Two-Dimensional Gas of Massless Dirac Fermions in Graphene

From the tight-binding approximation, we solve for the spectrum of energy bands in monolayer graphene. Near the points K and K' where the energy dispersion is 0, we expand the Hamiltonian and reveal a linear Hamiltonian that resembles the Dirac equation for massless fermions where the velocity is approximately one three-hundreths the speed of light. We observe a property analogous to spin in regular spin one-half systems known as pseudospin. Lastly, we identify the unique Landau levels in graphene that characterize graphene's quantized conductance in the quantum Hall effect. This was a term paper for the MIT quantum physics III undergraduate course,

Nature, 2009

In graphene, which is an atomic layer of crystalline carbon, two of the distinguishing properties of the material are the charge carriers' two-dimensional and relativistic character. The first experimental evidence of the two-dimensional nature of graphene came from the observation of a sequence of plateaus in measurements of its transport properties in the presence of an applied magnetic field 1,2. These are signatures of the so-called integer quantum Hall effect. However, as a consequence of the relativistic character of the charge carriers, the integer quantum Hall effect observed in graphene is qualitatively different from its semiconductor analogue 3. As a third distinguishing feature of graphene, it has been conjectured that interactions and correlations should be important in this material, but surprisingly, evidence of collective behaviour in graphene is lacking. In particular, the quintessential collective quantum behaviour in two dimensions, the fractional quantum Hall effect (FQHE), has so far resisted observation in graphene despite intense efforts and theoretical predictions of its existence 4-9. Here we report the observation of the FQHE in graphene. Our observations are made possible by using suspended graphene devices probed by two-terminal charge transport measurements 10. This allows us to isolate the sample from substrate-induced perturbations that usually obscure the effects of interactions in this system and to avoid effects of finite geometry. At low carrier density, we find a field-induced transition to an insulator that competes with the FQHE, allowing its observation only in the highest quality samples. We believe that these results will open the door to the physics of FQHE and other collective behaviour in graphene.

PRB 69, 075104 (04); V.P. Gusynin, S.G. Sharapov, PRB 71, 125124 (05), PRL 95, 146801 (05); cond-mat/0512157; V.P. Gusynin, S.G. Sharapov, J.P. Carbotte, cond-mat/0603267.

Physical Review B, 2006

The analytical expressions for both diagonal and off-diagonal ac and dc conductivities of graphene placed in an external magnetic field are derived. These conductivities exhibit rather unusual behavior as functions of frequency, chemical potential and applied field which is caused by the fact that the quasiparticle excitations in graphene are Dirac-like. One of the most striking effects observed in graphene is the odd integer quantum Hall effect. We argue that it is caused by the anomalous properties of the Dirac quasiparticles from the lowest Landau level. Other quantities such as Hall angle and Nernst signal also exhibit rather unusual behavior, in particular when there is an excitonic gap in the spectrum of the Dirac quasiparticle excitations.

Journal of Physics A: Mathematical and Theoretical, 2007

We study the Euclidean effective action per unit area and the charge density for a Dirac field in a two-dimensional spatial region, in the presence of a uniform magnetic field perpendicular to the 2D-plane, at finite temperature and density. In the limit of zero temperature we reproduce, after performing an adequate Lorentz boost, the Hall conductivity measured for different kinds of graphene samples, depending upon the phase choice in the fermionic determinant.

Physical Review Letters, 2005

Monolayer graphite films, or graphene, have quasiparticle excitations that can be described by 2 + 1 dimensional Dirac theory. We demonstrate that this produces an unconventional form of the quantized Hall conductivity σxy = −(2e 2 /h)(2n + 1) with n = 0, 1, . . ., that notably distinguishes graphene from other materials where the integer quantum Hall effect was observed. This unconventional quantization is caused by the quantum anomaly of the n = 0 Landau level and was discovered in recent experiments on ultrathin graphite films.

We show how the two-dimensional Dirac oscillator model can describe the dynamics of electrons in graphene. This model explains the origin of the left-handed chirality observed for electrons in monolayer and bilayer graphene. The relativistic dispersion relation observed for monolayer graphene is obtained directly from the energy spectrum, while the parabolic dispersion relation observed for the case of bilayer graphene is obtained in the non-relativistic limit. Additionally, if an external magnetic field is applied, the unusual Landau-level spectrum for monolayer graphene is obtained, but for bilayer graphene the model predicts the existence of a magnetic field-dependent gap. Finally, this model also leads to the existence of a chiral phase transition.

Nature Physics, 2006

T here are two known distinct types of the integer quantum Hall effect. One is the conventional quantum Hall effect, characteristic of two-dimensional semiconductor systems 1,2 , and the other is its relativistic counterpart observed in graphene, where charge carriers mimic Dirac fermions characterized by Berry's phase π, which results in shifted positions of the Hall plateaus 3-9 . Here we report a third type of the integer quantum Hall effect. Charge carriers in bilayer graphene have a parabolic energy spectrum but are chiral and show Berry's phase 2π affecting their quantum dynamics. The Landau quantization of these fermions results in plateaus in Hall conductivity at standard integer positions, but the last (zero-level) plateau is missing. The zero-level anomaly is accompanied by metallic conductivity in the limit of low concentrations and high magnetic fields, in stark contrast to the conventional, insulating behaviour in this regime. The revealed chiral fermions have no known analogues and present an intriguing case for quantum-mechanical studies. provides a schematic overview of the quantum Hall effect (QHE) behaviour observed in bilayer graphene by comparing it with the conventional integer QHE. In the standard theory, each filled single-degenerate Landau level contributes one conductance quantum e 2 /h towards the observable Hall conductivity (here e is the electron charge and h is Planck's constant). The conventional QHE is shown in , where plateaus in Hall conductivity σ xy make up an uninterrupted ladder of equidistant steps. In bilayer graphene, QHE plateaus follow the same ladder but the plateau at zero σ xy is markedly absent . Instead, the Hall conductivity undergoes a double-sized step across this region. In addition, longitudinal conductivity σ xx in bilayer graphene remains of the order of e 2 /h, even at zero σ xy . The origin of the unconventional QHE behaviour lies in the coupling between two graphene layers, which transforms massless Dirac fermions, characteristic of single-layer graphene 3-9 , into a new type of chiral quasiparticle. Such quasiparticles have an ordinary parabolic spectrum ε(p) = p 2 /2m with effective mass m, but

Solid State Communications, 2007

Graphene is the first example of truly two-dimensional crystals-it's just one layer of carbon atoms. It turns out to be a gapless semiconductor with unique electronic properties resulting from the fact that charge carriers in graphene demonstrate charge-conjugation symmetry between electrons and holes and possess an internal degree of freedom similar to "chirality" for ultrarelativistic elementary particles. It provides unexpected bridge between condensed matter physics and quantum electrodynamics (QED). In particular, the relativistic Zitterbewegung leads to the minimum conductivity of order of conductance quantum e 2 /h in the limit of zero doping; the concept of Klein paradox (tunneling of relativistic particles) provides an essential insight into electron propagation through potential barriers; vacuum polarization around charge impurities is essential for understanding of high electron mobility in graphene; index theorem explains anomalous quantum Hall effect.

Nature, 2005

When electrons are confined in two-dimensional materials, quantum-mechanically enhanced transport phenomena such as the quantum Hall effect can be observed. Graphene, consisting of an isolated single atomic layer of graphite, is an ideal realization of such a two-dimensional system. However, its behaviour is expected to differ markedly from the well-studied case of quantum wells in conventional semiconductor interfaces. This difference arises from the unique electronic properties of graphene, which exhibits electron-hole degeneracy and vanishing carrier mass near the point of charge neutrality 1,2 . Indeed, a distinctive half-integer quantum Hall effect has been predicted 3-5 theoretically, as has the existence of a non-zero Berry's phase (a geometric quantum phase) of the electron wavefunction-a consequence of the exceptional topology of the graphene band structure 6,7 . Recent advances in micromechanical extraction and fabrication techniques for graphite structures 8-12 now permit such exotic two-dimensional electron systems to be probed experimentally. Here we report an experimental investigation of magneto-transport in a high-mobility single layer of graphene. Adjusting the chemical potential with the use of the electric field effect, we observe an unusual halfinteger quantum Hall effect for both electron and hole carriers in graphene. The relevance of Berry's phase to these experiments is confirmed by magneto-oscillations. In addition to their purely scientific interest, these unusual quantum transport phenomena may lead to new applications in carbon-based electronic and magneto-electronic devices.