Graphene: Corrugations, defects, scattering mechanisms, and chemical functionalizationMikhail KatsnelsonTheory of Condensed MatterInstitute for Molecules and MaterialsRadboud University of Nijmegen

Outline

Introduction: electronic structureIntrinsic ripples in 2D: Application to grapheneDirac fermions in curved space: Pseudomagnetic fields and their effect on electronic structureElectronic structure of point defectsScattering mechanismsChemical functionalization: graphane etc.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 CarbonGraphene: prototype truly 2D crystalNanotubesFullerenesDiamond, Graphite

Crystallography of grapheneTwo sublattices

Tight-binding description of the electronic structureOperators a and b for sublattices A and B(Wallace 1947)

Band structure of graphenesp2 hybridization, bands crossingthe neutrality point

Massless Dirac fermionsIf Umklapp-processes K-K are neglected:and doping is small:2D Dirac massless fermions with the Hamiltonian 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 thesymmetries 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 dependenceof the cyclotron masson the charge-carrierconcentration

+ anomalous QHE (Berry phase)

E =ckAnomalous Quantum Hall Effect(McClure1956)

Anomalous QHE in single- andbilayer grapheneSingle-layer: half-integerquantization since zero-energy Landau level has twice smaller degeneracy(Novoselov et al 2005, Zhanget al 2005)Bilayer: integer quantizationbut no zero- plateau(chiral fermions withparabolic gapless spectrum)(Novoselov et al 2006)

Half-integer quantum Hall effect and index theoremAtiyah-Singer index theorem: number of chiralmodes with zero energy for massless Dirac fermions with gauge fieldsSimplest case: 2D, electromagnetic field (magnetic flux in units of the flux quantum)Magnetic field can be inhomogeneous!!!

Ripples on graphene: Dirac fermions in curved space

Freely suspendedgraphene membraneis 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 FlexibleMembranesD. R. Nelson, T. Piran & S. Weinberg (Editors), Statistical Mechanics of membranes and SurfacesWorld Sci., 2004Continuum medium theory

Statistical Mechanics of FlexibleMembranes IIElastic energyDeformation tensor

Harmonic ApproximationCorrelation function of height fluctuationsCorrelation function of normalsIn-plane components:

Anharmonic effects In harmonic approximation: Long-range order of normals is destroyedCoupling 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 beflat

Anharmonic coupling (bending-stretching) isessential; bending fluctuations grow with thesample size L as L, 0.6

Ripples with various size, broad distribution,power-law correlation functions of normals

Computer simulations

Bond order potential for carbon: LCBOPII(Fasolino & Los 2003): fitting to energy of different molecules and solids, elasticmoduli, phase diagram, thermodynamics, etc.

Method: classical Monte-Carlo, crystallites withN = 240, 960, 2160, 4860, 8640, and 19940

Temperatures: 300 K , 1000 K, and 3500 K (Fasolino, Los & MIK, Nature Mater.6, 858 (2007)

A snapshot for room temperatureBroad distribution of ripple sizes + some typicallength due to intrinsic tendency of carbon to bebonded

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) 0.85= 1- /2 In agreementwith phenom. 0.8. (J. Los et al,2009)

Chemical bonds I

Chemical bonds IIRT: tendencyto formation of single and double bonds instead ofequivalent conjugated bonds

Bending for chemical reasons

Pseudomagnetic fields due to ripplesNearest-neighbour approximation: changes ofhopping integrals Vector potentialsK and K points are shiftedin opposite directions;Umklapp processes restore time-reversal symmetrySuppression of weaklocalization?

Midgap states due to ripples Guinea, MIK & Vozmediano, PR B 77, 075422 (2008)Periodic pseudomagnetic field due to structuremodulation

Zero-energy LLis not broadened,in contrast with the others

In agreement withexperiment

(A.Giesbers, U.Zeitler, MIK et.al., PRL 2007)

Midgap states: Ab initio IWehling, Balatsky, Tsvelik, MIK & Lichtenstein, EPL 84, 17003 (2008)DFT (GGA), VASP

Midgap states: Ab initio II

Electronic structure of point defectsGreens function in the presence of defects:Equation for T-matrix:U is scattering potentialImpurity potential T-matrix Greens function local DOS

Dirac spectrum Greens function for massless Dirac caseE = vkGreens function for idealcase (continuum model) :Contains logarithmic divergence at smallenergy

Results: TB model, singleand double impurity(Wehling et al, PR B 75, 125425 (2007))

Electronic structure of graphene with adsorbed moleculesUse 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) adsorbedmolecules (Wehling et al, Nano Lett. 8, 173 (2008))

- Density functional (LDA and GGA) PAW method, VASP code

Electronic structure: resultsNO2

N2O4Single 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 computationalresults. Adsorption energies for monomerand 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 caseresonant scatterers

Adsorption energies

General problem: GGA underestimates them(no VdW contributions), LDA overestimatesFor different equilibrium configurations: GGA, monomer: 85 meV, 67 meV LDA: 170-180 meVEquilibrium distances from graphene: 0.34-0.35nm GGA, dimer: 67 meV, 50 meV, 44 meVLDA: 110-280 meVEquilibrium distances from graphene: 0.38-0.39nm General conclusion: adsorption energies are closefor the cases of monomer and dimer

Water or graphene: role of substrate

Wehling, MIK & Lichtenstein, Appl. Phys. Lett. 93, 202110 (2008) Different configurationsof water on graphene orbetween graphene andSiO2

Water or graphene: role of substrate II

Just water:no resonancesnear the Dirac point

Water or graphene: role of substrate III

Water between graphene andsubstrate (e,f): interaction withsurface defects leads to SiOHgroups working as resonantscatterers

Charge-carrier scattering mechanisms in grapheneNovoselov et al, Nature 2005Conductivity is approx.proportional to charge-carrier concentration n(concentration-independentmobility).

Standard explanation(Nomura & MacDonald 2006):charge impurities

Scattering by point defects:Contribution to transport properties

Contribution of point defects to resistivity Justification of standard Boltzmann equation except very small doping: n > exp(-h/e2),or EF >> 1/|ln(kFa)| (M.Auslender and MIK, PRB 2007)

Radial Dirac equation

Scattering cross section

Wave functions beyond the range of actionof potentialScatteringcross section:

Scattering cross section II

Exact symmetry for massless fermions:As a consequence

Cylindrical potential well

A generic short-rangepotential: scatteringis very weak

Resonant scattering case

Much larger resistivityNonrelativistic case:The same result as for resonant scatteringfor massless Dirac fermions!

Charge impurities

Coulomb potential Scattering phases are energy independent.Scattering cross section 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 2007and others). Relativistic collapse for supercritical charges!!!

Experimental situation

Schedin et al, Nature Mater. 6, 652 (2007)It seems that mobility is not very sensitive tocharge impurities; linear-screening theoryoverestimate the effect 1.5-2 orders of magnitudeNonlinear screening (resume): if Ze2/vF = < -irrelevant, if > - up to =

Cannot explain a strong suppression of scattering

Experimental situation II

Ponomarenko et al, PRL 102,206603 (2009)Almost no sensitivityto screened medium(ethanol, = 25), glycerol,water (more complicated)and to dielectric constantof 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, constantpotential (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 barriersResonant impuritiessurvive, charged impurities form clusters?Still under discussions!

Main scattering mechanism:scattering by ripples?

Scattering by random vector potential:Random potential due to surface curvatureAssumption: intrinsic ripples due to thermalfluctuationsMIK & Geim, Phil. Trans. R. Soc. A 366, 195 (2008)

Main screening mechanism II

Harmonic ripples @ RTFor the caseThe same concentrationdependence as for charge impurities The problem: quenching mechanism?! T is replaced by a quenching temperature (substrate disorder, Coulomb forces, adsorbates!!!)

Hydrogen: from single atomto graphane

(Boukhvalov, MIK & Lichtenstein, PR B 77, 035427 (2008))Also for hydrogen storage, etc.Equilibrium structure forsingle hydrogen atomand for pairCrystal structure of graphene and graphane

Hydrogen: from single atomto graphane II

Gap values for completefunctionalization by otherspecies (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) Create ripple as a hemisphere; (2) put pair of H atoms; (3) optimize the structureHydrogenation of flat surface is not favorable with respect to H2

Role of ripples II

Ripples are stable withinregions A-B, C-D, E-F

Curvature vs geometric frustrations

Strong stabilization in E-F:resonance between rippleand hydrogen midgap states((b), with H dashed green,without H solid red)(c) opening a gap for six Hper ripple

Quenching of ripples by hydrogen (OH,) adsorption?!

Conclusions and final remarksGraphene as a prototype truly 2D crystal: ripple physics Main scattering mechanism: still under discussions; electronic structure calculations are of crucial importanceChemistry of graphene: graphane etc. Role of ripples: difference between graphene and graphite (graphene is more active?)