumpling instability of soft membranes 共Chaikin and Lubensky, 1995; Nelson et al., 2004兲. For L Ⰶ LT, one finds that Nph goes to zero exponentially with the size of the system indicating that systems with finite size can be flat at sufficiently low temperatures. Note that for ␬ ⬇ 1 eV, ␳ ⬇ 2200 kg/ m3, t = 3.4 Å 共␴ ⬇ 7.5⫻ 10−7 kg/ m2兲, and T ⬇ 300 K, we find LT ⬇ 1 Å, indicating that free-floating graphene should always crumple at room temperature due to thermal fluctuations associated with flexural phonons. Nevertheless, the previous discussion involves only the harmonic 共quadratic part兲 of the problem. Nonlinear effects such as large bending deformations 共Peliti 134 Castro Neto et al.: The electronic properties of graphene and Leibler, 1985兲, the coupling between flexural and in-plane modes 关or phonon-phonon interactions 共Le Doussal and Radzihovsky, 1992; Bonini et al., 2007兲兴, and the presence of topological defects 共Nelson and Peliti, 1987兲 can lead to strong renormalizations of the bending rigidity, driving the system toward a flat phase at low temperatures 共Chaikin and Lubensky, 1995兲. This situation has been confirmed in numerical simulations of free graphene sheets 共Adebpour et al., 2007; Fasolino et al., 2007兲. The situation is rather different if the system is under tension or in the presence of a substrate or scaffold that can hold the graphene sheet. In fact, static rippling of graphene flakes suspended on scaffolds has been observed for single layer as well as bilayers 共Meyer, Geim, Katsnelson, Novoselov, Booth, et al., 2007; Meyer, Geim, Katsnelson, Novoselov, Obergfell, et al., 2007兲. In this case the dispersion, in accordance with Eq. 共133兲, is at least linear in k, and the integral in Eq. 共136兲 converges in the infrared 共k → 0兲, indicating that the number of flexural phonons is finite and graphene does not crumple. We should note that these thermal fluctuations are dynamic and hence average to zero over time, therefore the graphene sheet is expected to be flat under these circumstances. Obviously, in the presence of a substrate or scaffold described by Eq. 共128兲, static deformations of the graphene sheet are allowed. Also, hydrocarbon molecules that are often present on top of free hanging graphene membranes could quench flexural fluctuations, making them static. Finally, one should note that in the presence of a metallic gate the electron-electron interactions lead to the coupling of the phonon modes to the electronic excitations in the gate. This coupling could be partially responsible for the damping of the phonon modes due to dissipative effects 共Seoanez et al., 2007兲 as observed in graphene resonators 共Bunch et al., 2007兲. IV. DISORDER IN GRAPHENE Graphene is a remarkable material from an electronic point of view. Because of the robustness and specificity of the sigma bonding, it is very hard for alien atoms to replace the carbon atoms in the honeycomb lattice. This is one of the reasons why the electron mean free path in graphene can be so long, reaching up to 1 ␮m in the existing samples. Nevertheless, graphene is not immune to disorder and its electronic properties are controlled by extrinsic as well as intrinsic effects that are unique to this system. Among the intrinsic sources of disorder, highlight are surface ripples and topological defects. Extrinsic disorder can come about in many different forms: adatoms, vacancies, charges on top of graphene or in the substrate, and extended defects such as cracks and edges. It is easy to see that from the point of view of single electron physics 关that is, terms that can be added to Eq. 共5兲兴, there are two main terms to which disorder couples. The first one is a local change in the single site energy, Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 Hdd = 兺 Vi共a†i ai + b†i bi兲, 共140兲 i where Vi is the strength of the disorder potential on site Ri, which is diagonal in the sublattice indices and hence, from the point of view of the Dirac Hamiltonian 共18兲, can be written as Hdd = 冕 d 2r Va共r兲⌿̂†a共r兲⌿̂a共r兲, 兺 a=1,2 共141兲 which acts as a chemical potential shift for the Dirac fermions, it that is, shifts locally the Dirac point. Because the density of states vanishes in single-layer graphene, and as a consequence the lack of electrostatic screening, charge potentials may be rather important in determining the spectroscopic and transport properties 共Ando, 2006b; Adam et al., 2007; Nomura and MacDonald, 2007兲. Of particular importance is the Coulomb impurity problem, where Va共r兲 = e2 1 , ⑀0 r 共142兲 with ⑀0 the dielectric constant of the medium. The solution of the Dirac equation for the Coulomb potential in 2D has been studied analytically 共DiVincenzo and Mele, 1984; Biswas et al., 2007; Novikov, 2007a; Pereira, Nilsson, and Castro Neto, 2007; Shytov et al., 2007兲. Its solution has many of the features of the 3D relativistic hydrogen atom problem 共Baym, 1969兲. As in the case of the 3D problem, the nature of the eigenfunctions depends strongly on graphene’s dimensionless coupling constant, g= Ze2 . ⑀ 0v F 共143兲 Note that the coupling constant can be varied by either changing the charge of the impurity Z or modifying the dielectric environment and changing ⑀0. For g ⬍ gc = 21 , the solutions of this problem are given in terms of Coulomb wave functions with logarithmic phase shifts. The local density of states 共LDOS兲 is affected close to the impurity due the electron-hole asymmetry generated by the Coulomb potential. The local charge density decays like 1 / r3 plus fast oscillations of the order of the lattice spacing 关in the continuum limit this gives rise to a Dirac delta function for the density 共Kolezhuk et al., 2006兲兴. As in 3D QED, the 2D problem becomes unstable for g ⬎ gc = 21 leading to supercritical behavior and the socalled fall of electron to the center 共Landau and Lifshitz, 1981兲. In this case, the LDOS is strongly affected by the presence of the Coulomb impurity with the appearance of bound states outside the band and scattering resonances within the band 共Pereira, Nilsson, and Castro Neto, 2007兲 and the local electronic density decays monotonically like 1 / r2 at large distances. Schedin et al. 共2007兲 argued that without a high vacuum environment, these Coulomb effects can be strongly suppressed by large effective dielectric constants due to the presence of a nanometer thin layer of 135 Castro Neto et al.: The electronic properties of graphene absorbed water 共Sabio et al., 2007兲. In fact, experiments in ultrahigh vacuum conditions 共Chen, Jang, Fuhrer, et al., 2008兲 display scattering features in the transport that can be associated with charge impurities. Screening effects that affect the strength and range of the Coulomb interaction are rather nontrivial in graphene 共Fogler, Novikov, and Shklovskii, 2007; Shklovskii, 2007兲 and, therefore, important for the interpretation of transport data 共Bardarson et al., 2007; Nomura et al., 2007; SanJose et al., 2007; Lewenkopf et al., 2008兲. Another type of disorder is the one that changes the distance or angles between the pz orbitals. In this case, the hopping energies between different sites are modified, leading to a new term to the original Hamiltonian 共5兲, † † 共aa兲 † Hod = 兺 兵␦t共ab兲 ij 共ai bj + H.c.兲 + ␦tij 共ai aj + bi bj兲其, i,j 共144兲 or in Fourier space, Hod = ei共k−k⬘兲·R −i␦ 兺 ak† bk⬘ 兺ជ ␦t共ab兲 i i k,k⬘ ជ ·k aa ⬘ + H.c. A共r兲 = Ax共r兲 + iAy共r兲. 共149兲 In terms of the Dirac Hamiltonian 共18兲, we can rewrite Eq. 共146兲 as Hod = 冕 ជ 共r兲⌿̂ 共r兲 + ␾共r兲⌿̂†共r兲⌿̂ 共r兲兴, d2r关⌿̂†1共r兲␴ · A 1 1 1 共150兲 ជ = 共A , A 兲. This result shows that changes in the where A x y ជ hopping amplitude lead to the appearance of vector A and scalar ⌽ potentials in the Dirac Hamiltonian. The presence of a vector potential in the problem indicates ជ = 共c / ev 兲 ⵜ ⫻ A ជ should that an effective magnetic field B F also be present, naively implying a broken time-reversal symmetry, although the original problem was timereversal invariant. This broken time-reversal symmetry is not real since Eq. 共150兲 is the Hamiltonian around only one of the Dirac cones. The second Dirac cone is related to the first by time-reversal symmetry, indicating that the effective magnetic field is reversed in the second cone. Therefore, there is no global broken symmetry but a compensation between the two cones. i,␦ab i共k−k⬘兲·Ri−i␦ab·k⬘ + 共ak† ak⬘ + bk† bk⬘兲 兺 ␦t共aa兲 , 共145兲 i e ជ i,␦ជ aa 共␦t共aa兲 where ␦t共ab兲 ij ij 兲 is the change of the hopping energy between orbitals on lattice sites Ri and Rj on the same 共different兲 sublattices 共we have written R = R + ␦ជ , where j i ␦ជ ab is the nearest-neighbor vector and ␦ជ aa is the nextnearest-neighbor vector兲. Following the procedure of Sec. II.B, we project out the Fourier components of the operators close to the K and K⬘ points of the BZ using Eq. 共17兲. If we assume that ␦tij is smooth over the lattice spacing scale, that is, it does not have a Fourier component with momentum K − K⬘ 共so the two Dirac cones are not coupled by disorder兲, we can rewrite Eq. 共145兲 in real space as Hod = 冕 d2r兵A共r兲a†1共r兲b1共r兲 + H.c. + ␾共r兲关a†1共r兲a1共r兲 + b†1共r兲b1共r兲兴其, 共146兲 with a similar expression for cone 2 but with A replaced by A*, where A共r兲 = 兺 ␦t共ab兲共r兲e−i␦ab·K , 共147兲 ␾共r兲 = 兺 ␦t共aa兲共r兲e−i␦aa·K . 共148兲 ជ ␦ជ ab ជ ␦ជ aa Note that whereas ␾共r兲 = ␾*共r兲, because of the inversion symmetry of the two triangular sublattices that make up the honeycomb lattice, A is complex because of a lack of inversion symmetry for nearest-neighbor hopping. Hence, Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 A. Ripples Graphene is a one-atom-thick system, the extreme case of a soft membrane. Hence, like soft membranes, it is subject to distortions of its structure due to either thermal fluctuations 共as we discussed in Sec. III兲 or interaction with a substrate, scaffold, and absorbands 共Swain and Andelman, 1999兲. In the first case, the fluctuations are time dependent 共although with time scales much longer than the electronic ones兲, while in the second case the distortions act as quenched disorder. In both cases, disorder occurs because of the modification of the distance and relative angle between the carbon atoms due to the bending of the graphene sheet. This type of off-diagonal disorder does not exist in ordinary 3D solids, or even in quasi-1D or quasi-2D systems, where atomic chains and atomic planes, respectively, are embedded in a 3D crystalline structure. In fact, graphene is also different from other soft membranes because it is 共semi兲metallic, while previously studied membranes were insulators. The problem of bending graphitic systems and its effect on the hybridization of the ␲ orbitals has been studied in the context of classical minimal surfaces 共Lenosky et al., 1992兲 and applied to fullerenes and carbon nanotubes 共Tersoff, 1992; Kane and Mele, 1997; Zhong-can et al., 1997; Xin et al., 2000; Tu and Ou-Yang, 2002兲. In order to understand the effect of bending on graphene, consider the situation shown in Fig. 25. The bending of the graphene sheet has three main effects: the decrease of the distance between carbon atoms, a rotation of the pZ orbitals 关compression or dilation of the lattice are energetically costly due to the large spring constant of graphene ⬇57 eV/ Å2 共Xin et al., 2000兲兴, and a rehybridization between ␲ and ␴ orbitals 共Eun-Ah Kim 136 Castro Neto et al.: The electronic properties of graphene z z z1 α topography. Ab initio band-structure calculations also give support to this scenario 共Dharma-Wardana, 2007兲. The connection between the ripples and the electronic problem comes from the relation between the height field h共r兲 and the local curvature of the graphene sheet R, β z2 l + + − d − x 2 ⬇ ⵜ2h共r兲, R共r兲 R R θ FIG. 25. Bending the surface of graphene by a radius R and its effect on the pz orbitals. 共154兲 hence we see that due to bending the electrons are subject to a potential that depends on the structure of a graphene sheet 共Eun-Ah Kim and Castro Neto, 2008兲, V共r兲 ⬇ V0 − ␣关ⵜ2h共r兲兴2 , and Castro Neto, 2008兲. Bending by a radius R decreases the distance between the orbitals from ᐉ to d = 2R sin关ᐉ / 共2R兲兴 ⬇ ᐉ − ᐉ3 / 24R2 for R Ⰷ ᐉ. The decrease in the distance between the orbitals increases the overlap between the two lobes of the pZ orbital 共Harrison, 1980兲: 0 0 关1 + ᐉ2 / 共12R2兲兴, where a = ␲, ␴, and Vppa is the Vppa ⬇ Vppa overlap for a flat graphene sheet. The rotation of the pZ orbitals can be understood within the Slater-Koster formalism, namely, the rotation can be decomposed into a pz − pz 共␲ bond兲 plus a px − px 共␴ bond兲 hybridization with energies Vpp␲ and Vpp␴, respectively 共Harrison, V共␪兲 = Vpp␲ cos2共␪兲 − Vpp␴ sin2共␪兲 ⬇ Vpp␲ − 共Vpp␲ 1980兲: + Vpp␴兲关ᐉ / 共2R兲兴2, leading to a decrease in the overlap. Furthermore, the rotation leads to rehybridization between ␲ and ␴ orbitals leading to a further shift in energy of the order of 共Eun-Ah Kim and Castro Neto, 2 2 2008兲 ␦⑀␲ ⬇ 共Vsp ␴ + Vpp␴兲 / 共⑀␲ − ⑀a兲. In the presence of a substrate, as discussed in Sec. III, elasticity theory predicts that graphene can be expected to follow the substrate in a smooth way. Indeed, by minimizing the elastic energy 共126兲–共128兲 with respect to the height h, we get 共Swain and Andelman, 1999兲 ␬ⵜ4h共r兲 − ␥ⵜ2h共r兲 + vh共r兲 = vs共r兲, 共151兲 which can be solved by Fourier transform, h共k兲 = where ᐉt = ᐉc = s共k兲 , 1 + 共ᐉtk兲2 + 共ᐉck兲4 冉冊 冉冊 ␥ 共152兲 1/2 , v ␬ 1/4 . 共153兲 v Equation 共152兲 gives the height configuration in terms of the substrate profile, and ᐉt and ᐉc provide the length scales for elastic distortion of graphene on a substrate. Hence, disorder in the substrate translates into disorder in the graphene sheet 共albeit restricted by elastic constraints兲. This picture has been confirmed by STM measurements on graphene 共Ishigami et al., 2007; Stolyarova et al., 2007兲 in which strong correlations were found between the roughness of the substrate and the graphene Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 共155兲 where ␣ 共␣ ⬇ 10 eV Å 兲 is the constant that depends on microscopic details. The conclusion from Eq. 共155兲 is that Dirac fermions are scattered by ripples of the graphene sheet through a potential that is proportional to the square of the local curvature. The coupling between geometry and electron propagation is unique to graphene, and results in additional scattering and resistivity due to ripples 共Katsnelson and Geim, 2008兲. 2 B. Topological lattice defects Structural defects of the honeycomb lattice like pentagons, heptagons, and their combinations such as Stone-Wales defect 共a combination of two pentagonheptagon pairs兲 are also possible in graphene and can lead to scattering 共Cortijo and Vozmediano, 2007a, 2007b兲. These defects induce long-range deformations, which modify the electron trajectories. We consider first a disclination. This defect is equivalent to the deletion or inclusion of a wedge in the lattice. The simplest one in the honeycomb lattice is the absence of a 60° wedge. The resulting edges can be glued in such a way that all sites remain threefold-coordinated. The honeycomb lattice is recovered everywhere, except at the apex of the wedge, where a fivefold ring, a pentagon, is formed. One can imagine a situation in which the nearest-neighbor hoppings are unchanged. Nevertheless, the existence of a pentagon implies that the two sublattices of the honeycomb structure can no longer be defined. A trajectory around the pentagon after a closed circuit has to change the sublattice index. The boundary conditions imposed at the edges of a disclination are shown in Fig. 26, identifying sites from different sublattices. In addition, the wave functions at the K and K⬘ points are exchanged when moving around the pentagon. Far away from the defect, a slow rotation of the spinorial wave function components can be described by a gauge field that acts on the valley and sublattice indices 共González et al., 1992, 1993b兲. This gauge field is technically non-Abelian, although a transformation can be defined that makes the resulting Dirac equation equivalent to one with an effective Abelian gauge field 共González et al., 1993b兲. The final continuum equation gives a reasonable description of the electronic spectrum of fullerenes Castro Neto et al.: The electronic properties of graphene 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 FIG. 26. 共Color online兲 Sketch of the boundary conditions associated to a disclination 共pentagon兲 in the honeycomb lattice. with different sizes 共González et al., 1992, 1993b兲, and other structures that contain pentagons 共LeClair, 2000; Osipov et al., 2003; Kolesnikov and Osipov, 2004, 2006; Lammert and Crespi, 2004兲. It is easy to see that a heptagon leads to the opposite effective field. An in-plane dislocation, that is, the inclusion of a semi-infinite row of sites, can be considered as induced by a pentagon and a heptagon together. The nonAbelian field described above is canceled away from the core. A closed path implies a shift by one 共or more兲 lattice spacing. The wave functions at the K and K⬘ points acquire phases, e±2␲i/3, under a translation by one lattice unit. Hence, the description of a dislocation in the continuum limit requires an 共Abelian兲 vortex of charge ±2␲ / 3 at its core. Dislocations separated over distances of the order of d lead to an effective flux through an area of perimeter l of the order of 共Morpurgo and Guinea, 2006兲 ⌽⬃ d , k Fl 2 共156兲 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 mixing of the K and K⬘ wave functions, leading to a gauge field like the one induced by a pentagon 共González et al., 2001兲. Graphene samples with disclinations and dislocations are feasible in particular substrates 共Coraux et al., 2008兲, and gauge fields related to the local curvature are then expected to play a crucial role in such structures. The resulting electronic structure can be analyzed using the theory of quantum mechanics in curved space 共Birrell and Davies, 1982; Cortijo and Vozmediano, 2007a, 2007b; de Juan et al., 2007兲. 137 time ␶elas ⯝ 共vFni兲−1. Hence, the regions with few impurities can be considered low-density metals in the dirty −1 limit as ␶elas ⯝ ⑀ F. The Dirac equation allows for localized solutions that satisfy many possible boundary conditions. It is known that small circular defects result in localized and semilocalized states 共Dong et al., 1998兲, that is, states whose wave function decays as 1 / r as a function of the distance from the center of the defect. A discrete version of these states can be realized in a nearest-neighbor tight-binding model with unitary scatterers such as vacancies 共Pereira et al., 2006兲. In the continuum, the Dirac equation 共19兲 for the wave function ␺共r兲 = „␾1共r兲 , ␾2共r兲… can be written, for E = 0, as ⳵w␾1共w,w*兲 = 0, ⳵w*␾2共w,w*兲 = 0, 共157兲 where w = x + iy is a complex number. These equations indicate that at zero energy the components of the wave function can only be either holomorphic or antiholomorphic with respect to the variable w 关that is, ␾1共w , w*兲 = ␾1共w*兲 and ␾2共w , w*兲 = ␾2共w兲兴. Since the boundary conditions require that the wave function vanishes at infinity, the only possible solutions have the form ⌿K共rជ兲 ⬀ „1 / 共x + iy兲n , 0… or ⌿K⬘共rជ兲 ⬀ „0 , 1 / 共x − iy兲n…. The wave functions in the discrete lattice must be real, and at large distances the actual solution found near a vacancy tends to a superposition of two solutions formed from wave functions from the two valleys with equal weight, in a way similar to the mixing at armchair edges 共Brey and Fertig, 2006b兲. The construction of the semilocalized state around a vacancy in the honeycomb lattice can be extended to other discrete models, which leads to the Dirac equation in the continuum limit. One particular case is the nearest-neighbor square lattice with half flux per plaquette, or the nearest-neighbor square lattice with two flavors per site. The latter has been extensively studied in relation to the effects of impurities on the electronic structure of d-wave superconductors 共Balatsky et al., 2006兲, and numerical results are in agreement with the existence of this solution. As the state is localized on one sublattice only, the solution can be generalized for the case of two vacancies. D. Localized states near edges, cracks, and voids 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 leads to a change in the electronic density of the order of ni. The corresponding shift in the Fermi energy is ⑀F ⯝ vF冑ni. In addition, impurities lead to a finite elastic and to an elastic scattering mean free path lelas ⯝ an−1/2 i Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 Localized states can be defined at edges where the number of atoms in the two sublattices is not compensated. The number of them depend on details of the edge. The graphene edges can be strongly deformed, due to the bonding of other atoms to carbon atoms at the edges. These atoms should not induce states in the graphene ␲ band. In general, a boundary inside the graphene material will exist, as shown in Fig. 27, beyond which the sp2 hybridization is well defined. If this is the 138 Castro Neto et al.: The electronic properties of graphene carbon in regions of size of the order of 共L / a兲2. The resulting system can be considered a metal with a low density of carriers, ncarrier ⬀ a / L per unit cell, and an elastic mean free path lelas ⯝ L. Then, we obtain vF ⑀F ⯝ FIG. 27. 共Color online兲 Sketch of a rough graphene surface. The full line gives the boundary beyond which the lattice can be considered undistorted. The number of midgap states changes depending on the difference in the number of removed sites for two sublattices. case, the number of midgap states near the edge is roughly proportional to the difference in sites between the two sublattices near this boundary. Along a zigzag edge there is one localized state per three lattice units. This implies that a precursor structure for localized states at the Dirac energy can be found in ribbons or constrictions of small lengths 共Muñoz-Rojas et al., 2006兲, which modifies the electronic structure and transport properties. Localized solutions can also be found near other defects that contain broken bonds or vacancies. These states do not allow an analytical solution, although, as discussed above, the continuum Dirac equation is compatible with many boundary conditions, and it should describe well localized states that vary slowly over distances comparable to the lattice spacing. The existence of these states can be investigated by analyzing the scaling of the spectrum near a defect as a function of the size of the system L 共Vozmediano et al., 2005兲. A number of small voids and elongated cracks show states whose energy scales as L−2, while the energy of extended states scales as L−1. A state with energy scaling L−2 is compatible with continuum states for which the modulus of the wave function decays as r−2 as a function of the distance from the defect. E. Self-doping The band-structure calculations discussed in the previous sections show that the electronic structure of a single graphene plane is not strictly symmetrical in energy 共Reich et al., 2002兲. The absence of electron-hole symmetry shifts the energy of the states localized near impurities above or below the Fermi level, leading to a transfer of charge from or to the clean regions. Hence, the combination of localized defects and the lack of perfect electron-hole symmetry around the Dirac points leads to the possibility of self-doping, in addition to the usual scattering processes. Extended lattice defects, like edges, grain boundaries, or microcracks, are likely to induce a number of electronic states proportional to their length L / a, where a is of the order of the lattice constant. Hence, a distribution of extended defects of length L at a distance equal to L itself gives rise to a concentration of L / a carriers per Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 1 ␶elas 冑aL , ⯝ vF , L 共158兲 and, therefore, 共␶elas兲−1 Ⰶ ⑀F when a / L Ⰶ 1. Hence, the existence of extended defects leads to the possibility of self-doping but maintaining most of the sample in the clean limit. In this regime, coherent oscillations of transport properties are expected, although the observed electronic properties may correspond to a shifted Fermi energy with respect to the nominally neutral defect-free system. One can describe the effects that break electron-hole symmetry near the Dirac points in terms of a finite nextnearest-neighbor hopping between ␲ orbitals t⬘ in Eq. 共5兲. Consider, for instance, the electronic structure of a ribbon of width L terminated by zigzag edges, which, as discussed, lead to surface states for t⬘ = 0. The translational symmetry along the axis of the ribbon allows us to define bands in terms of a wave vector parallel to this axis. On the other hand, the localized surface bands, extending from k储 = 共2␲兲 / 3 to k储 = −共2␲兲 / 3, acquire a dispersion of the order of t⬘. Hence, if the Fermi energy remains unchanged at the position of the Dirac points 共⑀Dirac = −3t⬘兲, this band will be filled, and the ribbon will no longer be charge neutral. In order to restore charge neutrality, the Fermi level needs to be shifted by an amount of the order of t⬘. As a consequence, some extended states near the Dirac points are filled, leading to the phenomenon of self-doping. The local charge is a function of distance to the edges, setting the Fermi energy so that the ribbon is globally neutral. Note that the charge transferred to the surface states is localized mostly near the edges of the system. The charge transfer is suppressed by electrostatic effects, as large deviations from charge neutrality have an associated energy cost 共Peres, Guinea, and Castro Neto, 2006a兲. In order to study these charging effects, we add to the free-electron Hamiltonian 共5兲 the Coulomb energy of interaction between electrons, HI = 兺 Ui,jninj , 共159兲 i,j where ni = 兺␴共ai,† ␴ai,␴ + bi,† ␴bi,␴兲 is the number operator at site Ri, and Ui,j = e2 ⑀0兩Ri − Rj兩 共160兲 is the Coulomb interaction between electrons. We expect, on physics grounds, that an electrostatic potential builds up at the edges, shifting the position of the surface states, and reducing the charge transferred to or 139 0.06 0.04 0.5 0.02 0.45 0 0.4 -0.02 0.5005 0 10 20 Charge density Electrostatic potential Castro Neto et al.: The electronic properties of graphene 0.5 -0.04 0 100 200 300 400 500 Position 0.0005 Doping 0.0004 0.0003 0.0002 0.0001 0 0 200 400 800 600 1000 1200 K K K’ K’ 1400 Width FIG. 28. 共Color online兲 Displaced electronic charge close to a graphene zigzag edge. Top: self-consistent analysis of the displaced charge density 共in units of number of electrons per carbon兲 shown as a continuous line, and the corresponding electrostatic potential 共in units of t兲 shown as a dashed line, for a graphene ribbon with periodic boundary conditions along the zigzag edge 共with a length of L = 960a兲 and with a circumference of size W = 80冑3a. Inset: The charge density near the edge. Due to the presence of the edge, there is a displaced charge in the bulk 共bottom panel兲 that is shown as a function of width W. Note that the displaced charge vanishes in the bulk limit 共W → ⬁兲, in agreement with Eq. 共161兲. Adapted from Peres, Guinea, and Castro Neto, 2006a. from them. The potential at the edge induced by a constant doping ␦ per carbon atom is ⬃共␦e2 / a兲W / a 共␦e2 / a is the Coulomb energy per carbon兲, and W the width of the ribbon 共W / a is the number of atoms involved兲. The charge transfer is stopped when the potential shifts the localized states to the Fermi energy, that is, when t⬘ ⬇ 共e2 / a兲共W / a兲␦. The resulting self-doping is therefore ␦⬃ t ⬘a 2 , e 2W 共161兲 which vanishes when W → ⬁. We treat the Hamiltonian 共159兲 within the Hartree approximation 共that is, we replace HI by HM.F. = 兺iVini, where Vi = 兺jUi,j具nj典, and solve the problem selfconsistently for 具ni典兲. Numerical results for graphene ribbons of length L = 80冑3a and different widths are shown in Fig. 28 共t⬘ / t = 0.2 and e2 / a = 0.5t兲. The largest width studied is ⬃0.1 ␮m, and the total number of carbon atoms in the ribbon is ⬇105. Note that as W increases, the self-doping decreases indicating that, for a perfect graphene plane 共W → ⬁兲, the self-doping effect disappears. For realistic parameters, we find that the amount of self-doping is 10−4 − 10−5 electrons per unit cell for sizes 0.1– 1 ␮m. F. Vector potential and gauge field disorder As discussed in Sec. IV, lattice distortions modify the Dirac equation that describes the low-energy band structure of graphene. We consider here deformations that change slowly on the lattice scale, so that they do not Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 FIG. 29. 共Color online兲 Gauge field induced by a simple elastic strain. Top: the hopping along the horizontal bonds is assumed to be changed on the right-hand side of the graphene lattice, defining a straight boundary between the unperturbed and perturbed regions 共dashed line兲. Bottom: the modified hopping acts like a constant gauge field, which displaces the Dirac cones in opposite directions at the K and K⬘ points of the Brillouin zone. The conservation of energy and momentum parallel to the boundary leads to a deflection of electrons by the boundary. mix the two inequivalent valleys. As shown earlier, perturbations that hybridize the two sublattices lead to terms that change the Dirac Hamiltonian from vF␴ · k to vF␴ · k + ␴ · A. Hence, the vector A can be thought of as if induced by an effective gauge field A. In order to preserve time-reversal symmetry, this gauge field must have opposite signs at the two Dirac cones AK = −AK⬘. A simple example is a distortion that changes the hopping between all bonds along a given axis of the lattice. We assume that the sites at the ends of those bonds define the unit cell, as shown in Fig. 29. If the distortion is constant, its only effect is to displace the Dirac points away from the BZ corners. The two inequivalent points are displaced in opposite directions. This uniform distortion is the equivalent of a constant gauge field, which does not change the electronic spectrum. The situation changes if one considers a boundary that separates two domains where the magnitude of the distortion is different. The shift of Dirac points leads to a deflection of the electronic trajectories that cross the boundary, also shown in Fig. 29. The modulation of the gauge field leads to an effective magnetic field, which is of opposite sign for the two valleys. We have shown in Sec. IV.B how topological lattice defects, such as disclinations and dislocations, can be described by an effective gauge field. Those defects can only exist in graphene sheets that are intrinsically curved, and the gauge field only depends on the topology of the lattice. Changes in the nearest-neighbor hopping also lead to effective gauge fields. We next consider two physical processes that induce effective gauge fields: 共i兲 changes in the hopping induced by hybridization between ␲ and ␴ bands, which arise in curved sheets, and 140 Castro Neto et al.: The electronic properties of graphene TABLE II. Estimates of the effective magnetic length, and effective magnetic fields generated by the deformations. The intrinsic curvature entry also refers to the contribution from topological defects. lB 冉冊 l l h Intrinsic curvature Extrinsic curvature Elastic strains B 共T兲 h = 1 nm, l = 10 nm, a = 0.1 nm l 冑 冑 l 0.006 1 al ␤ h2 6 ab A共h兲 x =− 3Eaba2 2 2 关共⳵xh兲 − 共⳵2yh兲2兴, 8 3Eaba2 2 共⳵xh + ⳵2yh兲⳵xh⳵yh, 4 A共h兲 y = 0.06 t l3 E ah2 potential is generated for nearest-neighbor hopping 共u = ␦ជ 兲 共Eun-Ah Kim and Castro Neto, 2008兲, where the coupling constant Eab depends on microscopic details 共Eun-Ah Kim and Castro Neto, 2008兲. The flux of effective magnetic field through a ripple of lateral dimension l and height h is given by ⌽⬇ 共ii兲 changes in the hopping due to modulation in the bond length, which is associated with elastic strain. The strength of these fields depends on parameters that describe the value of the ␲-␴ hybridization, and the dependence of hopping on the bond length. A comparison of the relative strengths of the gauge fields induced by intrinsic curvature, ␲-␴ hybridization 共extrinsic curvature兲, and elastic strains, arising from a ripple of typical height and size, is given in Table II. Eaba2h2 , v Fl 3 共166兲 where the radius of curvature is R−1 ⬇ hl−2. For a ripple with l ⬇ 20 nm, h ⬇ 1 nm, taking Eab / vF ⬇ 10 Å−1, we find ⌽ ⬇ 10−3⌽0. 2. Elastic strain The elastic free energy for graphene can be written in terms of the in-plane displacement u共r兲 = 共ux , uy兲 as F关u兴 = 1 2 冕 冋 d2r 共B − G兲 Ni · Nj ⬇ 1 + Ni · 关共uij · ⵜ兲Ni兴 + 21 Ni · 关共uij · ⵜ兲2Ni兴, 共162兲 where we assume smoothly varying N共r兲. We use Eq. 共125兲 in terms of the height field h共r兲 关N共r兲 ⬇ z − ⵜh共r兲 − 共ⵜh兲2z / 2兴 to rewrite Eq. 共162兲 as 共163兲 Hence, bending of the graphene sheet leads to a modification of the hopping amplitude between different sites of the form t0ij 关共uij · ⵜ兲 ⵜ h共ri兲兴2 , 2 共164兲 where t0ij is the bare hopping energy. A similar effect leads to changes in the electronic states of carbon nanotubes 共Kane and Mele, 1997兲. Using the results of Sec. IV, namely, Eq. 共147兲, we can now show that a vector Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 uii 2 + 2G 兺 i,j=1,2 册 u2ij , 共167兲 As discussed in Sec. IV.A, when the ␲ orbitals are not parallel, the hybridization between them depends on their relative orientation. The angle ␪i determines the relative orientation of neighboring orbitals at some position ri in the graphene sheet. The value of ␪i depends on the local curvature of the layer. The relative angle of rotation of two pz orbitals at position ri and rj can be written as cos共␪i − ␪j兲 = Ni · Nj, where Ni is the unit vector perpendicular to the surface, defined in Eq. 共125兲. If rj = ri + uij, we can write ␦tij ⬇ − 冉兺 冊 i=1,2 1. Gauge field induced by curvature Ni · Nj ⬇ 1 − 21 关共uij · ⵜ兲 ⵜ h共ri兲兴2 . 共165兲 where B is the bulk modulus, G is the shear modulus, and uij = 冉 1 ⳵ui ⳵uj + 2 ⳵xj ⳵xj 冊 共168兲 is the strain tensor 共x1 = x and x2 = y兲. There are many types of static deformation of the honeycomb lattice that can affect the propagation of Dirac fermions. The simplest one is due to changes in the area of the unit cell due to either dilation or contraction. Changes in the unit-cell area lead to local changes in the density of electrons and, therefore, local changes in the chemical potential in the system. In this case, their effect is similar to the one found in Eq. 共148兲, and we have ␾dp共r兲 = g共uxx + uyy兲, 共169兲 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 modify the relative orientation of atoms also lead to an effective gauge field, which acts on each K point separately, as first discussed in relation to carbon nanotubes 共Suzuura and Ando, 2002b; Mañes, 2007兲. Consider two carbon atoms located in two different sublattices in the same unit cell at Ri. The change in the local bond length can be written as Castro Neto et al.: The electronic properties of graphene ␦ui = ␦ជ ab a · 关uA共Ri兲 − uB共Ri + ␦ជ ab兲兴. 共170兲 The local displacements of the atoms in the unit cell can be related to u共r兲 by 共Ando, 2006a兲 共␦ជ ab · ⵜ兲u = ␬−1共uA − uB兲, 共171兲 where ␬ is a dimensionless quantity that depends on microscopic details. Changes in the bond length lead to changes in the hopping amplitude, tij ⬇ t0ij + ⳵tij ␦ui , ⳵a 共172兲 and we can write ␦t共ab兲共r兲 ⬇ ␤ ␦u共r兲 a , 共173兲 where ␤= ⳵t共ab兲 . ⳵ ln共a兲 共174兲 Substituting Eq. 共170兲 into Eq. 共173兲 and the final result into Eq. 共147兲, one finds 共Ando, 2006a兲 3 A共s兲 x = 4 ␤␬共uxx − uyy兲, 3 A共s兲 y = 2 ␤␬uxy . 共175兲 We assume that the strains induced by a ripple of dimension l and height h scale as uij ⬃ 共h / l兲2. Then, using ␤ / vF ⬇ a−1 ⬃ 1 Å−1, we find that the total flux through a ripple is ⌽⬇ h2 . al 共176兲 For ripples such that h ⬃ 1 nm and l ⬃ 20 nm, this estimate gives ⌽ ⬃ 10−1⌽0, in reasonable agreement with the estimates by Morozov et al. 共2006兲. The strain tensor must satisfy some additional constraints, as it is derived from a displacement vector field. These constraints are called Saint-Venant compatibility conditions 共Landau and Lifshitz, 1959兲, Wijkl = ⳵uij ⳵ukl ⳵uil ⳵ujk + − − = 0. ⳵ x k⳵ x l ⳵ x i⳵ x j ⳵ x j⳵ x k ⳵ x i⳵ x l 共177兲 An elastic deformation changes the distances in the crystal lattice and can be considered as a change in the metric, gij = ␦ij + uij . 共178兲 The compatibility equations 共177兲 are equivalent to the condition that the curvature tensor derived from Eq. 共178兲 is zero. Hence, a purely elastic deformation cannot induce intrinsic curvature in the sheet, which only arises from topological defects. The effective fields associated with elastic strains can be large 共Morozov et al., 2006兲, leading to significant changes in the electronic wave functions. An analysis of the resulting state, and the posRev. Mod. Phys., Vol. 81, No. 1, January–March 2009 141 sible instabilities that may occur, can be found in Guinea et al. 共2008兲. 3. Random gauge fields The preceding discussion suggests that the effective fields associated with lattice defects can modify significantly the electronic properties. This is the case when the fields do not change appreciably on scales comparable to the 共effective兲 magnetic length. The general problem of random gauge fields for Dirac fermions has been extensively analyzed before the current interest in graphene, as the topic is also relevant for the IQHE 共Ludwig et al., 1994兲 and d-wave superconductivity 共Nersesyan et al., 1994兲. The one-electron nature of this two-dimensional problem makes it possible, at the Dirac energy, to map it onto models of interacting electrons in one dimension, where many exact results can be obtained 共Castillo et al., 1997兲. The low-energy density of states ␳共␻兲 acquires an anomalous exponent ␳共␻兲 ⬀ 兩␻兩1−⌬, where ⌬ ⬎ 0. The density of states is enhanced near the Dirac energy, reflecting the tendency of disorder to close gaps. For sufficiently large values of the random gauge field, a phase transition is also possible 共Chamon et al., 1996; Horovitz and Doussal, 2002兲. Perturbation theory shows that random gauge fields are a marginal perturbation at the Dirac point, leading to logarithmic divergences. These divergences tend to have the opposite sign with respect to those induced by the Coulomb interaction 共see Sec. V.B兲. As a result, a renormalization-group 共RG兲 analysis of interacting electrons in a random gauge field suggests the possibility of nontrivial phases 共Stauber et al., 2005; Aleiner and Efetov, 2006; Altland, 2006; Dell’Anna, 2006; Foster and Ludwig, 2006a, 2006b; Nomura et al., 2007; Khveshchenko, 2008兲, where interactions and disorder cancel each other. G. Coupling to magnetic impurities Magnetic impurities in graphene can be introduced chemically by deposition and intercalation 共Calandra and Mauri, 2007; Uchoa et al., 2008兲, or self-generated by the introduction of defects 共Kumazaki and Hirashima, 2006, 2007兲. The energy dependence of the density of states in graphene leads to changes in the formation of a Kondo resonance between a magnetic impurity and the graphene electrons. The vanishing of the density of states at the Dirac energy implies that a Kondo singlet in the ground state is not formed unless the exchange coupling exceeds a critical value, of the order of the electron bandwidth, a problem already studied in connection with magnetic impurities in d-wave superconductors 共Cassanello and Fradkin, 1996, 1997; Polkovnikov et al., 2001; Polkovnikov, 2002; Fritz et al., 2006兲. For weak exchange couplings, the magnetic impurity remains unscreened. An external gate changes the chemical potential, allowing for a tuning of the Kondo resonance 共Sengupta and Baskaran, 2008兲. The situation changes significantly if the scalar potential in- 142 Castro Neto et al.: The electronic properties of graphene duced by the magnetic impurity is taken into account. This potential that can be comparable to the bandwidth allows the formation of midgap states and changes the phase shift associated with spin scattering 共Hentschel and Guinea, 2007兲. These phase shifts have a weak logarithmic dependence on the chemical potential, and a Kondo resonance can exist, even close to the Dirac energy. The RKKY interaction between magnetic impurities is also modified in graphene. At finite fillings, the absence of intravalley backscattering leads to a reduction of the Friedel oscillations, which decay as sin共2kFr兲 / 兩r兩3 共Ando, 2006b; Cheianov and Fal’ko, 2006; Wunsch et al., 2006兲. This effect leads to an RKKY interaction at finite fillings, which oscillate and decay as 兩r兩−3. When intervalley scattering is included, the interaction reverts to the usual dependence on distance in two dimensions 兩r兩−2 共Cheianov and Fal’ko, 2006兲. At half filling extended defects lead to an RKKY interaction with an 兩r兩−3 dependence 共Vozmediano et al., 2005; Dugaev et al., 2006兲. This behavior is changed when the impurity potential is localized on atomic scales 共Brey et al., 2007; Saremi, 2007兲, or for highly symmetrical couplings 共Saremi, 2007兲. H. Weak and strong localization In sufficiently clean systems, where the Fermi wavelength is much shorter than the mean free path kFl Ⰷ 1, electronic transport can be described in classical terms, assuming that electrons follow well-defined trajectories. At low temperatures, when electrons remain coherent over long distances, quantum effects lead to interference corrections to the classical expressions for the conductivity, the weak-localization correction 共Bergman, 1984; Chakravarty and Schmid, 1986兲. These corrections are usually due to the positive interference between two paths along closed loops, traversed in opposite directions. As a result, the probability that the electron goes back to the origin is enhanced, so that quantum corrections decrease the conductivity. These interferences are suppressed for paths longer than the dephasing length l␾ determined by interactions between the electron and environment. Interference effects can also be suppressed by magnetic fields that break down time-reversal symmetry and add a random relative phase to the process discussed above. Hence, in most metals, the conductivity increases when a small magnetic field is applied 共negative magnetoresistance兲. Graphene is special in this respect, due to the chirality of its electrons. The motion along a closed path induces a change in the relative weight of the two components of the wave function, leading to a new phase, which contributes to the interference processes. If the electron traverses a path without being scattered from one valley to the other, this 共Berry兲 phase changes the sign of the amplitude of one path with respect to the time-reversed path. As a consequence, the two paths interfere destructively, leading to a suppression of backscattering 共Suzuura and Ando, 2002a兲. Similar processes take place in Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 materials with strong spin-orbit coupling, as the spin direction changes along the path of the electron 共Bergman, 1984; Chakravarty and Schmid, 1986兲. Hence, if scattering between valleys in graphene can be neglected, one expects a positive magnetoresistance, i.e., weak antilocalization. In general, intravalley and intervalley elastic scattering can be described in terms of two different scattering times ␶intra and ␶inter, so that if ␶intra Ⰶ ␶inter, one expects weak antilocalization processes, while if ␶inter Ⰶ ␶intra, ordinary weak localization will take place. Experimentally, localization effects are always strongly suppressed close to the Dirac point but can be partially or, in rare cases, completely recovered at high carrier concentrations, depending on a particular singlelayer sample 共Morozov et al., 2006; Tikhonenko et al., 2008兲. Multilayer samples exhibit an additional positive magnetoresistance in higher magnetic fields, which can be attribued to classical changes in the current distribution due to a vertical gradient of concentration 共Morozov et al., 2006兲 and antilocalization effects 共Wu et al., 2007兲. The propagation of an electron in the absence of intervalley scattering can be affected by the effective gauge fields induced by lattice defects and curvature. These fields can suppress the interference corrections to the conductivity 共Morozov et al., 2006; Morpurgo and Guinea, 2006兲. In addition, the description in terms of free Dirac electrons is only valid near the neutrality point. The Fermi energy acquires a trigonal distortion away from the Dirac point, and backward scattering within each valley is no longer completely suppressed 共McCann et al., 2006兲, leading to further suppression of antilocalization effects at high dopings. Finally, the gradient of external potentials induces a small asymmetry between the two sublattices 共Morpurgo and Guinea, 2006. This effect will also contribute to reduce antilocalization, without giving rise to localization effects. The above analysis has to be modified for a graphene bilayer. Although the description of the electronic states requires a two-component spinor, the total phase around a closed loop is 2␲, and backscattering is not suppressed 共Kechedzhi et al., 2007兲. This result is consistent with experimental observations, which show the existence of weak localization effects in a bilayer 共Gorbachev et al., 2007兲. When the Fermi energy is at the Dirac point, a replica analysis shows that the conductivity approaches a universal value of the order of e2 / h 共Fradkin, 1986a, 1986b兲. This result is valid when intervalley scattering is neglected 共Ostrovsky et al., 2006, 2007; Ryu et al., 2007兲. Localization is induced when these terms are included 共Aleiner and Efetov, 2006; Altland, 2006兲, as also confirmed by numerical calculations 共Louis et al., 2007兲. Interaction effects tend to suppress the effects of disorder. The same result, namely, a conductance of the order of e2 / h, is obtained for disordered graphene bilayers where a self-consistent calculation leads to universal conductivity at the neutrality point 共Nilsson, Castro Neto, Guinea, et al., 2006; Katsnelson, 2007c; Nilsson et al., 2008兲. In a biased graphene bilayer, the presence of impurities leads 143 Castro Neto et al.: The electronic properties of graphene to the appearance of impurity tails in the density of states due to the creation of midgap states, which are sensitive to the applied electric field that opens the gap between the conduction and valence bands 共Nilsson and Castro Neto, 2007兲. We point out that most calculations of transport properties assume self-averaging, that is, one can exchange a problem with a lack of translational invariance by an effective medium system with damping. This procedure only works when the disorder is weak and the system is in the metallic phase. Close to the localized phase this procedure breaks down, the system divides itself into regions of different chemical potential and one has to think about transport in real space, usually described in terms of percolation 共Cheianov, Fal’ko, Altshuler, et al., 2007; Shklovskii, 2007兲. Single electron transistor 共SET兲 measurements of graphene show that this seems to be the situation in graphene at half filling 共Martin et al., 2008兲. Finally, we point out that graphene stacks suffer from another source of disorder, namely, c axis disorder, which is due to either impurities between layers or rotation of graphene planes relative to each other. In either case, the in-plane and out-of-plane transport is directly affected. This kind of disorder has been observed experimentally by different techniques 共Bar et al., 2007; Hass, Varchon, Millán-Otoya, et al., 2008兲. In the case of the bilayer, the rotation of planes changes substantially the spectrum restoring the Dirac fermion description 共Lopes dos Santos et al., 2007兲. The transport properties in the out-of-plane direction are determined by the in† terlayer current operator ĵn,n+1 = it 兺 共cA,n,s cA,n+1,s † − cA,n+1,scA,n,s兲, where n is a layer index and A is a generic index that defines the sites coupled by the interlayer hopping t. If we only consider hopping between nearest-neighbor sites in consecutive layers, these sites belong to one of the two sublattices in each layer. In a multilayer with Bernal stacking, these connected sites are the ones where the density of states vanishes at zero energy, as discussed above. Hence, even in a clean system, the number of conducting channels in the direction perpendicular to the layers vanishes at zero energy 共Nilsson, Castro Neto, Guinea, and Peres, 2006; Nilsson et al., 2008兲. This situation is reminiscent of the in-plane transport properties of a single-layer graphene. Similar to the latter case, a self-consistent Born approximation for a small concentration of impurities leads to a finite conductivity, which becomes independent of the number of impurities. I. Transport near the Dirac point In clean graphene, the number of channels available for electron transport decreases as the chemical potential approaches the Dirac energy. As a result, the conductance through a clean graphene ribbon is at most 4e2 / h, where the factor of 4 stands for the spin and valley degeneracy. In addition, only one out of every three possible clean graphene ribbons has a conduction chanRev. Mod. Phys., Vol. 81, No. 1, January–March 2009 nel at the Dirac energy. The other two-thirds are semiconducting, with a gap of the order of vF / W, where W is the width. This result is a consequence of the additional periodicity introduced by the wave functions at the K and K⬘ points of the Brillouin zone, irrespective of the boundary conditions. A wide graphene ribbon allows for many channels, which can be approximately classified by the momentum perpendicular to the axis of the ribbon, ky. At the Dirac energy, transport through these channels is inhibited by the existence of a gap ⌬ky = vFky. Transport through these channels is suppressed by a factor of the order of e−kyL, where L is the length of the ribbon. The number of transverse channels increases as W / a, where W is the width of the ribbon and a is a length of the order of the lattice spacing. The allowed values of ky are ⬀ny / W, where ny is an integer. Hence, for a ribbon such that W Ⰷ L, there are many channels that satisfy kyL Ⰶ 1. Transport through these channels is not strongly inhibited, and their contribution dominates when the Fermi energy lies near the Dirac point. The conductance arising from these channels is given by 共Katsnelson, 2006b; Tworzydlo et al., 2006兲 G⬃ e2 W h 2␲ 冕 dkye−kyL ⬃ e2 W . h L 共179兲 The transmission at normal incidence ky = 0 is 1, in agreement with the absence of backscattering in graphene, for any barrier that does not induce intervalley scattering 共Katsnelson et al., 2006兲. The transmission of a given channel scales as T共ky兲 = 1 / cosh2共kyL / 2兲. Equation 共179兲 shows that the contribution from all transverse channels leads to a conductance that scales, similar to a function of the length and width of the system, as the conductivity of a diffusive metal. Moreover, the value of the effective conductivity is of the order of e2 / h. It can also be shown that the shot noise depends on current in the same way as in a diffusive metal. A detailed analysis of possible boundary conditions at the contacts and their influence on evanescent waves can be found in Robinson and Schomerus 共2007兲 and Schomerus 共2007兲. The calculations leading to Eq. 共179兲 can be extended to a graphene bilayer. The conductance is, again, a summation of terms arising from evanescent waves between the two contacts, and it has the dependence on sample dimensions of a 2D conductivity of the order of e2 / h 共Snyman and Beenakker, 2007兲, although there is a prefactor twice as big as the one in single-layer graphene. The calculation of the conductance of clean graphene in terms of transmission coefficients, using the Landauer method, leads to an effective conductivity that is equal to the value obtained for bulk graphene using diagrammatic methods, the Kubo formula 共Peres, Guinea, and Castro Neto, 2006b兲, in the limit of zero impurity concentration and zero doping. Moreover, this correspondence remains valid for the case of a bilayer without and with trigonal warping effects 共Koshino and Ando, 2006; Cserti, Csordés, and Dévid, 2007兲. 144 Castro Neto et al.: The electronic properties of graphene Disorder at the Dirac energy changes the conductance of graphene ribbons in two opposite directions 共Louis et al., 2007兲: 共i兲 a sufficiently strong disorder, with shortrange 共intervalley兲 contributions, leads to a localized regime, where the conductance depends exponentially on the ribbon length, and 共ii兲 at the Dirac energy, disorder allows midgap states that can enhance the conductance mediated by evanescent waves discussed above. A fluctuating electrostatic potential also reduces the effective gap for the transverse channels, further enhancing the conductance. The resonant tunneling regime mediated by midgap states was suggested by analytical calculations 共Titov, 2007兲. The enhancement of the conductance by potential fluctuations can also be studied semianalytically. In the absence of intervalley scattering, it leads to an effective conductivity that grows with ribbon length 共San-Jose et al., 2007兲. In fact, analytical and numerical studies 共Bardarson et al., 2007; Nomura et al., 2007; SanJose et al., 2007; Lewenkopf et al., 2008兲, show that the conductivity obeys a universal scaling with the lattice size L, ␴共L兲 = 2e2 关A ln共L/␰兲 + B兴, h 共180兲 where ␰ is a length scale associated with a range of interactions and A and B are numbers of the order of unit 关A ⬇ 0.17 and B ⬇ 0.23 for a graphene lattice in the shape of a square of size L 共Lewenkopf et al., 2008兲兴. Note, therefore, that the conductivity is always of the order of e2 / h and has a weak dependence on size. J. Boltzmann equation description of dc transport in doped graphene It was shown experimentally that the dc conductivity of graphene depends linearly on the gate potential 共Novoselov et al., 2004; Novoselov, Geim, Morozov, et al., 2005; Novoselov, Jiang, Schedin, et al., 2005兲, except very close to the neutrality point 共see Fig. 30兲. Since the gate potential depends linearly on the electronic density n, one has a conductivity ␴ ⬀ n. As shown by Shon and Ando 共1998兲, if the scatterers are short ranged, the dc conductivity should be independent of the electronic density, at odds with the experimental result. It has been shown 共Ando, 2006b; Nomura and MacDonald, 2006, 2007兲 that, by considering a scattering mechanism based on screened charged impurities, it is possible to obtain from a Boltzmann equation approach a conductivity varying linearly with the density, in agreement with the experimental result 共Ando, 2006b; Novikov, 2007b; Peres, Lopes dos Santos, and Stauber, 2007; Trushin and Schliemann, 2007; Katsnelson and Geim, 2008兲. The Boltzmann equation has the form 共Ziman, 1972兲 − vk · ⵱rf共⑀k兲 − e共E + vk ⫻ H兲 · ⵜkf共⑀k兲 = − 冏 冏 ⳵fk ⳵t . scatt 共181兲 The solution of the Boltzmann equation in its general form is difficult and one needs, therefore, to rely upon Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 FIG. 30. 共Color online兲 Changes in conductivity ␴ of graphene with varying gate voltage Vg and carrier concentration n. Here ␴ is proportional to n. Note that samples with higher mobility 共⬎1 m2 / V s兲 normally show a sublinear dependence, presumably indicating the presence of different types of scatterers. Inset: Scanning-electron micrograph of one of experimental devices 共in false colors matching those seen in visible optics兲. The scale of the micrograph is given by the width of the Hall bar, which is 1 ␮m. Adapted from Novoselov, Geim, Morozov, et al., 2005. some approximation. The first step in the usual approximation scheme is to write the distribution as f共⑀k兲 = f0共⑀k兲 + g共⑀k兲, where f0共⑀k兲 is the steady-state distribution function and g共⑀k兲 is assumed small. Inserting this ansatz in into Eq. 共181兲 and keeping only terms that are linear in the external fields, one obtains the linearized Boltzmann equation 共Ziman, 1972兲, which reads − ⳵ f 0共 ⑀ k兲 vk · ⳵⑀k =− 冏 冏 ⳵fk ⳵t 冋冉 − 冊 冉 ⑀k − ␨ 1 ⵱ rT + e E − ⵱ r␨ T e 冊册 + vk · ⵱rgk + e共vk ⫻ H兲 · ⵱kgk . 共182兲 scatt The second approximation has to do with the form of the scattering term. The simplest approach is to introduce a relaxation-time approximation, − 冏 冏 ⳵fk ⳵t → scatt gk , ␶k 共183兲 where ␶k is the relaxation time, assumed to be momentum-dependent. This momentum dependence is determined phenomenologically in such way that the dependence of the conductivity upon the electronic density agrees with experimental data. The Boltzmann equation is certainly not valid at the Dirac point, but since many experiments are performed at finite carrier density, controlled by an external gate voltage, we expect the Boltzmann equation to give reliable results if an appropriate form for ␶k is used 共Adam et al., 2007兲. 145 Castro Neto et al.: The electronic properties of graphene We compute the Boltzmann relaxation time ␶k for two different scattering potentials: 共i兲 a Dirac delta function potential and 共ii兲 an unscreened Coulomb potential. The relaxation time ␶k is defined as 冕 冕 1 = ni ␶k k⬘dk⬘ S共k,k⬘兲共1 − cos ␪兲, 共2␲兲2 d␪ 共184兲 K. Magnetotransport and universal conductivity where ni is impurity concentration per unit of area, and the transition rate S共k , k⬘兲 is given, in the Born approximation, by S共k,k⬘兲 = 2␲兩Hk⬘,k兩2 1 ␦共k⬘ − k兲, vF 共185兲 where vFk is the dispersion of Dirac fermions in graphene and Hk⬘,k is defined as Hk⬘,k = 冕 dr␺k* 共r兲US共r兲␺k共r兲, ⬘ 共186兲 with US共r兲 the scattering potential and ␺k共r兲 is the electronic spinor wave function of a clean graphene sheet. If the potential is short ranged 共Shon and Ando, 1998兲 of the form US = v0␦共r兲, the Boltzmann relaxation time is given by ␶k = 4vF 1 . niv20 k 共187兲 On the other hand, if the potential is the Coulomb potential, given by US共r兲 = eQ / 4␲⑀0⑀r for charged impurities of charge Q, the relaxation time is given by ␶k = vF u20 k, 共188兲 where u20 = niQ2e2 / 16⑀20⑀2. As argued below, the phenomenology of Dirac fermions implies that the scattering in graphene must be of the form 共188兲. Within the relaxation time approximation, the solution of the linearized Boltzmann equation when an electric field is applied to the sample is gk = − ⳵ f 0共 ⑀ k兲 e␶kvk · E, ⳵⑀k tential is one possible mechanism of producing a scattering rate of the form 共188兲, but we do not exclude that other mechanisms may exist 关see, for instance, Katsnelson and Geim 共2008兲兴. 共189兲 The description of the magnetotransport properties of electrons in a disordered honeycomb lattice is complex because of the interference effects associated with the Hofstadter problem 共Gumbs and Fekete, 1997兲. We simplify our problem by describing electrons in the honeycomb lattice as Dirac fermions in the continuum approximation, introduced in Sec. II.B. Furthermore, we focus only on the problem of short-range scattering in the unitary limit since in this regime many analytical results are obtained 共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兲. The problem of magnetotransport in the presence of Coulomb impurities, as discussed, is still an open research problem. A similar approach was considered by Abrikosov in the quantum magnetoresistance study of nonstoichiometric chalcogenides 共Abrikosov, 1998兲. In the case of graphene, the effective Hamiltonian describing Dirac fermions in a magnetic field 共including disorder兲 can be written as H = H0 + Hi, where H0 is given by Eq. 共5兲 and Hi is the impurity potential given by 共Peres, Guinea, and Castro Neto, 2006a兲 Ni Hi = V 兺 ␦共r − rj兲I. The formulation of the problem in second quantization requires the solution of H0, which was done in Sec. II.I. The field operators, close to the K point, are defined as 共the spin index is omitted for simplicity兲 ⌿共r兲 = 兺 k and the electric current reads 共spin and valley indexes included兲 + 冉 冊 兺冑 冉 eikx 冑L 4 兺 e v kg k . A k 共190兲 Since at low temperatures the following relation −⳵f0共⑀k兲 / ⳵⑀k → ␦共␮ − vFk兲 holds, one can see that assuming Eq. 共188兲 where k is measured relatively to the Dirac point, the electronic conductivity turns out to be ␴xx = 2 e2 ␲vF2 e2 ␮2 = 2 n, h u20 h u20 共191兲 where u0 is the strength of the scattering potential 共with dimensions of energy兲. The electronic conductivity depends linearly on the electron density, in agreement with the experimental data. We stress that the Coulomb poRev. Mod. Phys., Vol. 81, No. 1, January–March 2009 0 ␾0共y兲 eikx n,k,␣ J= 共192兲 j=1 ck,−1 ␾n共y − klB2 兲 2 2L ␾n+1共y − klB兲 冊 ck,n,␣ , 共193兲 where ck,n,␣ destroys an electron in band ␣ = ± 1, with 2 ; ck,−1 destroys an energy level n and guiding center klB electron in the zero Landau level; the cyclotron frequency is given by Eq. 共96兲. The sum over n = 0 , 1 , 2 , . . . is cut off at n0 given by E共1 , n0兲 = W, where W is of the order of the electronic bandwidth. In this representation, H0 becomes diagonal, leading to Green’s functions of the form 共in the Matsubara representation兲 G0共k,n, ␣ ;i␻兲 = where 1 , i␻ − E共␣,n兲 共194兲 146 Castro Neto et al.: The electronic properties of graphene E共␣,n兲 = ␣␻c冑n ωc = 0.14 eV, B = 12 T 共195兲 DOS per u.c. (1/eV) are the Landau levels for this problem 共␣ = ± 1 labels the two bands兲. Note that G0共k , n , ␣ ; i␻兲 is effectively k independent, and E共␣ , −1兲 = 0 is the zero-energy Landau level. When expressed in the Landau basis, the scattering Hamiltonian 共192兲 connects Landau levels of negative and positive energy. 0.008 ni = 0.001 0.006 ni = 0.005 ni = 0.0009 0.004 0.002 0 ωc = 0.1 eV, B = 6 T 1. The full self-consistent Born approximation (FSBA) In order to describe the effect of impurity scattering on the magnetoresistance of graphene, the Green’s function for Landau levels in the presence of disorder needs to be computed. In the context of the 2D electron gas, an equivalent study was performed by Ohta and Ando 共Ohta, 1971a, 1971b; Ando, 1974a, 1974b, 1974c, 1975; Ando and Uemura, 1974兲 using the averaging procedure over impurity positions of Duke 共1968兲. Below, the averaging procedure over impurity positions is performed in the standard way, namely, having determined the Green’s function for a given impurity configuration 共r1 , . . . , rNi兲, the position averaged Green’s function is determined from 具G共p,n, ␣ ;i␻ ;r1, . . . ,rNi兲典 ⬅ G共p,n, ␣ ;i␻兲 冋兿 冕 册 Ni = L−2Ni drj j=1 ⫻G共p,n, ␣ ;i␻ ;r1, . . . ,rNi兲. 共196兲 In the presence of Landau levels, the average over impurity positions involves the wave functions of the onedimensional harmonic oscillator. After lengthy algebra, the Green’s function in the presence of vacancies, in the FSBA, can be written as G共p,n, ␣ ; ␻ + 0+兲 = 关␻ − E共n, ␣兲 − ⌺1共␻兲兴−1 , 共197兲 G共p,− 1; ␻ + 0+兲 = 关␻ − ⌺2共␻兲兴−1 , 共198兲 where ⌺1共␻兲 = − ni关Z共␻兲兴−1 , 共199兲 ⌺2共␻兲 = − ni关gcG共p,− 1; ␻ + 0+兲/2 + Z共␻兲兴−1 , 共200兲 Z共␻兲 = gcG共p,− 1; ␻ + 0+兲/2 + gc 兺 G共p,n, ␣ ; ␻ + 0+兲/2, 共201兲 n,␣ 2 is the degeneracy of a Landau level per and gc = Ac / 2␲lB unit cell. One should note that the Green’s functions do not depend upon p explicitly. The self-consistent solution of Eqs. 共197兲–共201兲 gives the density of states, the electron self-energy, and the change of Landau level energy position due to disorder. The effect of disorder on the density of states of Dirac fermions in a magnetic field is shown in Fig. 31. For reference, we note that E共1 , 1兲 = 0.14 eV for B = 14 T, Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 DOS per u.c. (1/eV) 0.008 ni = 0.001 ni = 0.0008 ni = 0.0006 0.006 0.004 0.002 0 -3 -2 -1 0 ω / ωc 1 2 3 FIG. 31. 共Color online兲 Density of states of Dirac fermions in a magnetic field. Top: electronic density of states 共DOS兲 ␳共␻兲 as a function of ␻ / ␻c 共␻c = 0.14 eV兲 in a magnetic field B = 12 T for different impurity concentrations ni. Bottom: ␳共␻兲 as a function of ␻ / ␻c 共␻c = 0.1 eV is the cyclotron frequency兲 in a magnetic field B = 6 T. The solid line shows the DOS in the absence of disorder. The position of the Landau levels in the absence of disorder is shown as vertical lines. The two arrows in the top panel show the position of the renormalized Landau levels 共see Fig. 32兲 given by the solution of Eq. 共202兲. Adapted from Peres, Guinea, and Castro Neto, 2006a. and E共1 , 1兲 = 0.1 eV for B = 6 T. From Fig. 31 we see that, for a given ni, the effect of broadening due to impurities is less effective as the magnetic field increases. It is also clear that the position of Landau levels is renormalized relatively to the nondisordered case. The renormalization of the Landau level position can be determined from poles of Eqs. 共197兲 and 共198兲, ␻ − E共␣,n兲 − Re ⌺共␻兲 = 0. 共202兲 Due to the importance of scattering at low energies, the solution to Eq. 共202兲 does not represent exact eigenstates of the system since the imaginary part of the selfenergy is nonvanishing. However, these energies do determine the form of the density of states, as discussed below. In Fig. 32, the graphical solution to Eq. 共202兲 is given for two different energies 关E共−1 , n兲, with n = 1 , 2兴. It is clear that the renormalization is important for the first Landau level. This result is due to the increase in scattering at low energies, which is already present in the case of zero magnetic field. The values of ␻ satisfying Eq. 共202兲 show up in the density of states as the energy values where the oscillations due to the Landau level quantization have a maximum. In Fig. 31, the position of the renormalized Landau levels is shown in the top panel 共marked by two arrows兲, corresponding to the bare energies E共−1 , n兲, with n = 1 , 2. The importance of this renormalization decreases with the decrease in the number of impurities. This is clear in Fig. 31, where a 147 Castro Neto et al.: The electronic properties of graphene ni = 0.001 ni=0.001 0 0.4 -0.2 0.2 ω−E(-1,1) -0.4 ω−E(-1,2) 2 π KB(ω) h / e 0.2 0.6 0 -6 -4 -2 0 2 4 6 20 0.4 Re Σ1(ω) / ωc -Im Σ1(ω) / ωc 20 B = 12 T B=6T 0.8 -6 -4 -2 0 2 4 10 5 5 -0.2 0 B = 10 T 15 10 0 -0.4 6 0.8 B = 12 T 15 0.2 0.4 20 0 -0.4 -0.2 0 0.2 0.4 0.2 0.4 20 0 -0.2 0.2 -0.4 2 0.2 0.4 π KB(ω) h / e ω Re Σ2(ω) / ωc -Im Σ2(ω) / ωc 0.4 0.6 15 15 B=8T 10 10 5 5 B=6T 0 -6 -4 -2 0 2 ω / ωc 4 6 -6 -4 -2 0 2 ω / ωc 4 6 0 -0.4 FIG. 32. 共Color online兲 Imaginary 共right兲 and real 共left兲 parts of ⌺1共␻兲 共top兲 and ⌺2共␻兲 共bottom兲, in units of ␻c, as a function of ␻ / ␻c. The right panels also show the intercept of ␻ − E共␣ , n兲 with Re⌺共␻兲 as required by Eq. 共202兲. Adapted from Peres, Guinea, and Castro Neto, 2006a. visible shift toward low energies is evident when ni has a small 10% change, from ni = 10−3 to ni = 9 ⫻ 10−4. Studying of the magnetoresistance properties of the system requires calculation of the conductivity tensor. We compute the current-current correlation function, and from it the conductivity tensor is derived. The details of the calculations are presented in Peres, Guinea, and Castro Neto 共2006a兲. If, however, we neglect the real part of the self-energy, assume for Im ⌺i共␻兲 = ⌫ 共i = 1 , 2兲 a constant value, and consider that E共1 , 1兲 Ⰷ ⌫, these results reduce to those of Gorbar et al. 共2002兲. It is instructive to consider first the case ␻ , T → 0, which leads to 关␴xx共0 , 0兲 = ␴0兴 ␴0 = 冋 e2 4 Im ⌺1共0兲/Im ⌺2共0兲 − 1 h␲ 1 + 关Im ⌺1共0兲/␻c兴2 + 册 n0 + 1 , n0 + 1 + 关Im ⌺1共0兲/␻c兴2 共203兲 冋 册 关Im ⌺1共0兲兴2 e2 4 1− , h␲ 共vF⌳兲2 + 关Im ⌺1共0兲兴2 共204兲 where we have introduced the energy cutoff vF⌳. Either when Im ⌺1共0兲 ⯝ Im ⌺2共0兲 and ␻c Ⰷ Im ⌺1共0兲 关or n0 2 Ⰷ Im ⌺1共0兲 / ␻c, ␻c = E共0 , 1兲 = 冑2vF / lB 兴 or when ⌳vF Ⰷ Im ⌺1共0兲, in the absence of an applied field, Eqs. 共203兲 and 共204兲 reduce to ␴0 = 4 e2 , ␲h 共205兲 which is the so-called universal conductivity of graphene 共Fradkin, 1986a, 1986b; Lee, 1993; Ludwig et al., 1994; Nersesyan et al., 1994; Ziegler, 1998; Yang and Nayak, 2002; Katsnelson, 2006b; Peres, Guinea, and Castro Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 0 ω (eV) 0.2 0.4 0 -0.4 -0.2 0 ω (eV) FIG. 33. 共Color online兲 Conductivity kernel K共␻兲 共in units of e2 / ␲h兲 as a function of energy ␻ for different magnetic fields and ni = 10−3. The horizontal lines mark the universal limit of the conductivity per cone ␴0 = 2e2 / ␲h. The vertical lines show the position of the Landau levels in the absence of disorder. Adapted from Peres, Guinea, and Castro Neto, 2006a. Neto, 2006a; Tworzydlo et al., 2006兲. This result was obtained previously by Ando and collaborators using the second-order self-consistent Born approximation 共Shon and Ando, 1998; Ando et al., 2002兲. Because the dc magnetotransport properties of graphene are normally measured with the possibility of tuning its electronic density by a gate potential 共Novoselov et al., 2004兲, it is important to compute the conductivity kernel, since this has direct experimental relevance. In the case ␻ → 0, we write the conductivity ␴xx共0 , T兲 as ␴xx共0,T兲 = where we include a factor of 2 due to the valley degeneracy. In the absence of a magnetic field 共␻c → 0兲, the above expression reduces to ␴0 = -0.2 e2 ␲h 冕 ⬁ −⬁ d⑀ ⳵f共⑀兲 KB共⑀兲, ⳵⑀ 共206兲 where the conductivity kernel KB共⑀兲 is given in the Appendix of Peres, Guinea, and Castro Neto 共2006a兲. The magnetic field dependence of kernel KB共⑀兲 is shown in Fig. 33. One can observe that the effect of disorder is the creation of a region where KB共⑀兲 remains constant before it starts to increase in energy with superimposed oscillations coming from the Landau levels. The same effect, but with the absence of the oscillations, was identified at the level of the self-consistent density of states plotted in Fig. 31. Together with ␴xx共0 , T兲, the Hall conductivity ␴xy共0 , T兲 allows the calculation of the resistivity tensor 共109兲. We now focus on the optical conductivity ␴xx共␻兲 共Peres, Guinea, and Castro Neto, 2006a; Gusynin et al., 2007兲. This quantity can be probed by reflectivity experiments in the subterahertz to midinfrared frequency range 共Bliokh, 2005兲. This quantity is shown in Fig. 34 for different magnetic fields. It is clear that the first peak is controlled by E共1 , 1兲 − E共1 , −1兲, and we have checked that it does not obey any particular scaling form as a function of ␻ / B. On the other hand, as the effect of 148 Castro Neto et al.: The electronic properties of graphene ni = 0.001 3 B = 12 T uop = B = 10 T 1 冑2 共uA − uB兲, 共207兲 π σ(ω) h / e 2 3 c 2.5 b 2.5 a 2 2 1 2 3 4 ω / ωc 5 6 7 π σ(ω) h / e 2 0 2.5 B=6T 2 0 1 2 3 4 ω / ωc 5 6 7 0 2.7 2.6 2.5 2.4 2.3 2.2 2.1 2 1 2 3 4 ω / ωc 5 6 7 B = 12 T B = 10 T B=6T 0.22 0.24 0.28 0.26 1/2 1/2 ω / B (eV/ T ) FIG. 34. 共Color online兲 Frequency-dependent conductivity per cone ␴共␻兲 共in units of e2 / ␲h兲 at T = 10 K and ni = 10−3, as a function of the energy ␻ 共in units of ␻c兲 for different values of the magnetic field B. The vertical arrows in the upper left panel, labeled a, b, and c, show the positions of the transitions between different Landau levels: E共1 , 1兲 − E共−1 , 0兲, E共2 , 1兲 − E共−1 , 0兲, and E共1 , 1兲 − E共1 , −1兲, respectively. The horizontal continuous lines show the value of the universal conductivity. The lower right panel shows the conductivity for different values of magnetic field as a function of ␻ / 冑B. Adapted from Peres, Guinea, and Castro Neto, 2006a. scattering becomes less important, high-energy conductivity oscillations start obeying the scaling ␻ / 冑B, as shown 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 deformations of the graphene sheet due to bending and strain couple to the Dirac fermions via vector potentials. Just as bending has to do with the flexural modes of the graphene sheet 共as discussed in Sec. III兲, strain fields are related to optical and acoustic modes 共Wirtz and Rubio, 2004兲. Given the local displacements of the atoms in each sublattice uA and uB, the electron-phonon coupling has essentially the form discussed previously for static fields. Coupling to acoustic modes is the most straightforward one, since it already appears in the elastic theory. If uac is the acoustic phonon displacement, then the relation between this displacement and the atom displacement is given by Eq. 共171兲, and its coupling to electrons is given by the vector potential 共175兲 in the Dirac equation 共150兲. For optical modes, the situation is slightly different since the optical mode displacement is 共Ando, 2006a, 2007b兲 Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 that is, the bond length deformation vector. To calculate the coupling to electrons, we can proceed as previously and compute the change in the nearest-neighbor hopping energy due to the lattice distortion through Eqs. 共172兲, 共173兲, 共170兲, and 共207兲. Once again the electronphonon interaction becomes a problem of coupling electrons with a vector potential as in Eq. 共150兲, where the components of the vector potential are A共op兲 =− x 冑 3 ␤ op u , 2 a2 y =− A共op兲 y 冑 3 ␤ op u , 2 a2 x 共208兲 where ␤ = ⳵t / ⳵ ln共a兲 was defined in Eq. 共174兲. Note that ជ op = −冑3 / 2共␤ / a2兲␴ជ ⫻ u . A similar expreswe can write A op ជ replaced by −A ជ. sion is valid close to the K’ point with A Optical phonons are important in graphene research because of Raman spectroscopy. The latter has played an important role in the study of carbon nanotubes 共Saito et al., 1998兲 because of the 1D character of these systems, namely, the presence of van Hove singularities in the 1D spectrum leads to colossal enhancements of the Raman signal that can be easily detected, even for a single isolated carbon nanotube. In graphene, the situation is different since its 2D character leads to a much smoother density of states 共except for the van Hove singularity at high energies of the order of the hopping energy t ⬇ 2.8 eV兲. Nevertheless, graphene is an open surface and hence is readily accessible by Raman spectroscopy. In fact, it has played an important role because it allows the identification of the number of planes 共Ferrari et al., 2006; Gupta et al., 2006; Graf et al., 2007; Malard et al., 2007; Pisana et al., 2007; Yan et al., 2007兲 and the study of the optical phonon modes in graphene, particularly the ones in the center of the BZ with momentum q ⬇ 0. Similar studies have been performed in graphite ribbons 共Cancado et al., 2004兲. We consider the effect of Dirac fermions on the optical modes. If one treats the vector potential, electronphonon coupling, Eqs. 共150兲 and 共208兲 up to secondorder perturbation theory, its main effect is the polarization of the electron system by creating electronhole pairs. In the QED language, the creation of electron-hole pairs is called pair 共electron-antielectron兲 production 共Castro Neto, 2007兲. Pair production is equivalent to a renormalization of the phonon propagator by a self-energy that is proportional to the polarization function of the Dirac fermions. The renormalized phonon frequency is given by 共Ando, 2006a, 2007b; Lazzeri and Mauri, 2006; Castro Neto and Guinea, 2007; Saha et al., 2007兲 149 Castro Neto et al.: The electronic properties of graphene 2␤2 ␹共q, ␻0兲, a 2␻ 0 共209兲 where ␻0 is the bare phonon frequency, and the electron-phonon polarization function is given by ␹共q, ␻兲 = 冕 兺 s,s⬘=±1 f关Es共k + q兲兴 − f关Es⬘共k兲兴 dk , 2 共2␲兲 ␻0 − Es共k + q兲 + Es⬘共k兲 + i␩ 2 0.5 ∆Ω0
ΛΩ0  ⍀0共q兲 ⬇ ␻0 − 0 -0.5 -1 -1.5 共210兲 where Es共q兲 is the Dirac fermion dispersion 共s = + 1 for the upper band and s = −1 for the lower band兲, and f关E兴 is the Fermi-Dirac distribution function. For Raman spectroscopy, the response of interest is at q = 0, where only the interband processes such that ss⬘ = −1 共that is, processes between the lower and upper cones兲 contribute. The electron-phonon polarization function can be calculated using the linearized Dirac fermion dispersion 共7兲 and the low-energy density of states 共15兲, ␹共0, ␻0兲 = 6冑3 ␲vF2 ⫻ 冉 冕 ␲vF2 + 冋 冊 1.5 2 (b) 共211兲 v F⌳ − ␮ 冉冏 冏 ␻0/2 + ␮ ␻0 ln + i␲␪共␻0/2 − ␮兲 4 ␻0/2 − ␮ 冊册 , 共212兲 where the cutoff-dependent term is a contribution coming from the occupied states in the lower ␲ band and hence is independent of the chemical potential value. This contribution can be fully incorporated into the bare value of ␻0 in Eq. 共209兲. Hence the relative shift in the phonon frequency can be written as 冉 冏 冏 冊 ␻0/2 + ␮ ␭ ␦␻0 ␮ ⬇− + ln + i␲␪共␻0/2 − ␮兲 , − ␻0 4 ␻0 ␻0/2 − ␮ 共213兲 where ␭= 1 Μ
Ω0 dEE共f关− E兴 − f关E兴兲 0 where we have introduced the cutoff momentum ⌳ 共⬇1 / a兲 so that the integral converges in the ultraviolet. At zero temperature T = 0 we have f关E兴 = ␪共␮ − E兲 and we assume electron doping ␮ ⬎ 0, so that f关−E兴 = 1 共for the case of hole doping, ␮ ⬍ 0, obtained by electron-hole symmetry兲. The integration in Eq. 共211兲 gives ␹共0, ␻0兲 = 0.5 vF⌳ 1 1 − , ␻0 + 2E + i␩ ␻ − 2E + i␩ 6冑3 0 (a) 36冑3 ␤2 ␲ 8Ma2␻0 共214兲 is the dimensionless electron-phonon coupling. Note that Eq. 共213兲 has a real and an imaginary part. The real part represents the actual shift in frequency, while the imaginary part gives the damping of the phonon mode due to pair production 共see Fig. 35兲. There is a clear Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 FIG. 35. Kohn anomaly in graphene. Top: the continuous line is the relative phonon frequency shift as a function of ␮ / ␻0, and the dashed line is the damping of the phonon due to electron-hole pair creation. Bottom: 共a兲 electron-hole process that leads to phonon softening 共␻0 ⬎ 2␮兲, and 共b兲 electron-hole process that leads to phonon hardening 共␻0 ⬍ 2␮兲. change in behavior depending on whether ␮ is larger or smaller than ␻0 / 2. For ␮ ⬍ ␻0 / 2, there is a decrease in the phonon frequency implying that the lattice is softening, while for ␮ ⬎ ␻0 / 2, the lattice hardens. The interpretation for this effect is also given in Fig. 35. On the one hand, if the frequency of the phonon is larger than twice the chemical potential, real electron-hole pairs are produced, leading to stronger screening of the electron-ion interaction, and hence to a softer phonon mode. At the same time, the phonons become damped and decay. On the other hand, if the frequency of the phonon is smaller than twice the chemical potential, the production of electron-hole pairs is halted by the Pauli principle and only virtual excitations can be generated, leading to polarization and lattice hardening. In this case, there is no damping and the phonon is long lived. This result has been observed experimentally by Raman spectroscopy 共Pisana et al., 2007; Yan et al., 2007兲. Electron-phonon coupling has also been investigated theoretically for a finite magnetic field 共Ando, 2007a; Goerbig et al., 2007兲. In this case, resonant coupling occurs due to the large degeneracy of the Landau levels, and different Raman transitions are expected as compared with the zero-field case. The coupling of electrons to flexural modes on a free-standing graphene sheet was discussed by Mariani and von Oppen 共2008兲. 150 Castro Neto et al.: The electronic properties of graphene B. Electron-electron interactions Of all disciplines of condensed-matter physics, the study of electron-electron interactions is probably the most complex since it involves understanding the behavior of a macroscopic number of variables. Hence, the problem of interacting systems is a field in constant motion and we shall not try to give here a comprehensive survey of the problem for graphene. Instead, we focus on a small number of topics that are of current discussion in the literature. Since graphene is a truly 2D system, it is informative to compare it with the more standard 2DEG that has been studied extensively in the past 25 years since the development of heterostructures and the discovery of the quantum Hall effect 关for a review, see Stone 共1992兲兴. At the simplest level, metallic systems have two main kinds of excitations: electron-hole pairs and collective modes such as plasmons. Electron-hole pairs are incoherent excitations of the Fermi sea and a direct result of Pauli’s exclusion principle: an electron inside the Fermi sea with momentum k is excited outside the Fermi sea to a new state with momentum k + q, leaving a hole behind. The energy associated with such an excitation is ␻ = ⑀k+q − ⑀k, and for states close to the Fermi surface 共k ⬇ kF兲 their energy scales linearly with the excitation momentum ␻q ⬇ vFq. In a system with nonrelativistic dispersion such as normal metals and semiconductors, the electron-hole continuum is made out of intraband transitions only and exists even at zero energy since it is always possible to produce electron-hole pairs with arbitrarily low energy close to the Fermi surface, as shown in Fig. 36共a兲. Besides that, the 2DEG can also sustain collective excitations such as plasmons that have dispersion ␻plasmon共q兲 ⬀ 冑q, and exist outside the electron-hole continuum at sufficiently long wavelengths 共Shung, 1986a兲. In systems with relativisticlike dispersion, such as graphene, these excitations change considerably, especially when the Fermi energy is at the Dirac point. In this case, the Fermi surface shrinks to a point and hence intraband excitations disappear and only interband transitions between the lower and upper cones can exist 关see Fig. 36共b兲兴. Therefore, neutral graphene has no electronhole excitations at low energy, instead each electronhole pair costs energy and hence the electron hole occupies the upper part of the energy versus momentum diagram. In this case, plasmons are suppressed and no coherent collective excitations can exist. If the chemical potential is moved away from the Dirac point, then intraband excitations are restored and the electron-hole continuum of graphene shares features of the 2DEG and undoped graphene. The full electron-hole continuum of doped graphene is shown in Fig. 36共c兲, and in this case plasmon modes are allowed. As the chemical potential is moved away from the Dirac point, graphene resembles more and more the 2DEG. These features in the elementary excitations of graphene reflect its screening properties as well. In fact, the polarization and dielectric functions of undoped Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 FIG. 36. 共Color online兲 Electron-hole continuum and collective modes of 共a兲 a 2DEG, 共b兲 undoped graphene, and 共c兲 doped graphene. graphene are different from the ones of the 2DEG 共Lindhard function兲. In the random-phase approximation 共RPA兲, the polarization function can be calculated analytically 共Shung, 1986a; González et al., 1993a, 1994兲, ⌸共q, ␻兲 = q2 4冑vF2 q2 − ␻2 , 共215兲 and hence, for ␻ ⬎ vFq, the polarization function is imaginary, indicating the damping of electron-hole pairs. Note that the static polarization function 共␻ = 0兲 vanishes linearly with q, indicating the lack of screening in the system. This polarization function has also been calculated in the presence of a finite chemical potential 共Shung, 1986a, 1986b; Ando, 2006b; Wunsch et al., 2006; Hwang and Das Sarma, 2007兲. Undoped, clean graphene is a semimetal, with a vanishing density of states at the Fermi level. As a result, the linear Thomas-Fermi screening length diverges, and the long-range Coulomb interaction is not screened. At finite electron density n, the Thomas-Fermi screening length reads ␭TF ⬇ 1 1 1 1 , = 4 ␣ k F 4 ␣ 冑␲ n 共216兲 where ␣= e2 ⑀ 0v F 共217兲 is the dimensionless coupling constant in the problem 关the analog of Eq. 共143兲 in the Coulomb impurity problem兴. Going beyond the linear Thomas-Fermi regime, it 151 Castro Neto et al.: The electronic properties of graphene has been shown that the Coulomb law is modified 共Katsnelson, 2006a; Fogler, Novikov, and Shklovskii, 2007; Zhang and Fogler, 2008兲. The Dirac Hamiltonian in the presence of interactions can be written as H ⬅ − ivF e2 + 2⑀0 冕 冕 vq k FIG. 37. 共Color online兲 Hartree-Fock self-energy diagram that leads to a logarithmic renormalization of the Fermi velocity. d2r⌿̂†共r兲␴ · ⵜ⌿̂共r兲 1 ␳ˆ 共r兲␳ˆ 共r⬘兲, d rd r⬘ 兩r − r⬘兩 2 2 共218兲 where ␳ˆ 共r兲 = ⌿̂†共r兲⌿̂共r兲 共219兲 is the electronic density. One can observe that the Coulomb interaction, unlike in QED, is assumed to be instantaneous since vF / c ⬇ 1 / 300 and hence retardation effects are very small. Moreover, photons propagate in 3D space whereas electrons are confined to the 2D graphene sheet. Hence, the Coulomb interaction breaks the Lorentz invariance of the problem and makes the many-body situation different from the one in QED 共Baym and Chin, 1976兲. Furthermore, the problem depends on two parameters: vF and e2 / ⑀0. Under a dimensional scaling, r → ␭r , t → ␭t , ⌿ → ␭−1⌿, both parameters remain invariant. In RG language, the Coulomb interaction is a marginal variable, whose strength relative to the kinetic energy does not change upon a change in scale. If the units are chosen such that vF is dimensionless, the value of e2 / ⑀0 will also be rendered dimensionless. This is the case in theories considered renormalizable in quantum field theory. The Fermi velocity in graphene is comparable to that in half-filled metals. In solids with lattice constant a, the total kinetic energy per site 1 / ma2, where m is the bare mass of the electron, is of the same order of magnitude as the electrostatic energy e2 / ⑀0a. The Fermi velocity for fillings of the order of unity is vF ⬃ 1 / ma. Hence, e2 / ⑀0vF ⬃ 1. This estimate is also valid in graphene. Hence, unlike in QED, where ␣QED = 1 / 137, the coupling constant in graphene is ␣ ⬃ 1. Despite the fact that the coupling constant is of the order of unity, a perturbative RG analysis can be applied. RG techniques allow us to identify stable fixed points of the model, which may be attractive over a broader range than the one where a perturbative treatment can be rigorously justified. Alternatively, an RG scheme can be reformulated as the process of piecewise integration of high-energy excitations 共Shankar, 1994兲. This procedure leads to changes in the effective lowenergy couplings. The scheme is valid if the energy of the renormalized modes is much larger than the scales of interest. The Hartree-Fock correction due to Coulomb interactions between electrons 共given by Fig. 37兲 gives a logarithmic correction to the electron self-energy 共González et al., 1994兲, Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 k+q ⌺HF共k兲 = 冉冊 ␣ ⌳ , k ln 4 k 共220兲 where ⌳ is a momentum cutoff which sets the range of validity of the Dirac equation. This result remains true even to higher order in perturbation theory 共Mishchenko, 2007兲 and is also obtained in large N expansions 共Rosenstein et al., 1989, 1991; Son, 2007兲 共where N is the number of flavors of Dirac fermions兲, with the only modification the prefactor in Eq. 共220兲. This result implies that the Fermi velocity is renormalized toward higher values. As a consequence, the density of states near the Dirac energy is reduced, in agreement with the general trend of repulsive interactions to induce or increase gaps. This result can be understood from the RG point of view by studying the effect of reducing the cutoff from ⌳ to ⌳ − d⌳ and its effect on the effective coupling. It can be shown that ␣ obeys 共González et al., 1994兲 ⌳ ⳵␣ ␣2 =− . 4 ⳵⌳ 共221兲 Therefore, the Coulomb interaction becomes marginally irrelevant. These features are confirmed by a full relativistic calculation, although the Fermi velocity cannot surpass the velocity of light 共González et al., 1994兲. This result indicates that strongly correlated electronic phases, such as ferromagnetism 共Peres et al., 2005兲 and Wigner crystals 共Dahal et al., 2006兲, are suppressed in clean graphene. A calculation of higher-order self-energy terms leads to a wave-function renormalization, and to a finite quasiparticle lifetime, which grows linearly with quasiparticle energy 共González et al., 1994, 1996兲. The wavefunction renormalization implies that the quasiparticle weight tends to zero as its energy is reduced. A strongcoupling expansion is also possible, assuming that the number of electronic flavors justifies an RPA expansion, keeping only electron-hole bubble diagrams 共González et al., 1999兲. This analysis confirms that the Coulomb interaction is renormalized toward lower values. The enhancement in the Fermi velocities leads to a widening of the electronic spectrum. This is consistent with measurements of the gaps in narrow single-wall nanotubes, which show deviations from the scaling with R−1, where R is the radius, expected from the Dirac equation 共Kane and Mele, 2004兲. The linear dependence of the inverse quasiparticle lifetime with energy is consistent with photoemission experiments in graphite, for energies larger with respect to the interlayer interactions 共Xu et al., 1996; Zhou, Gweon, and Lanzara, 2006; Zhou, Gweon, et al., 2006; Bostwick, Ohta, Seyller, et al., 2007; 152 Castro Neto et al.: The electronic properties of graphene Sugawara et al., 2007兲. Note that in graphite, bandstructure effects modify the lifetimes at low energies 共Spataru et al., 2001兲. The vanishing of the quasiparticle peak at low energies can lead to an energy-dependent renormalization of the interlayer hopping 共Vozmediano et al., 2002, 2003兲. Other thermodynamic properties of undoped and doped graphene can also be calculated 共Barlas et al., 2007; Vafek, 2007兲. Nonperturbative calculations of the long-range interaction effects in undoped graphene show that a transition to a gapped phase is also possible, when the number of electronic flavors is large 共Khveshchenko, 2001; Luk’yanchuk and Kopelevich, 2004; Khveshchenko and Shively, 2006兲. The broken symmetry phase is similar to the excitonic transition found in materials where it becomes favorable to create electron-hole pairs that then form bound excitons 共excitonic transition兲. Undoped graphene cannot have well-defined plasmons, as their energies fall within the electron-hole continuum, and therefore have a significant Landau damping. At finite temperatures, however, thermally excited quasiparticles screen the Coulomb interaction, and an acoustic collective charge excitation can exist 共Vafek, 2006兲. Doped graphene shows a finite density of states at the Fermi level, and the long-range Coulomb interaction is screened. Accordingly, there are collective plasma interactions near q → 0, which disperse as ␻p ⬃ 冑兩q兩, since the system is 2D 共Shung, 1986a, 1986b; Campagnoli and Tosatti, 1989兲. The fact that the electronic states are described by the massless Dirac equation implies that ␻P ⬀ n1/4, where n is the carrier density. The static dielectric constant has a continuous derivative at 2kF, unlike in the case of the 2D electron gas 共Ando, 2006b; Wunsch et al., 2006; Sarma et al., 2007兲. This fact is associated with the suppressed backward scattering in graphene. The simplicity of the band structure of graphene allows analytical calculation of the energy and momentum dependence of the dielectric function 共Wunsch et al., 2006; Sarma et al., 2007兲. The screening of the long-range Coulomb interaction implies that the low-energy quasiparticles show a quadratic dependence on energy with respect to the Fermi energy 共Hwang et al., 2007兲. One way to probe the strength of the electronelectron interactions is via the electronic compressibility. Measurements of the compressibility using a singleelectron transistor 共SET兲 show little sign of interactions in the system, being well fitted by the noninteracting result that, contrary to the two-dimensional electron gas 共2DEG兲 共Eisenstein et al., 1994; Giuliani and Vignale, 2005兲, is positively divergent 共Polini et al., 2007; Martin et al., 2008兲. Bilayer graphene, on the other hand, shares characteristics of the single layer and the 2DEG with a nonmonotonic dependence of the compressibility on the carrier density 共Kusminskiy et al., 2008兲. In fact, bilayer graphene very close to half filling has been predicted to be unstable toward Wigner crystallization 共Dahal et al., 2007兲, just like the 2DEG. Furthermore, according to Hartree-Fock calculations, clean bilayer graphene is unstable toward ferromagnetism 共Nilsson et al., 2006a兲. Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 1. Screening in graphene stacks The electron-electron interaction leads to the screening of external potentials. In a doped stack, the charge tends to accumulate near the surfaces, and its distribution is determined by the dielectric function of the stack in the out-of-plane direction. The same polarizability describes the screening of an external field perpendicular to the layers, similar to the one induced by a gate in electrically doped systems 共Novoselov et al., 2004兲. The self-consistent distribution of charge in a biased graphene bilayer has been studied by McCann 共2006兲. From the observed charge distribution and selfconsistent calculations, an estimate of the band-structure parameters and their relation with the induced gap can be obtained 共Castro, Novoselov, Morozov, et al., 2007兲. In the absence of interlayer hopping, the polarizability of a set of stacks of 2D electron gases can be written as a sum of the screening by the individual layers. Using the accepted values for the effective masses and carrier densities of graphene, this scheme gives a first approximation to screening in graphite 共Visscher and Falicov, 1971兲. The screening length in the out-of-plane direction is of about two graphene layers 共Morozov et al., 2005兲. This model is easily generalizable to a stack of semimetals described by the 2D Dirac equation 共González et al., 2001兲. At half filling, the screening length in all directions diverges, and the screening effects are weak. Interlayer hopping modifies this picture significantly. The hopping leads to coherence 共Guinea, 2007兲. The out-of-plane electronic dispersion is similar to that of a one-dimensional conductor. The out-of-plane polarizability of a multilayer contains intraband and interband contributions. The subbands in a system with Bernal stacking have a parabolic dispersion, when only the nearest-neighbor hopping terms are included. This band structure leads to an interband susceptibility described by a sum of terms like those in Eq. 共228兲, which diverges at half filling. In an infinite system, this divergence is more pronounced at k⬜ = ␲ / c, that is, for a wave vector equal to twice the distance between layers. This effect enhances Friedel-like oscillations in the charge distribution in the out-of-plane direction, which can lead to the changes in the sign of the charge in neighboring layers 共Guinea, 2007兲. Away from half filling, a graphene bilayer behaves, from the point of view of screening, in a way very similar to the 2DEG 共Wang and Chakraborty, 2007b兲. C. Short-range interactions In this section, we discuss the effect of short-range Coulomb interactions on the physics of graphene. The simplest carbon system with a hexagonal shape is the benzene molecule. The value of the Hubbard interaction among ␲ electrons was, for this system, computed long ago by Parr et al. 共1950兲, yielding U = 16.93 eV. For comparison purposes, in polyacetylene the value for the Hubbard interaction is U ⯝ 10 eV and the hopping energy is t ⬇ 2.5 eV 共Baeriswyl et al., 1986兲. These two ex- 153 Castro Neto et al.: The electronic properties of graphene amples show that the value of the on-site Coulomb interaction is fairly large for ␲ electrons. As a first guess for graphene, one can take U to be of the same order as for polyacethylene, with the hopping integral t ⯝ 2.8 eV. Of course in pure graphene the electron-electron interaction is not screened, since the density of states is zero at the Dirac point, and one should work out the effect of Coulomb interactions by considering the bare Coulomb potential. On the other hand, as shown before, defects induce a finite density of states at the Dirac point, which could lead to an effective screening of the long-range Coulomb interaction. We assume that the bare Coulomb interaction is screened in graphene and that Coulomb interactions are represented by the Hubbard interaction. This means that we must add to the Hamiltonian 共5兲 a term of the form HU = U 兺 关a†↑共Ri兲a↑共Ri兲a†↓共Ri兲a↓共Ri兲 Ri + b†↑共Ri兲b↑共Ri兲b†↓共Ri兲b↓共Ri兲兴. 共222兲 The simplest question one can ask is whether this system shows a tendency toward some kind of magnetic order driven by the interaction U. Within the simplest Hartree-Fock approximation 共Peres et al., 2004兲, the instability line toward ferromagnetism is given by U F共 ␮ 兲 = 2 , ␳共␮兲 共223兲 which is merely the Stoner criterion. Similar results are obtained in more sophisticated calculations 共Herbut, 2006兲. At half filling the value for the density of states is ␳共0兲 = 0 and the critical value for UF is arbitrarily large. Therefore, we do not expect a ferromagnetic ground state at the neutrality point of one electron per carbon atom. For other electronic densities, ␳共␮兲 becomes finite producing a finite value for UF. We note that the inclusion of t⬘ does not change these findings, since the density of states remains zero at the neutrality point. The critical interaction strength toward an antiferromagnetic ground state is given by 共Peres et al., 2004兲 UAF共␮兲 = 2 共1/N兲 兺 1/兩E+共k兲兩 k,␮⬎0 , 共224兲 where E+共k兲 is given in Eq. 共6兲. This result gives a finite UAF at the neutrality point 共Sorella and Tosatti, 1992; Martelo et al., 1997兲, UAF共0兲 = 2.23t. 共225兲 Quantum Monte Carlo calculations 共Sorella and Tosatti, 1992; Paiva et al., 2005兲, however, raise its value to UAF共0兲 ⯝ 5t. 共226兲 Taking for graphene the same value for U as in polyacetylene and t = 2.8 eV, one obtains U / t ⯝ 3.6, which puts the system far from the transition toward an antiferromagnet ground state. Yet another possibility is that the system may be in a sort of a quantum spin liquid Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 共Lee and Lee, 2005兲 关as originally proposed by Pauling 共1972兲 in 1956兴 since mean-field calculations give a critical value for U to be of the order of U / t ⯝ 1.7. Whether this type of ground state really exists and whether quantum fluctuations push this value of U toward larger values is not known. 1. Bilayer graphene: Exchange The exchange interaction can be large in an unbiased graphene bilayer with a small concentration of carriers. It was shown that the exchange contribution to the electronic energy of a single graphene layer does not lead to a ferromagnetic instability 共Peres et al., 2005兲. The reason for this is a significant contribution from the interband exchange, which is a term usually neglected in doped semiconductors. This contribution depends on the overlap of the conduction and valence wave functions, and it is modified in a bilayer. The interband exchange energy is reduced in a bilayer 共Nilsson et al., 2006b兲, and a positive contribution that depends logarithmically on the bandwidth in graphene is absent in its bilayer. As a result, the exchange energy becomes negative, and scales as n3/2, where n is the carrier density, similar to the 2DEG. The quadratic dispersion at low energies implies that the kinetic energy scales as n2, again as in the 2DEG. This expansion leads to E = Ekin + Eexc ⬇ ␲vF2 n2 e2n3/2 − . 8t⬜ 27冑␲⑀0 共227兲 Writing n↑ = 共n + s兲 / 2, n↓ = 共n − s兲 / 2, where s is the magnetization, Eq. 共227兲 predicts a second-order transition to a ferromagnetic state for n = 4e4t2 / 81␲3vF4 ⑀0. Higher-order corrections to Eq. 共227兲 lead to a first-order transition at slightly higher densities 共Nilsson et al., 2006b兲. For a ratio ␥1 / ␥0 ⬇ 0.1, this analysis implies that a graphene bilayer should be ferromagnetic for carrier densities such that 兩n兩 ⱗ 4 ⫻ 1010 cm−2. A bilayer is also the unit cell of Bernal graphite, and the exchange instability can also be studied in an infinite system. Taking into account nearest-neighbor interlayer hopping only, bulk graphite should also show an exchange instability at low doping. In fact, there is experimental evidence for a ferromagnetic instability in strongly disordered graphite 共Esquinazi et al., 2002, 2003; Kopelevich and Esquinazi, 2007兲. The analysis described above can be extended to the biased bilayer, where a gap separates the conduction and valence bands 共Stauber et al., 2007兲. The analysis of this case is somewhat different, as the Fermi surface at low doping is a ring, and the exchange interaction can change its bounds. The presence of a gap further reduces 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 ring shaped, the biased bilayer is unstable toward ferromagnetism. 154 Castro Neto et al.: The electronic properties of graphene FIG. 38. 共Color online兲 Sketch of the expected magnetization for a graphene bilayer at half filling. 2. Bilayer graphene: Short-range interactions The band structure of a graphene bilayer, at half filling, leads to logarithmic divergences in different response functions at q = 0. The two parabolic bands that are tangent at k = 0 lead to a susceptibility given by that ␹共qជ , ␻兲 ⬀ 冕 兩qជ 兩⬍⌳ d 2k 1 ␻ − 共vF2 /t兲兩k兩2 冉冑 冊 ⬀ ln ⌳ ␻t/vF2 , 共228兲 where ⌳ ⬃ 冑t2 / vF2 is a high momentum cutoff. These logarithmic divergences are similar to the ones that show up when the Fermi surface of a 2D metal is near a saddle point in the dispersion relation 共González et al., 1996兲. A full treatment of these divergences requires a RG approach 共Shankar, 1994兲. Within a simpler meanfield treatment, however, it is easy to note that the divergence of the bilayer susceptibility gives rise to an instability toward an antiferromagnetic phase, where the carbon atoms that are not connected to the neighboring layers acquire a finite magnetization, while the magnetization of atoms with neighbors in the contiguous layers remains zero. A scheme of the expected ordered state is shown in Fig. 38. D. Interactions in high magnetic fields The formation of Landau levels enhances the effect of interactions due to the quenching of the kinetic energy. This effect is most pronounced at low fillings, when only the lowest levels are occupied. New phases may appear at low temperatures. We consider here phases different from the fractional quantum Hall effect, which has not been observed in graphene so far. The existence of new phases can be inferred from the splitting of the valley or spin degeneracy of the Landau levels, which can be observed in spectroscopy measurements 共Sadowski et al., 2006; Henriksen, Tung, Jiang, et al., 2007兲, or in the appearance of new quantum Hall plateaus 共Zhang et al., 2006; Abanin, Novoselov, Zeitler, et al., 2007; Giesbers et al., 2007; Goswami et al., 2007; Jiang, Zhang, Stormer, et al., 2007兲. Interactions can lead to new phases when their effect overcomes that of disorder. An analysis of the competition between disorder and interactions has been found by Nomura and MacDonald 共2007兲. The energy splitting of the different broken symmetry phases, in a clean system, is determined by lattice effects, so that it is reduced by factors of order a / lB, where a is a length of the order of the lattice spacing and lB is the magnetic length 共AliRev. Mod. Phys., Vol. 81, No. 1, January–March 2009 cea and Fisher, 2006; Goerbig et al., 2006, 2007; Wang et al., 2008兲. The combination of disorder and a magnetic field may also lift the degeneracy between the two valleys, favoring valley-polarized phases 共Abanin, Lee, and Levitov, 2007兲. In addition to phases with enhanced ferromagnetism or with broken valley symmetry, interactions at high magnetic fields can lead to excitonic instabilities 共Gusynin et al., 2006兲 and Wigner crystal phases 共Zhang and Joglekar, 2007兲. When only the n = 0 state is occupied, the Landau levels have all their weight in a given sublattice. Then, the breaking of valley degeneracy can be associated with a charge-density wave, which opens a gap 共Fuchs and Lederer, 2007兲. It is interesting to note that in these phases new collective excitations are possible 共Doretto and Morais Smith, 2007兲. Interactions modify the edge states in the quantum Hall regime. A novel phase can appear when the n = 0 is the last filled level. The Zeeman splitting shifts the electronlike and holelike chiral states, which disperse in opposite directions near the boundary of the sample. The resulting level crossing between an electronlike level with spin antiparallel to the field, and a holelike level with spin parallel to the field, may lead to Luttingerliquid features in the edge states 共Fertig and Brey, 2006; Abanin, Novoselov, Zeitler, et al., 2007兲. VI. CONCLUSIONS Graphene is a unique system in many ways. It is truly 2D, has unusual electronic excitations described in terms of Dirac fermions that move in a curved space, is an interesting mix of a semiconductor 共zero density of states兲 and a metal 共gaplessness兲, and has properties of soft matter. The electrons in graphene seem to be almost insensitive to disorder and electron-electron interactions and have very long mean free paths. Hence, graphene’s properties are different from what is found in usual metals and semiconductors. Graphene has also a robust but flexible structure with unusual phonon modes that do not exist in ordinary 3D solids. In some sense, graphene brings together issues in quantum gravity and particle physics, and also from soft and hard condensed matter. Interestingly enough, these properties can be easily modified with the application of electric and magnetic fields, addition of layers, control of its geometry, and chemical doping. Moreover, graphene can be directly and relatively easily probed by various scanning probe techniques from mesoscopic down to atomic scales, because it is not buried inside a 3D structure. This makes graphene one of the most versatile systems in condensed-matter research. Besides the unusual basic properties, graphene has the potential for a large number of applications 共Geim and Novoselov, 2007兲, from chemical sensors 共Chen, Lin, Rooks, et al., 2007; Schedin et al., 2007兲 to transistors 共Nilsson et al., 2006b; Oostinga et al., 2007兲. Graphene can be chemically and/or structurally modified in order to change its functionality and henceforth its potential applications. Moreover, graphene can be easily obtained Castro Neto et al.: The electronic properties of graphene from graphite, a material that is abundant on the Earth’s surface. This particular characteristic makes graphene one of the most readily available materials for basic research since it frees economically challenged research institutions in developing countries from the dependence of expensive sample-growing techniques. Many of graphene’s properties are currently subject of intense research and debate. Understanding the nature of the disorder and how it affects the transport properties 共a problem of fundamental importance for applications兲, the effect of phonons on electronic transport, the nature of electron-electron interactions, and how they modify its physical properties are research areas that are still in their infancy. In this review, we have only touched the surface. Whereas many papers have been written on monolayer graphene in the past few years, only a small fraction actually deal with multilayers. The majority of the theoretical and experimental efforts have concentrated on the single layer, perhaps because of its simplicity and the natural attraction that a one atom thick material, which can be produced by simple methods in almost any laboratory, creates. Nevertheless, few-layer graphene is equally interesting and unusual with a technological potential, perhaps larger than the single layer. Indeed, the theoretical understanding and experimental exploration of multilayers is far behind the single layer. This is a fertile and open field of research for the future. Finally, we have focused entirely on pure carbon graphene where the band structure is dominated by the Dirac description. Nevertheless, chemical modification of graphene can lead to entirely new physics. Depending on the nature of chemical dopants and how they are introduced into the graphene lattice 共adsorption, substitution, or intercalation兲, there can be many results. Small concentrations of adsorbed alkali metal can be used to change the chemical potential while adsorbed transition elements can lead to strong hybridization effects that affect the electronic structure. In fact, the introducion of d- and f-electron atoms in the graphene lattice may produce a significant enhancement of the electron-electron interactions. Hence, it is easy to envision a plethora of many-body effects that can be induced by doping and have to be studied in the context of Dirac electrons: Kondo effect, ferromagnetism, antiferromagnetism, and charge- and spin-density waves. The study of chemically induced many-body effects in graphene would add a new chapter to the short but fascinating history of this material. Only time will tell, but the potential for more amazement is lurking on the horizon. ACKNOWLEDGMENTS We have benefited immensely from discussions with many colleagues and friends in the last few years, but we would like to especially thank Boris Altshuler, Eva Andrei, Alexander Balatsky, Carlo Beenakker, Sankar Das Sarma, Walt de Heer, Millie Dresselhaus, Vladimir Falko, Andrea Ferrari, Herb Fertig, Eduardo Fradkin, Ernie Hill, Mihail Katsnelson, Eun-Ah Kim, Philip Kim, Rev. Mod. Phys., Vol. 81, No. 1, January–March 2009 155 Valery Kotov, Alessandra Lanzara, Leonid Levitov, Allan MacDonald, Sergey Morozov, Johan Nilsson, Vitor Pereira, Philip Phillips, Ramamurti Shankar, João Lopes dos Santos, Shan-Wen Tsai, Bruno Uchoa, and Maria Vozmediano. N.M.R.P. acknowledges financial support from POCI 2010 via project PTDC/FIS/64404/2006. F.G. was supported by MEC 共Spain兲 Grant No. FIS200505478-C02-01 and EU Contract No. 12881 共NEST兲. A.H.C.N was supported through NSF Grant No. DMR0343790. K.S.N. and A.K.G. were supported by EPSRC 共UK兲 and the Royal Society. 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