WDS'9 Proceedings of Contributed Papers, Part III, 97, 9. ISBN 978-8-778-- MATFYZPRESS Quantum Oscillations in Graphene in the Presence of Disorder D. Iablonskyi Taras Shevchenko National University of Kyiv, Kyiv, Ukraine. Abstract. 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 (DEG) 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., ]. 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, 97]. 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, 98]. This opens the possibility of investigating relativistic phenomena at a speed of 6 m/s (the Fermi velocity of graphene), /th the speed of light. This difference in the nature of the quasiparticles in graphene from conventional DEG (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.a). A unit cell contains two atoms, one of type A and one of type B. The vectors ( ) ( ) a = a,, a = a,, () shown there are primitive translations, where the lattice constant a = a = a = a CC and a CC is the distance between two nearest carbon atoms. The correspond- ing reciprocal lattice whose vectors are b = π a (,/ ) and b = π a (, / ) is shown in Fig.b together with the reduced (symmetrical and extended) Brillouin zone. The reciprocal vectors satisfy the relation a i b j = πδ ij. Any A atom at the position n = a n +a n, where n,n are integers, is connected to its nearest neighbors on B sites by the three vectors δ i : δ = (a a )/, δ = a / + a /, δ = δ δ = a / a /. () The carbon atoms in the graphene plane are connected by strong covalent σ - bonds due to the sp hybridization of the atomic s,p x,p y orbitals. The p z (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 [ ( ) ] ie a n,σ exp hc δ ia b n+δ,σ + h.c., () n,δ i,σ 97
a b a b K K K a A b B Figure. (a) Graphene hexagonal lattice constructed as a superposition of two triangular lattices A and B, with basis vectors a, for lattice A and vectors δ i, i =,,, connecting A to B. (b) The green hexagon is a Brillouin zone (BZ) and pink diamond is the extended BZ for the honeycomb lattice. The reciprocal lattice vectors are b,. where t is the nearest neighbor hopping parameter, a n,σ and b n+δ,σ are the Fermi operators of electrons with spin σ =, on A and B sublattices, respectively. Since we are interested in the current response, the vector potential A is introduced in the Hamiltonian () by means of the Peierls substitution a n,σb m,σ a n,σ exp ( ie hc n m Adr ) b m,σ, that introduces the phase factor exp( ie hc δ ia)in the hopping term. Accordingly, because the graphene structure contains two atoms per unit cell (two sublattices), the spectrum of quasiparticles excitations has two branches (bands) E(k) = ±t + cos k xa + cos k xa cos ky a. () The Lagrangian density of noninteracting quasiparticles in a single graphene sheet in the continuum limit reads L = Ψ σ (t,r)[iγ ( h t iµ σ ) + i hv F γ D x + i hv F γ D y ]Ψ σ (t,r) () σ=± with D α = α +ie/ hca α, α = x,y written in the coordinate representation. In what follows, we consider the setup when the external constant magnetic field B = A is applied perpendicular to the graphene plane along the positive z axis. The magnetic field also enters the Zeeman term included via µ σ. The theoretical explanation of the basic experiments which proved that Dirac quasiparticles exist in graphene is grounded in the Dirac Lagrangian (). Self-consistent Born approximation (SCBA) The Landau level energy eigenvalues are given by E m = sgn(m) hω m, where ω = v F eb/ hc is the cyclotron frequency of Dirac electrons and m =, ±, ±,... The Landau level spectrum for Dirac electrons is significantly different ) from the spectrum for electrons in normal metals which is given as E m = hω c (m + where ω c = eb/m c c is the cyclotron frequency and m =,,,... Since the gap between the relativistic Landau levels decreases for higher Landau levels, disorder will have stronger effect in contrast to the equally spaced non-relativistic Landau levels. Hence, in the semiclassical limit of higher Landau levels, the damping will be stronger for the relativistic case. For the sake of simplicity consider the short range impurity potential. Following Ando [Ando, 97], the self-consistency equation (Fig.) for the self-energy Σ() in SCBA can be written as Σ() = h ω π F τ N c m= N c E m Σ() (6) 98
where τ being the mean free time of quasiparticles (this parameter controls the concentration of the impurities). Note that we have introduced a cutoff N c, which is of the order of the bandwidth. It determines the limit of the applicability of the linearized Dirac spectra. Figure. The Dyson equation and self-energy in the self-consistent Born approximation. Let Σ() = hω(a + ib) (7) Ignoring the arguments of a and b, self-consistency equation becomes hω c a ib = h [ N c F τ m= (a + ib) (a + ib) m ] a + ib (8) The sum in (8) can be calculated using the Poison summation formula, and, to leading orders in ( h/ F τ), a and b are given by a [ ( ) h h hω F τ b F τ e π ( ωcτ F ) πyf sin( )] B + φ [ + e π ( ωcτ F ) cos( )] πyf F ωτ B + φ where φ arctan( h/ F τ), y = ( h/ F τ), F() = A() hc/πe and the area of the Fermi pocket is given by A() = π(/ hv F ). Density of states (DOS) From definition density of states is (9) () ρ() = πω N c m= N c IG m, G m = m Σ() () where Ω = πl and l = hc/eb. Without impurities, the density of states consists of Dirac-delta peaks located at zero frequency and at Landau levels. By introducing impurities in the system, we expect the broadening and shift of these levels, and it can be determined from the solution of the self consistency equations (see Fig.). Conductivity Using the Kubo formula the longitudinal dc conductivity σ xx becomes σ xx = d f() K xx() () where f() is the Fermi-Dirac distribution function and the kernel K xx () is K xx () = (e hω) π h N f Ig m ( + i)ig m+ ( + i) () m= 99
. ρ(e) for E=sgn(n)hω n / with Σ=Σ(E) ρ(e) for E=sgn(n)hω n / with Σ= iγ ρ(e) for E=hω c (n+/) with Σ=Σ(E) ρ(e) for E=hω c (n+/) with Σ= iγ Density of States, πl hωρ(e).... Energy, E/hω Figure. Density of states of graphene and DEG with the self-energy in SCBA and with the self-energy is a constant. The remaining parameters are v F = 6 m/s, B = T, τ = 9 s. In the above equation g m () = Σ ( Σ) m( hω) () At T =, σ xx (T = ) = K xx ( F ). After using the Poisson s summation formula the conductivity (see Fig. and Fig.) can be written as { ( )} σ xx σ + e π πyf(f ) ωcτ cos () B where Conclusion σ = σ + (ω c τ), σ = e N f F τ π h (6) We have studied the effect of impurities in two dimensional Dirac fermions in the presence of quantizing. The energy spectrum depends on the level index n as n, as opposed to the (n+ /) linear dependence in normal metals. The self-energy in the full Born-approximation obeys self-consistency conditions, resulting in important magnetic field and frequency dependence of scattering rate and level shift. By increasing the field, oscillations become visible, corresponding to Landau levels. By further increasing the field, these become separated from each other, and clean gaps appear between the levels, in which intragap states, small islands show up at high field. The broadening of Landau levels and the intragap features differ from previous studies assuming a constant scattering rate, and should be detected experimentally in graphene. Both the electric conductance shows Shubnikov-de Haas oscillation in magnetic field, which disappear for small fields. These are periodic in /B, similarly to normal metals, in spite of the different Landau quantization. Besides oscillations, σ decreases with field, since the larger the cyclotron frequency, the smaller the probability of finding states around µ. Oscillations are also present as a function of chemical potential.
6 σ xx h/e 6 7 8 9 B(T) Figure. Conductivity of graphene as a function of the magnetic field. The inset shows the conductivity as a function of the inverse magnetic field. 9 8 7 6 σ xx h/e..... E F /hω Figure. Conductivity of graphene as a function of the Fermi energy. References K. S. Novoselov, A. K. Geim, S. V. Morozov, et al. Electric Field Effect in Atomically Thin Carbon Films. Science.. Vol. 6, no. 696. Pp. 666669. P. R. Wallace. The band theory of graphite. Phys. Rev. 97. Vol. 77. Pp. 66. G. W. Semenoff. Condensed-matter simulation of a three-dimensional anomaly. Phys. Rev. Lett. 98. Vol.. Pp. 9. T. Ando. Theoory of quantum transport in a two-dimensional electron system under magnetic fields. Jornal of the Physical Society of Japan. 97. 7,.