ELECTRONIC TRANSPORT IN GRAPHENE

Transcription

1 ELECTRONIC TRANSPORT IN GRAPHENE J. González Instituto de Estructura de la Materia, CSIC, Spain

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3 ELECTRONIC TRANSPORT IN GRAPHENE Graphene has opened the way to understand the behaior of an electron system in D = z remarable properties are obsered from the theoretical point of iew z it has spared great expectations of reaching ery large mobilities rom E. Stolyaroa et al., Proc. Natl. Acad. Sci. 14, 99 (7) But some challenges hae to be faced: z samples hae significant corrugation z the interaction with the substrate and boundary conditions modify significantly the transport properties rom E. Stolyaroa et al., Proc. Natl. Acad. Sci. 14, 99 (7)

4 ELECTRONIC TRANSPORT IN GRAPHENE The first experimental obserations and measurement of unusual transport properties pointed at the existence of a conical dispersion of uasiparticles in graphene the electric field effect shows that a substantial concentration of electrons (holes) can be induced by changes in the gate oltage n V g rom K. S. Nooselo et al., Nature 438, 197 (5) the response to a magnetic field is also unusual, as obsered in particular in the uantum Hall effect σ xy 4e = ( N + 1/ ) h rom K. S. Nooselo et al., Nature 438, 197 (5)

5 ELECTRONIC TRANSPORT IN GRAPHENE The obsered properties were actually consistent with the dispersion expected for electrons in a honeycomb lattice The undoped electron system has isolated ermi points at the corners of an hexagonal Brillouin zone The conical dispersion is a genuine property of two component fermions (Dirac fermions) with hamiltonian H = x + i y x i y We hae to introduce a Dirac fermion for each independent ermi point, at which ε ( ) = ±, n( ε ) ε

6 ELECTRONIC TRANSPORT IN GRAPHENE A direct eidence of the conical dispersion has been obtained with angle resoled photoemission spectroscopy A. Bostwic et al.,, Nature Phys. 3, 36 (7) These experiments are also useful to proide a measure of the interaction effects in graphene A. Bostwic et al.,, Nature Phys. 3, 36 (7)

7 QUANTUM HALL EECT IN GRAPHENE The peculiar uantization of the Hall conductiity can be explained satisfactorily by the coupling of the Dirac fermions to gauge fields: H tb = t r, r + ψ ( r ) exp( i( e / hc) A dl) ψ ( r) r r which corresponds to the gauge prescription H = σ σ ( A) c e This leads to Landau leels uantized according to the expression E N = sgn( N ) eh N B

8 QUANTUM HALL EECT IN GRAPHENE The uantum Hall effect is actually uite robust and should persist een in the presence of curature of the samples.. In the case of the shells of MWNTs with radius R = nm, for instance, we can predict the seuence of band structures for magnetic field strength B =, 5, 1, T : S. Bellucci,, J. G.,. Guinea, P. Onorato and E. Perfetto, J. Phys.: Condens.. Matter 19,, (7)

9 ELECTRONIC TRANSPORT IN GRAPHENE Howeer, there are open uestions in the case of the conductiity. This is usually computed in Boltzmann transport theory in terms of the density of states D as σ ( μ) = e D( μ) τ ( μ) The linear dependence obsered experimentally on gate oltage implies that σ should be proportional to the electron density. in the case of short range range scatterers, t 1/, implies that σ = const. in the case of charged long range scatterers, t, implies that, implies that σ μ² This argument seems to be also in agreement with recent experiments where the influence of the bacground dielectric constant is shown. The best fit to the conductiity is gien by 1 σ ( μ) = e n( μ) μ σ l s rom C. Jang et al.,, Phys. Re. Lett.. 11, (8)

10 JOSEPHSON EECT IN GRAPHENE There hae been also obserations of supercurrents,, when graphene is contacted with superconducting electrodes rom H. B. Heersche et al., Nature 446,, 56 (7) The reason why supercurrents may exist at the Dirac point is that Cooper pairs hae a nonanishing propagation een at anishing charge density D () (, ) = T = 8 1 rom J. G. and E. Perfetto, Phys. Re. B 76, (7)

11 ELECTRONIC TRANSPORT IN GRAPHENE The scattering by impurities is uite unconentinal in graphene,, due to the chirality of electrons. When a uasiparticle encircles a closed path in momentum space, it pics up a Berry phase of π Ψ iπ e Ψ In the absence of scatterers that may induce a large momentum transfer, transfer, bacscattering is then suppressed (H. Suzuura and T. Ando, Phys. Re. Lett.. 89,6663 (). This also explains the peculiar properties of electrons when tunneling neling across potential barriers: the transmission probability is eual to 1 at normal incidence, and for bacscattering M. I. Katsnelson,, K. S. Nooselo,, and A. K. Geim,, Nature Physics,, 6 (6)

12 MANY BODY EECTS IN GRAPHENE Graphene is a system with remarable many body properties, starting with the behaior of its electron hole excitations. The polarization is Π (, ) = 8 In the undoped system, there are no electron hole excitations nor plasmons into which the electrons can decay (J. G.,. Guinea and M.A.H. Vozmediano, Nucl. Phys. B44, 595 (1994)) The single particle properties are significantly renormalized due to the strong Coulomb interaction: 1 G = 1 G Σ σ γ ( g) log( E c / ) σ β( g) log( E c / ) (J. G.,. Guinea and M. A. H. Vozmediano, Phys. Re. B 59, 474 (1999)) with g e / 16

13 MANY BODY EECTS IN GRAPHENE The imaginary part of the self energy energy is e Im Σ(, ) log ( ) Howeer, this does nor imply a linear QP decay, as reflected in the singular behaior Im Σ(, + ε ) Im Σ(, ε ) = (J. G.,. Guinea and M. A. H. Vozmediano, Phys. Re. Lett. 77,, 3589 (1996)) In the doped system, the decay of uasiparticles is possible due to intraband electron hole excitations: The QP decay rate is now: τ 1 ( ) log ( ) E. H. Hwang, Ben Yu Kuang Hu, and S. Das Sarma, Phys. Re. B 76,, (7)

14 MANY BODY EECTS IN GRAPHENE We now turn to phonons as the releant source of scattering at low carrier densities. At sufficiently large energy/temperature we hae the contribution of optical phonons. The QP decay rate is τ 1 = Im Σ(, Im ig d ) d + σ ( ) D(, ) ( ) + ( ) iε This gies rise to a decay rate linearly proportional to the QP energy, aboe the phonon energy. Using Boltzmann transport theory, we obtain a resistiity that does not depend on carrier density and is lineraly proportional to temperature σ ( μ) = e D( μ) τ ( μ) J. G., E. Perfetto,, Phys. Re. Lett.. (in press) E. H. Hwang and S. Das Sarma, Phys. Re. B 77, (8)

15 MANY BODY EECTS IN GRAPHENE There is also interesting physics below the scale of the out of of plane phonons. These couple to the electron charge and hae therefore a strong hybridization with electrone lectron hole pairs. In the RPA, D(, ) + iε g 8 / / We obsere the appearance of ery soft phonon modes near the K point of graphene,, which are right below the particle hole continuum The hybrid states gie rise to a cubic dependence of the QP decay rate on energy, that competes with the lower bound gien at ery low energies by the decay into acoustic phonons J. G., E. Perfetto,, Phys. Re. Lett.. (in press)

16 LOW LOW ENERGY ELECTRONIC PROPERTIES ENERGY ELECTRONIC PROPERTIES The existence of soft phonon modes changes significantly the low The existence of soft phonon modes changes significantly the low energy electronic energy electronic properties. We hae for instance the properties. We hae for instance the uasiparticle uasiparticle decay rate decay rate We hae now uite different behaiors depending on the energy ra We hae now uite different behaiors depending on the energy range: nge: 8 / / ), ( = g Q ( ) ), ( ), ( ) ( ) ( ) ( Im ), ( Im ) ( 1 γ Ω Ω Ω + + Σ = Q d d g D i d d ig a δ φ ε τ < > τ g g, where, where (consistent with C. (consistent with C. H. Par H. Par et al. et al., Phys. Re. Phys. Re. Lett Lett , 8684 (7) ), 8684 (7) ) J. G. and E. J. G. and E. Perfetto Perfetto, Phys. Re. Phys. Re. Lett Lett. (in press). (in press)

17 LOW ENERGY ELECTRONIC PROPERTIES or comparison, we may derie the uasiparticle decay rate in the case of a screened Coulomb interaction: τ 1 = Im Σ(, Im ie d ) d ( a) + γ ( ) V (, ) ( ) + ( ) iε e 4 d + δ dω φ Ω 1 + l Ω In the limit δ, we hae a finite uasiparticle decay rate, with two different regimes τ 1 e e l 3 > l < l 1 1 (J. G.,. Guinea and M. A. H. Vozmediano, Phys. Re. Lett. 77,, 3589 (1996), also consistent with E. H. Hwang, B. Y. K. Hu,, and S. Das Sarma,, Phys. Re. B 76,, (7))

18 RESISTIVITY AND MOBILITY IN GRAPHENE The theoretical results hae to matched with the experimental measures of the resistiity. It is assumed that the resistiity has a T independent contribution from impurities, another from acoustic phonon scattering, and some extra contribution giing the nonlinear behaior ρ V, T ) = ρ ( V ) + ρ ( T ) + ρ ( V ( g imp g ac nl g, T ) J. H. Chen et al., Nature Nanotech. 3, 6 (8) J. H. Chen et al., Nature Nanotech. 3, 6 (8) The hope is to be able to remoe the contribution from impurities to remain with the intrinsic source of resistiity (phonons), in which case the mobility would dierge at low carrier density as 1 μ = ne ρ

19 To conclude, Graphene seems a uite exciting material from the experimental as well as from the theoretical point of iew from the theoretical point of iew, the uestion of the minimum conductiity is still a matter of debate from a practical point of iew, one has to be able to tailor the graphene structure at the nanoscale,, as well as to suppress the extrinsic scattering mechanisms that t are the main source of resistiity at present