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Graphene: Corrugations, defects, scattering mechanisms, and chemical functionalization

Explore the intricate world of graphene, covering intrinsic ripples, Dirac fermions in curved space, scattering mechanisms, chemical functionalization, and more. Discover the electronic structure, point defects, and collaborative experiments, shedding light on this fascinating material.

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Graphene: Corrugations, defects, scattering mechanisms, and chemical functionalization

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  1. Graphene: Corrugations, defects, scattering mechanisms, and chemical functionalization Mikhail Katsnelson Theory of Condensed Matter Institute for Molecules and Materials Radboud University of Nijmegen

  2. Outline • Introduction: electronic structure • Intrinsic ripples in 2D: Application to graphene • Dirac fermions in curved space: Pseudomagnetic fields and their effect on electronic structure • Electronic structure of point defects • Scattering mechanisms • Chemical functionalization: graphane etc. • Conclusions

  3. 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

  4. Allotropes of Carbon Diamond, Graphite Graphene: prototype truly 2D crystal Fullerenes Nanotubes

  5. Crystallography of graphene Two sublattices

  6. Tight-binding description of the electronic structure Operators a and b for sublattices A and B (Wallace 1947)

  7. Band structure of graphene sp2 hybridization, π bands crossing the neutrality point

  8. Massless Dirac fermions If 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

  9. 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.).

  10. 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”)

  11. EN=[2ec2B(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 Anomalous Quantum Hall Effect E =ck (McClure 1956)

  12. Anomalous QHE in single- andbilayer graphene Single-layer: half-integer quantization since zero- energy Landau level has twice smaller degeneracy (Novoselov et al 2005, Zhang et al 2005) Bilayer: integer quantization but no zero-νplateau (chiral fermions with parabolic gapless spectrum) (Novoselov et al 2006)

  13. 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 (magnetic flux in units of the flux quantum) Magnetic field can be inhomogeneous!!!

  14. 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

  15. Statistical Mechanics of FlexibleMembranes D. R. Nelson, T. Piran & S. Weinberg (Editors), Statistical Mechanics of membranes and Surfaces World Sci., 2004 Continuum medium theory

  16. Statistical Mechanics of FlexibleMembranes II Elastic energy Deformation tensor

  17. Harmonic Approximation Correlation function of height fluctuations Correlation function of normals In-plane components:

  18. 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)

  19. 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 withvarious size, broad distribution, power-law correlation functions of normals

  20. 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

  21. A snapshotfor room temperature Broad distribution of ripple sizes + some typical length due to intrinsic tendency of carbon to be bonded

  22. 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)

  23. Chemical bonds I

  24. Chemical bonds II RT: tendency to formation of single and double bonds instead of equivalent conjugated bonds Bending for “chemical” reasons

  25. Pseudomagnetic fields due to ripples Nearest-neighbour approximation: changes of hopping integrals K and K’ points are shifted in opposite directions; Umklapp processes restore time-reversal symmetry “Vector potentials” Suppression of weak localization?

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

  27. Zero-energy LL is not broadened, in contrast with the others In agreement with experiment (A.Giesbers, U.Zeitler, MIK et.al., PRL 2007)

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

  29. Midgap states: Ab initio II

  30. 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

  31. 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

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

  33. 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) andN2O4 (nonmagnetic) adsorbed • molecules (Wehling et al, Nano Lett. 8, 173 (2008)) • - Density functional (LDA and GGA) • PAW method, VASP code

  34. Electronic structure: results NO2 N2O4 Single molecule is paramagnetic, dimer is diamagnetic

  35. 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

  36. 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

  37. 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

  38. Water or graphene: role of substrate II Just water: no resonances near the Dirac point

  39. 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

  40. Charge-carrier scattering mechanisms in graphene Conductivity is approx. proportional to charge- carrier concentration n (concentration-independent mobility). Standard explanation (Nomura & MacDonald 2006): charge impurities Novoselov et al, Nature 2005

  41. 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)

  42. Radial Dirac equation

  43. Scattering cross section Wave functions beyond the range of action of potential Scattering cross section:

  44. Scattering cross section II Exact symmetry for massless fermions: As a consequence

  45. Cylindrical potential well A generic short-range potential: scattering is very weak

  46. Resonant scattering case Much larger resistivity Nonrelativistic case: The same result as for resonant scattering for massless Dirac fermions!

  47. 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!!!

  48. 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

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