Electronic properties of graphene in a strong magnetic field

M. O. Goerbig

Rev. Mod. Phys. 83, 1193 – Published 3 November 2011

Abstract

The basic aspects of electrons in graphene (two-dimensional graphite) exposed to a strong perpendicular magnetic field are reviewed. One of its most salient features is the relativistic quantum Hall effect, the observation of which has been the experimental breakthrough in identifying pseudorelativistic massless charge carriers as the low-energy excitations in graphene. The effect may be understood in terms of Landau quantization for massless Dirac fermions, which is also the theoretical basis for the understanding of more involved phenomena due to electronic interactions. The role of electron-electron interactions both in the weak-coupling limit, where the electron-hole excitations are determined by collective modes, and in the strong-coupling regime of partially filled relativistic Landau levels are presented. In the latter limit, exotic ferromagnetic phases and incompressible quantum liquids are expected to be at the origin of recently observed (fractional) quantum Hall states. Furthermore, the electron-phonon coupling in a strong magnetic field is discussed. Although the present review has a dominant theoretical character, a close connection with available experimental observation is intended.

20 More

Received 1 July 2010

DOI:https://doi.org/10.1103/RevModPhys.83.1193

Authors & Affiliations

M. O. Goerbig

Laboratoire de Physique des Solides, Université Paris-Sud, CNRS UMR 8502, F-91405 Orsay, France

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 83, Iss. 4 — October - December 2011

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
Electronic configurations for carbon in the ground state (left panel) and in the excited state (right panel).Reuse & Permissions
Figure 2
(a) Schematic view of the $s p^{2}$ hybridization. The orbitals form angles of 120°. (b) Benzene molecule ( $C_{6} H_{6}$ ). The six carbon atoms are situated at the corners of a hexagon and form covalent bonds with the H atoms. (c) Quantum-mechanical ground state of the benzene ring: a superposition of two configurations that differ by the positions of the $π$ bonds. (d) Graphene viewed as a tiling of benzene hexagons, where the H atoms are replaced by C atoms of neighboring hexagons and the $π$ electrons are delocalized over the whole structure.Reuse & Permissions
Figure 3
(a) Honeycomb lattice. The vectors $δ_{1}$ , $δ_{2}$ , and $δ_{3}$ connect NN carbon atoms, separated by a distance $a = 0.142 nm$ . The vectors $a_{1}$ and $a_{2}$ are basis vectors of the triangular Bravais lattice. (b) Reciprocal lattice of the triangular lattice. Its primitive lattice vectors are $a_{1}^{*}$ and $a_{2}^{*}$ . The shaded region represents the first Brillouin zone (BZ), with its center $Γ$ and the two inequivalent corners $K$ (black squares) and $K^{'}$ (white squares). The thick part of the border of the first BZ represents those points that are counted in its definition so that no points are doubly counted. The first BZ, defined in a strict manner, is thus the shaded region plus the thick part of the border. For completeness, we have also shown the three inequivalent crystallographic points $M$ , $M^{'}$ , and $M^{''}$ (white triangles).Reuse & Permissions
Figure 4
Tight-binding model for the honeycomb lattice.Reuse & Permissions
Figure 5
Energy dispersion as a function of the wave-vector components $k_{x}$ and $k_{y}$ , obtained within the tight-binding approximation, for $t_{NNN} / t = 0.1$ . The valence ( $π$ ) band is distinguished from the conduction ( $π^{*}$ ) band. The Fermi level is situated at the points where the $π$ band touches the $π^{*}$ band. The energy is measured in units of $t$ and the wave vector in units of $1 / a$ .Reuse & Permissions
Figure 6
Relation between band index $λ$ , valley pseudospin $ξ$ , and chirality $η$ in graphene.Reuse & Permissions
Figure 7
Contours of constant-(positive-) energy in reciprocal space. (a) Contours obtained from the full dispersion relation (21). The dashed line corresponds to the energy $t + t_{NNN}$ , which separates closed orbits around the $K$ and $K^{'}$ points (black lines, with energy $ϵ < t + t_{NNN}$ ) from those around the $Γ$ point (gray lines, with energy $ϵ > t + t_{NNN}$ ). (b) Comparison of the contours at energy $ϵ = 1$ , 1.5, and 2 eV around the $K^{'}$ point. The black lines correspond to the energies calculated from the full dispersion relation (21) and the gray ones to those calculated to second order within the continuum limit (54).Reuse & Permissions
Figure 8
Quinoid-type deformation of the honeycomb lattice; the bonds parallel to the deformation axis (double arrow) are modified. The shaded region indicates the unit cell of the oblique lattice, spanned by the lattice vectors $a_{1}$ and $a_{2}$ . Dashed and dash-dotted lines indicate next-nearest neighbors, with characteristic hopping integrals $t_{NNN}$ and $t_{NNN}^{'}$ , respectively, which are different due to the lattice deformation.Reuse & Permissions
Figure 9
Band dispersion of the quinoid-type deformed honeycomb lattice, for a lattice distortion of $δ a / a = - 0.4$ , with $t = 3 eV$ , $t_{NNN} / t = 0.1$ , $\partial t / \partial a = - 5 eV / Å$ , and $\partial t_{NNN} / \partial a = - 0.7 eV / Å$ . The inset shows a close up on one of the Dirac points $D^{'}$ .Reuse & Permissions
Figure 10
Topological semimetal-insulator transition in the model (64) driven by the gap parameter $Δ$ . (a) Two well-separated Dirac cones for $Δ ≪ 0$ , as for graphene. (b) The Dirac points move towards a single point when the modulus of the (negative) gap parameter is lowered. (c) The two Dirac points merge into a single point at the transition ( $Δ = 0$ ). The band dispersion remains linear in the $q_{y}$ direction while it becomes parabolic in the $q_{x}$ direction. (d) Beyond the transition ( $Δ > 0$ ), the (parabolic) bands are separated by a band gap $Δ$ (insulating phase). From 158.Reuse & Permissions
Figure 11
(a) Relativistic Landau levels as a function of the magnetic field. (b) Semiclassical picture of cyclotron motion described by the cyclotron coordinate $η$ , where the charged particle turns around the guiding center $R$ . The gray region depicts the uncertainty on the guiding center, as indicated by Eq. (98).Reuse & Permissions
Figure 12
Transmission spectroscopy on epitaxial multilayer graphene. The inset shows a representative transmission spectrum. The the positions of the absorption lines as a function of the square root of the magnetic field. The dashed lines correspond to transitions calculated at linear order, in agreement with the Dirac equation; one notes downward deviations in the high-energy limit. From 180.Reuse & Permissions
Figure 13
Diagrammatic representation of the interaction vertex [we use the shorthand notation $ν_{i} = (λ_{i} n_{i}, m_{i})$ for the quantum numbers]: (a) vertex associated with terms of the form $v_{λ_{1} n_{1}, \dots, λ_{4} n_{4}}^{ξ, ξ, ξ^{'}, - ξ^{'}} (q)$ or $v_{λ_{1} n_{1}, \dots, λ_{4} n_{4}}^{ξ, - ξ, ξ^{'}, ξ^{'}} (q)$ ; (b) vertex of umklapp type, $v_{λ_{1} n_{1} \dots λ_{4} n_{4}}^{ξ, ξ, - ξ, - ξ} (q)$ , (c) vertex of backscattering type, $v_{λ_{1} n_{1}, \dots, λ_{4} n_{4}}^{ξ, - ξ, - ξ, ξ} (q)$ ; and (d) vertex respecting the SU(2) valley-pseudospin symmetry $v_{λ_{1} n_{1}, \dots, λ_{4} n_{4}}^{ξ, - ξ, ξ^{'}, - ξ^{'}} (q)$ .Reuse & Permissions
Figure 14
Zero-field particle-hole excitation spectrum for doped graphene. (a) Possible intraband (I) and interband (II) single-pair excitations in doped graphene. The excitations close to the Fermi energy may have a wave-vector transfer comprised between $q = 0$ (Ia) and $q = 2 q_{F}$ (Ib), in terms of the Fermi wave vector $q_{F}$ . (b) Spectral function $Im Π_{0} (q, ω)$ in the wave-vector–energy plane. The regions corresponding to intraband and interband excitations are denoted by I and II, respectively.Reuse & Permissions
Figure 15
(a) Particle-hole bubble diagram (polarizability) in terms of Green’s functions $G (q, ω)$ (lines). (b) Spectral function $Im Π_{RPA} (q, ω)$ for doped graphene in the wave-vector–energy plane. The electron-electron interactions are taken into account within the random-phase approximation (RPA). We have chosen $α_{G} = 1$ here.Reuse & Permissions
Figure 16
Particle-hole excitation spectrum for graphene in a perpendicular magnetic field. We have chosen $N_{F} = 3$ in the CB and a LL broadening of (a) $δ = 0.05 ℏ v_{F} / l_{B}$ and (b) $δ = 0.2 ℏ v_{F} / l_{B}$ . The ultraviolet cutoff is chosen such that $N_{c} = 70$ .Reuse & Permissions
Figure 17
Particle-hole excitation spectrum for graphene in a perpendicular magnetic field. The Coulomb interaction is taken into account within the RPA. We have chosen $N_{F} = 3$ in the CB and a LL broadening of (a) $δ = 0.05 ℏ v_{F} / l_{B}$ and (b) $δ = 0.2 ℏ v_{F} / l_{B}$ . The ultraviolet cutoff is chosen such that $N_{c} = 70$ .Reuse & Permissions
Figure 18
Static polarization function $Π^{0} (q, ω = 0)$ for noninteracting electrons in graphene (a) without and (b) with a magnetic field. To compare the two cases, we have chosen a Fermi wave vector $q_{F} = \sqrt{2 N_{F} + 1} / l_{B} = \sqrt{7} / l_{B}$ that corresponds to $N_{F} = 3$ . The full black line represents the total polarizability, whereas the dotted and the dashed lines show the intraband and the interband contributions, respectively. From 186.Reuse & Permissions
Figure 19
Effective coupling constant $α_{G}^{*}$ as a function of the bare coupling $α_{G}$ . The dashed line indicates the asymptotic value $2 / π$ obtained for large values of the bare coupling ( $r_{s} ≫ 1$ ).Reuse & Permissions
Figure 20
Static dielectric function for graphene in the IQHE regime for $N_{F} = 1$ , 2, and 3 (in increasing order). From 186.Reuse & Permissions
Figure 21
Electron-phonon coupling in graphene. (a) Optical phonons in graphene, with a wave vector $q$ in the vicinity of the $Γ$ point at the center of the first BZ [see (c)]. The amplitude of the LO phonon is in the direction of propagation, that of the TO phonon perpendicular to it. The optical phonons modify the bond lengths of the honeycomb lattice. (b) As a consequence of the modified bond lengths, the electronic hopping is varied, and the electron-phonon coupling is off diagonal in the sublattice index.Reuse & Permissions
Figure 22
(a) Anticrossing of coupled phonon-magneto-exciton modes as a function of the magnetic-field. From 77. (b) Raman spectra as a function of the magnetic-field. The continuous white lines indicate the magnetic-field for which the phonon is in resonance with an inter-LL excitation of energy $Δ_{n}$ . Top: data for the full $B$ -field range. Bottom: zoom on the range from 0 to 10 T. From 62.Reuse & Permissions
Figure 23
(a) Completely filled topmost LL. Because of the Pauli principle, the only possible excitations are inter-LL transitions. (b) Partially filled LL. For sufficiently small values of $r_{s}$ (or $α_{G}$ ), the inter-LL excitations constitute high-energy degrees of freedom that may be omitted at low energies, where the relevant degrees couple states within the same LL.Reuse & Permissions
Figure 24
(a) Comparison between the relativistic (black curves) and nonrelativistic (gray curves) potentials for the LLs $n = 0$ , 1, and 5 in real space. The dashed line shows the potential in $n = 0$ , which is the same in both the relativistic and nonrelativistic cases. (b) Pseudopotentials for $n = 0$ , $n = 1$ relativistic, and $n = 1$ nonrelativistic. The lines are a guide to the eye. The open circles represent pseudopotentials with even relative pair angular momentum that are irrelevant in the case of completely spin-valley polarized electronic states. The energies are given in units of $e^{2} / ϵ l_{B}$ . From 76.Reuse & Permissions
Figure 25
Excitations of the SU(2) ferromagnetic state. (a) Spin waves. Such an excitation can be continuously deformed into the ferromagnetic ground state [spin represented on the Bloch sphere (right panel) by the thick arrow]: the gray curve can be shrunk into a single point. (b) Skyrmion with nonzero topological charge. The excitation consists of a reversed spin at the origin $z = 0$ , and the ferromagnetic state is recovered at large distances $| z / ζ | \to \infty$ . In contrast to spin-wave excitations, the spin explores the whole surface of the Bloch sphere and cannot be transformed by a continuous deformation into the majority spin (fat arrow). From 72.Reuse & Permissions
Figure 26
Bloch spheres for entangled spin-pseudospin systems. Bloch sphere for the (a) spin, (b) pseudospin, and (c) a third type of spin representing the entanglement. In the case of spin-pseudospin entanglement ( $| \cos α | \neq 1$ ), the (pseudo)spin magnetizations explore the interior of their respective spheres (black arrows). From 47.Reuse & Permissions
Figure 27
Possible scenarios for the lifted spin-valley degeneracy at $ν_{G} = 0$ . (a) $Δ_{Z} > Δ_{kek}$ in the bulk. When approaching the edge, the energy difference between the two valleys increases drastically, and two levels $(K^{'}, ↑)$ and $(K, ↓)$ cross the Fermi energy at the edge depicted by the dashed line (quantum Hall state). (b) $Δ_{kek} > Δ_{Z}$ in the bulk. The $K$ subbranches are already located above the Fermi energy, and those of $K^{'}$ below, such that the energy difference is simply increased when approaching the edge with no states crossing the Fermi energy (insulator).Reuse & Permissions

Reviews of Modern Physics