#### Graphene: Corrugations, defects, scattering mechanisms, and chemical functionalization Mikhail Katsnelson

Download Report#### Transcript Graphene: Corrugations, defects, scattering mechanisms, and chemical functionalization Mikhail Katsnelson

Graphene: Corrugations, defects, scattering mechanisms, and chemical functionalization Mikhail Katsnelson Theory of Condensed Matter Institute for Molecules and Materials Radboud University of Nijmegen Outline 1. Introduction: electronic structure 2. Intrinsic ripples in 2D: Application to graphene 3. Dirac fermions in curved space: Pseudomagnetic fields and their effect on electronic structure 4. Electronic structure of point defects 5. Scattering mechanisms 6. Chemical functionalization: graphane etc. 7. Conclusions Collaboration Andre Geim, Kostya Novoselov experiment!!! scattering mechanisms Tim Wehling, Sasha Lichtenstein adsorbates, ripples Danil Boukhvalov chemical functionalization Annalisa Fasolino, Jan Los, Kostya Zakharchenko atomistic simulations, ripples Paco Guinea ripples, scattering mechanisms Seb Lebegue, Olle Eriksson GW Allotropes of Carbon Diamond, Graphite Graphene: prototype truly 2D crystal Nanotubes Fullerenes Crystallography of graphene Two sublattices Tight-binding description of the electronic structure Operators a and b for sublattices A and B (Wallace 1947) Band structure of graphene sp2 hybridization, π bands crossing the neutrality point Massless Dirac fermions If Umklapp-processes K-K’ are neglected: and doping is small: 2D Dirac massless fermions with the Hamiltonian 0 H ic* x i y i x y 0 3 c 0a 2 * “Spin indices’’ label sublattices A and B rather than real spin Stability of the conical points (Manes, Guinea, Vozmediano, PRB 2007) Combination of time-reversal (T) and inversion (I) symmetry: Absence of the gap (topologically protected if the symmetries are not broken; with many-body effects, etc.). Experimental confirmation: Schubnikov – de Haas effect + anomalous QHE K. Novoselov et al, Nature 2005; Y. Zhang et al, Nature 2005 Square-root dependence of the cyclotron mass on the charge-carrier concentration + anomalous QHE (“Berry phase”) Anomalous Quantum Hall Effect E =ck EN =[2ec2B(N + ½ ½)]1/2 pseudospin ωC E =0 N =0 E =0 N =1 N =2 N =3 N =4 The lowest Landau level is at ZERO energy and shared equally by electrons and holes (McClure 1956) Anomalous QHE in single- and bilayer graphene Single-layer: half-integer quantization since zeroenergy Landau level has twice smaller degeneracy (Novoselov et al 2005, Zhang et al 2005) Bilayer: integer quantization but no zero-ν plateau (chiral fermions with parabolic gapless spectrum) (Novoselov et al 2006) Half-integer quantum Hall effect and “index theorem” Atiyah-Singer index theorem: number of chiral modes with zero energy for massless Dirac fermions with gauge fields Simplest case: 2D, electromagnetic field N N / 0 (magnetic flux in units of the flux quantum) Magnetic field can be inhomogeneous!!! Ripples on graphene: Dirac fermions in curved space Freely suspended graphene membrane is partially crumpled J. C. Meyer et al, Nature 446, 60 (2007) 2D crystals in 3D space cannot be flat, due to bending instability Statistical Mechanics of Flexible Membranes D. R. Nelson, T. Piran & S. Weinberg (Editors), Statistical Mechanics of membranes and Surfaces World Sci., 2004 Continuum medium theory Statistical Mechanics of Flexible Membranes II Elastic energy Deformation tensor Harmonic Approximation Correlation function of height fluctuations Correlation function of normals In-plane components: Anharmonic effects In harmonic approximation: Long-range order of normals is destroyed Coupling between bending and stretching modes stabilizes a “flat” phase (Nelson & Peliti 1987; Self-consistent perturbative approach: Radzihovsky & Le Doussal, 1992) Anharmonic effects II Harmonic approximation: membrane cannot be flat Anharmonic coupling (bending-stretching) is essential; bending fluctuations grow with the sample size L as Lς, ς ≈ 0.6 Ripples with various size, broad distribution, power-law correlation functions of normals Computer simulations (Fasolino, Los & MIK, Nature Mater.6, 858 (2007) Bond order potential for carbon: LCBOPII (Fasolino & Los 2003): fitting to energy of different molecules and solids, elastic moduli, phase diagram, thermodynamics, etc. Method: classical Monte-Carlo, crystallites with N = 240, 960, 2160, 4860, 8640, and 19940 Temperatures: 300 K , 1000 K, and 3500 K A snapshot for room temperature Broad distribution of ripple sizes + some typical length due to intrinsic tendency of carbon to be bonded To reach region of small q Larger samples (up to 40,000 atoms); Better MC sampling (movements of individual atoms + global wave distortions, 1000:1) η ≈ 0.85 ζ= 1- η/2 In agreement with phenom. η ≈ 0.8. (J. Los et al, 2009) Chemical bonds I Chemical bonds II RT: tendency to formation of single and double bonds instead of equivalent conjugated bonds Bending for “chemical” reasons Pseudomagnetic fields due to ripples Nearest-neighbour approximation: changes of hopping integrals “Vector potentials” K and K’ points are shifted in opposite directions; Umklapp processes restore time-reversal symmetry Suppression of weak localization? Midgap states due to ripples Guinea, MIK & Vozmediano, PR B 77, 075422 (2008) Periodic pseudomagnetic field due to structure modulation Zero-energy LL is not broadened, in contrast with the others In agreement with experiment (A.Giesbers, U.Zeitler, MIK et.al., PRL 2007) Midgap states: Ab initio I Wehling, Balatsky, Tsvelik, MIK & Lichtenstein, EPL 84, 17003 (2008) DFT (GGA), VASP Midgap states: Ab initio II Electronic structure of point defects Impurity potential → T-matrix → Green’s function → local DOS Green’s function in the presence of defects: Equation for T-matrix: U is scattering potential Dirac spectrum Green’s function for massless Dirac case E = ћvk Green’s function for ideal case (continuum model) : Contains logarithmic divergence at small energy Results: TB model, single and double impurity (Wehling et al, PR B 75, 125425 (2007)) Electronic structure of graphene with adsorbed molecules Use of graphene as a chemical sensor: one can feel individual molecules of NO2 measuring electric properties (Schedin et al, Nature Mater. 6, 652 (2007)) First-principle calculations of electronic structure for NO2 (magnetic) and N2O4 (nonmagnetic) adsorbed molecules (Wehling et al, Nano Lett. 8, 173 (2008)) - Density functional (LDA and GGA) - PAW method, VASP code Electronic structure: results NO 2 N 2O 4 Single molecule is paramagnetic, dimer is diamagnetic Fitting to experimental data Hall effect vs gate voltage at different temperatures: two impurity levels at - 300 meV (monomer) and - 60 meV (dimer) A good agreement with computational results. Adsorption energies for monomer and dimer are comparable. Magnetic molecules are stronger dopants than nonmagnetic onessince in the latter case impurity level is close to the Dirac point. Nonmagnetic molecules are in that case resonant scatterers Adsorption energies General problem: GGA underestimates them (no VdW contributions), LDA overestimates For different equilibrium configurations: GGA, monomer: 85 meV, 67 meV LDA: 170-180 meV Equilibrium distances from graphene: 0.34-0.35nm GGA, dimer: 67 meV, 50 meV, 44 meV LDA: 110-280 meV Equilibrium distances from graphene: 0.38-0.39nm General conclusion: adsorption energies are close for the cases of monomer and dimer Water or graphene: role of substrate Wehling, MIK & Lichtenstein, Appl. Phys. Lett. 93, 202110 (2008) Different configurations of water on graphene or between graphene and SiO2 Water or graphene: role of substrate II Just water: no resonances near the Dirac point Water or graphene: role of substrate III Water between graphene and substrate (e,f): interaction with surface defects leads to SiOH groups working as resonant scatterers Charge-carrier scattering mechanisms in graphene Conductivity is approx. proportional to chargecarrier concentration n (concentration-independent mobility). Standard explanation (Nomura & MacDonald 2006): charge impurities Novoselov et al, Nature 2005 Scattering by point defects: Contribution to transport properties Contribution of point defects to resistivity ρ Justification of standard Boltzmann equation except very small doping: n > exp(-πσh/e2), or EFτ >> 1/|ln(kFa)| (M.Auslender and MIK, PRB 2007) Radial Dirac equation Scattering cross section Wave functions beyond the range of action of potential Scattering cross section: Scattering cross section II Exact symmetry for massless fermions: As a consequence Cylindrical potential well A generic short-range potential: scattering is very weak Resonant scattering case Much larger resistivity Nonrelativistic case: The same result as for resonant scattering for massless Dirac fermions! Charge impurities Coulomb potential Scattering phases are energy independent. Scattering cross section σ is proportional to 1/k (concentration independent mobility as in experiment) (Perturbative: Nomura & MacDonald, PRL 2006; Ando, JPSJ 2006 – linear screening theory) Nonlinear screening (MIK, PRB 2006); exact solution of Coulomb-Dirac problem (Shytov, MIK & Levitov PRL 2007; Pereira & Castro Neto PRL 2007; Novikov PRB 2007 and others). Relativistic collapse for supercritical charges!!! Experimental situation Schedin et al, Nature Mater. 6, 652 (2007) It seems that mobility is not very sensitive to charge impurities; linear-screening theory overestimate the effect 1.5-2 orders of magnitude Nonlinear screening (resume): if Ze2/ħvF = β < ½ -irrelevant, if β > ½ - up to β = ½ Cannot explain a strong suppression of scattering Experimental situation II Ponomarenko et al, PRL 102, 206603 (2009) Almost no sensitivity to screened medium (ethanol, κ = 25), glycerol, water (more complicated) and to dielectric constant of substrate Explanation: clusterization of charge impurities??? MIK, Guinea and Geim, PR B 79, 195426 (2009) Clusterization For some charge impurities (e.g., Na, K…) barriers are low (< 0.1 eV) and there is tendency to clusterization Exp. review: Caragiu & Finberg, JPCM 17, R995 (2005) Calculations: Chan, Neaton & Cohen, PR B 77, 235430 (2008) Simplest model: just circular cluster, constant potential (shift of chemical potential) Correct concentration dependence, weakening of scattering in two order of magnitude due to clusterization! Clusterization II (Wehling, MIK & Lichtenstein 2009) Positions t (top of C atom) vs h (middle of hexagon): Covalent (neutral impurities usually have high barriers, Ionic (charged) impurities have lower barriers Resonant impurities survive, charged impurities form clusters? Still under discussions! Main scattering mechanism: scattering by ripples? MIK & Geim, Phil. Trans. R. Soc. A 366, 195 (2008) Scattering by random vector potential: Random potential due to surface curvature Assumption: intrinsic ripples due to thermal fluctuations Main screening mechanism II “Harmonic” ripples @ RT For the case The same concentration dependence as for charge impurities The problem: quenching mechanism?! T is replaced by a quenching temperature… (substrate disorder, Coulomb forces, adsorbates!!!) Hydrogen: from single atom to graphane (Boukhvalov, MIK & Lichtenstein, PR B 77, 035427 (2008)) Also – for hydrogen storage, etc. Equilibrium structure for single hydrogen atom and for pair Crystal structure of graphene and graphane Hydrogen: from single atom to graphane II Gap values for complete functionalization by other species (Boukhvalov & MIK ’08) GW for graphane: gap 5.4 eV (Lebegue, Klintenberg, Eriksson & MIK, PR B 79, 245117 (2009)) Towards graphane – experiment Role of ripples (Boukhvalov & MIK 2009) Hydrogenation of flat surface is not favorable with respect to H2 (1) Create ripple as a hemisphere; (2) put pair of H atoms; (3) optimize the structure Role of ripples II Ripples are stable within regions A-B, C-D, E-F Curvature vs geometric frustrations Strong stabilization in E-F: resonance between ripple and hydrogen midgap states ((b), with H – dashed green, without H – solid red) (c) opening a gap for six H per ripple Quenching of ripples by hydrogen (OH,…) adsorption?! Conclusions and final remarks Graphene as a prototype truly 2D crystal: ripple physics Main scattering mechanism: still under discussions; electronic structure calculations are of crucial importance Chemistry of graphene: graphane etc. Role of ripples: difference between graphene and graphite (graphene is more active?)