Quantum Oscillations in Graphene in the Presence of Disorder

We present a theoretical study of magnetotransport in a graphene monolayer taking into account the effects of disorder. The density of states (DOS) and conductivity found for graphene and compared to those found in dimensional electron gas (2DEG) or normal metals. The self-energy due to impurities is calculated self-consistently in Born approximation, and depends strongly on the frequency and field strength, resulting in asymmetric peaks in the density of states at the Landau level energies. The effect of the self-energy and Shubnikov-de Haas oscillations is considered. Introduction Recent advances of nanotechnology have made the creation and investigation of two dimensional carbon, called graphene, possible [Novoselov et al., 2004]. It is a monolayer of carbon atoms packed densely in a honeycomb structure. Monolayer graphene is a gapless semiconductor with conical touching of electron and hole bands [Wallece, 1947]. In spite of being few atom thick, these systems were found to be stable and ready for exploration. One of the most intriguing property of graphene is, that its charge carriers are well described by the relativistic Diracs equation, and are two-dimensional Dirac fermions [Semenoff, 1984]. This opens the possibility of investigating relativistic phenomena at a speed of ∼ 106m/s (the Fermi velocity of graphene), 1/300th the speed of light. This difference in the nature of the quasiparticles in graphene from conventional 2DEG (or normal metals) has given rise to a host of new and unusual phenomena. Theory The honeycomb lattice can be described in terms of two triangular sublattices, A and B (see Fig.1a). A unit cell contains two atoms, one of type A and one of type B. The vectors a1 = a ( 1 2 , √ 3 2 ) , a2 = a ( 1 2 ,− √ 3 2 ) , (1) shown there are primitive translations, where the lattice constant a = |a1| = |a2| = √ 3aCC and aCC is the distance between two nearest carbon atoms. The corresponding reciprocal lattice whose vectors are b1 = 2π a (1, 1/ √ 3) and b2 = 2π a (1,−1/ √ 3) is shown in Fig.1b together with the reduced (symmetrical and extended) Brillouin zone. The reciprocal vectors satisfy the relation ai · bj = 2πδij . Any A atom at the position n = a1n1 + a2n2, where n1, n2 are integers, is connected to its nearest neighbors on B sites by the three vectors δi: δ1 = (a1 − a2)/3, δ2 = a1/3 + 2a2/3, δ3 = −δ1 − δ2 = −2a1/3− a2/3. (2) The carbon atoms in the graphene plane are connected by strong covalent σ bonds due to the sp2 hybridization of the atomic 2s, 2px, 2py orbitals. The 2pz (π) orbitals are perpendicular to the plane and have a weak overlap. Therefore, we start with the simplest tight-binding description for π orbitals of carbon in terms of the Hamiltonian H = −t ∑