The electronic properties of graphene

A. H. Castro Neto, F. Guinea, N. M. R. Peres, K. S. Novoselov, and A. K. Geim

Rev. Mod. Phys. 81, 109 – Published 14 January 2009

Abstract

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 (surface) states in graphene depend on the edge termination (zigzag or armchair) 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.

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DOI:https://doi.org/10.1103/RevModPhys.81.109

Authors & Affiliations

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

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Vol. 81, Iss. 1 — January - March 2009

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Images

Figure 1
(Color online) Graphene (top left) is a honeycomb lattice of carbon atoms. Graphite (top right) can be viewed as a stack of graphene layers. Carbon nanotubes are rolled-up cylinders of graphene (bottom left). Fullerenes $(C_{60})$ are molecules consisting of wrapped graphene by the introduction of pentagons on the hexagonal lattice. From 75.Reuse & Permissions
Figure 2
(Color online) Honeycomb lattice and its Brillouin zone. Left: lattice structure of graphene, made out of two interpenetrating triangular lattices ( $a_{1}$ and $a_{2}$ are the lattice unit vectors, and $δ_{i}$ , $i = 1, 2, 3$ are the nearest-neighbor vectors). Right: corresponding Brillouin zone. The Dirac cones are located at the $K$ and $K^{'}$ points.Reuse & Permissions
Figure 3
(Color online) Electronic dispersion in the honeycomb lattice. Left: energy spectrum (in units of $t$ ) for finite values of $t$ and $t^{'}$ , with $t = 2.7 eV$ and $t^{'} = - 0.2 t$ . Right: zoom in of the energy bands close to one of the Dirac points.Reuse & Permissions
Figure 4
(Color online) 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. (13). $m_{0}$ is the free-electron mass. Adapted from 293.Reuse & Permissions
Figure 5
Density of states per unit cell as a function of energy (in units of $t$ ) computed from the energy dispersion (5), $t^{'} = 0.2 t$ (top) and $t^{'} = 0$ (bottom). 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 $ρ (ϵ) \propto ∣ ϵ ∣$ .Reuse & Permissions
Figure 6
(Color online) Klein tunneling in graphene. Top: schematic of the scattering of Dirac electrons by a square potential. Bottom: definition of the angles $ϕ$ and $θ$ used in the scattering formalism in regions I, II, and III.Reuse & Permissions
Figure 7
(Color online) Angular behavior of $T (ϕ)$ for two different values of $V_{0}$ : $V_{0} = 200 meV$ , dashed line; $V_{0} = 285 meV$ , solid line. The remaining parameters are $D = 110 nm$ (top), $D = 50 nm$ (bottom) $E = 80 meV$ , $k_{F} = 2 π ∕ λ$ , and $λ = 50 nm$ .Reuse & Permissions
Figure 8
(Color online) Energy spectrum (in units of $t$ ) for a graphene ribbon $600 a$ wide, as a function of the momentum $k$ along the ribbon (in units of $1 ∕ \sqrt{3} a$ ), in the presence of confining potential with $V_{0} = 1 eV$ , $λ = 180 a$ .Reuse & Permissions
Figure 9
(Color online) Lattice structure of bilayer graphene, its respective electronic hopping energies, and Brillouin zone. (a) Lattice structure of the bilayer with the various hopping parameters according to the SWM model. The $A$ sublattices are indicated by darker spheres. (b) Brillouin zone. Adapted from 240.Reuse & Permissions
Figure 10
(Color online) Band structure of bilayer graphene of $V = 0$ and $γ_{3} = 0$ .Reuse & Permissions
Figure 11
(Color online) Band structure of bilayer graphene for $V \neq 0$ and $γ_{3} = 0$ .Reuse & Permissions
Figure 12
(Color online) Sketch of the three inequivalent orientations of graphene layers with respect to each other.Reuse & Permissions
Figure 13
(Color online) Electronic bands of graphene multilayers. (a) Biased bilayer. (b) Trilayer with Bernal stacking. (c) Trilayer with orthorhombic stacking. (d) Stack with four layers where the top and bottom layers are shifted in energy with respect to the two middle layers by $+ 0.1 eV$ .Reuse & Permissions
Figure 14
(Color online) Ribbon geometry with zigzag edges.Reuse & Permissions
Figure 15
Sketch of a zigzag termination of a graphene bilayer. As discussed by 72), there is a band of surface states completely localized in the bottom layer, and another surface band which alternates between the two.Reuse & Permissions
Figure 16
(Color online) A piece of a honeycomb lattice displaying both zigzag and armchair edges.Reuse & Permissions
Figure 17
Electronic dispersion for graphene nanoribbons. Left: energy spectrum, as calculated from the tight-binding equations, for a nanoribbon with armchair (top) and zigzag (bottom) edges. The width of the nanoribbon is $N = 200$ unit cells. Only 14 eigenstates are depicted. Right: zoom of the low-energy states shown on the right.Reuse & Permissions
Figure 18
(Color online) SdH oscillations observed in longitudinal resistivity $ρ_{x x}$ 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 $Δ n = 4 B ∕ Φ_{0}$ , where $B$ is the applied field and $Φ_{0}$ is the flux quantum (293).Reuse & Permissions
Figure 19
(Color online) 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 $V_{H}$ and a magnetic flux $Φ$ .Reuse & Permissions
Figure 20
(Color online) 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 $σ_{x y}$ described by Eq. (111). Adapted from 293.Reuse & Permissions
Figure 21
(Color online) Fourteen energy levels of tight-binding electrons in graphene in the presence of a magnetic flux $Φ = Φ_{0} ∕ 701$ , 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 $μ$ .Reuse & Permissions
Figure 22
(Color online) Landau levels of graphene stacks shown in Fig. 13. The applied magnetic field is $1 T$ .Reuse & Permissions
Figure 23
(Color online) 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.Reuse & Permissions
Figure 24
(Color online) Suspended graphene sheet. (a) Bright-field transmission-electron-microscope image of a graphene membrane. Its central part (homogeneous and featureless region) is monolayer graphene. Adapted from 263. (b) 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 53.Reuse & Permissions
Figure 25
Bending the surface of graphene by a radius $R$ and its effect on the $p_{z}$ orbitals.Reuse & Permissions
Figure 26
(Color online) Sketch of the boundary conditions associated to a disclination (pentagon) in the honeycomb lattice.Reuse & Permissions
Figure 27
(Color online) 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.Reuse & Permissions
Figure 28
(Color online) Displaced electronic charge close to a graphene zigzag edge. Top: self-consistent analysis of the displaced charge density (in units of number of electrons per carbon) shown as a continuous line, and the corresponding electrostatic potential (in units of $t$ ) shown as a dashed line, for a graphene ribbon with periodic boundary conditions along the zigzag edge (with a length of $L = 960 a$ ) and with a circumference of size $W = 80 \sqrt{3} a$ . Inset: The charge density near the edge. Due to the presence of the edge, there is a displaced charge in the bulk (bottom panel) that is shown as a function of width $W$ . Note that the displaced charge vanishes in the bulk limit $(W \to \infty)$ , in agreement with Eq. (161). Adapted from 324.Reuse & Permissions
Figure 29
(Color online) 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 (dashed line). 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.Reuse & Permissions
Figure 30
(Color online) Changes in conductivity $σ$ of graphene with varying gate voltage $V_{g}$ and carrier concentration $n$ . Here $σ$ is proportional to $n$ . Note that samples with higher mobility $(> 1 m^{2} ∕ V s)$ normally show a sublinear dependence, presumably indicating the presence of different types of scatterers. Inset: Scanning-electron micrograph of one of experimental devices (in false colors matching those seen in visible optics). The scale of the micrograph is given by the width of the Hall bar, which is $1 μ m$ . Adapted from 293.Reuse & Permissions
Figure 31
(Color online) Density of states of Dirac fermions in a magnetic field. Top: electronic density of states (DOS) $ρ (ω)$ as a function of $ω ∕ ω_{c}$ $(ω_{c} = 0.14 eV)$ in a magnetic field $B = 12 T$ for different impurity concentrations $n_{i}$ . Bottom: $ρ (ω)$ as a function of $ω ∕ ω_{c}$ ( $ω_{c} = 0.1 eV$ is the cyclotron frequency) 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 (see Fig. 32) given by the solution of Eq. (202). Adapted from 324.Reuse & Permissions
Figure 32
(Color online) Imaginary (right) and real (left) parts of $Σ_{1} (ω)$ (top) and $Σ_{2} (ω)$ (bottom), in units of $ω_{c}$ , as a function of $ω ∕ ω_{c}$ . The right panels also show the intercept of $ω - E (α, n)$ with $Re Σ (ω)$ as required by Eq. (202). Adapted from 324.Reuse & Permissions
Figure 33
(Color online) Conductivity kernel $K (ω)$ (in units of $e^{2} ∕ π h$ ) as a function of energy $ω$ for different magnetic fields and $n_{i} = 10^{- 3}$ . The horizontal lines mark the universal limit of the conductivity per cone $σ_{0} = 2 e^{2} ∕ π h$ . The vertical lines show the position of the Landau levels in the absence of disorder. Adapted from 324.Reuse & Permissions
Figure 34
(Color online) Frequency-dependent conductivity per cone $σ (ω)$ (in units of $e^{2} ∕ π h$ ) at $T = 10 K$ and $n_{i} = 10^{- 3}$ , as a function of the energy $ω$ (in units of $ω_{c}$ ) 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: $E (1, 1) - E (- 1, 0)$ , $E (2, 1) - E (- 1, 0)$ , and $E (1, 1) - E (1, - 1)$ , 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 $ω ∕ \sqrt{B}$ . Adapted from 324.Reuse & Permissions
Figure 35
Kohn anomaly in graphene. Top: the continuous line is the relative phonon frequency shift as a function of $μ ∕ ω_{0}$ , and the dashed line is the damping of the phonon due to electron-hole pair creation. Bottom: (a) electron-hole process that leads to phonon softening $(ω_{0} > 2 μ)$ , and (b) electron-hole process that leads to phonon hardening $(ω_{0} < 2 μ)$ .Reuse & Permissions
Figure 36
(Color online) Electron-hole continuum and collective modes of (a) a 2DEG, (b) undoped graphene, and (c) doped graphene.Reuse & Permissions
Figure 37
(Color online) Hartree-Fock self-energy diagram that leads to a logarithmic renormalization of the Fermi velocity.Reuse & Permissions
Figure 38
(Color online) Sketch of the expected magnetization for a graphene bilayer at half filling.Reuse & Permissions

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