Graphene:%20Corrugations,%20defects,%20scattering%20mechanisms,%20and%20chemical%20functionalization

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, p 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
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-?
plateau (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 L?, ? 0.6 Ripples with
various 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 snapshot for room temperature
Broad distribution of ripple sizes some
typical length due to intrinsic tendency of
carbon to be bonded
22
(No Transcript)
23
To reach region of small q
Larger samples (up to 40,000 atoms) Better MC
sampling (movements of individual atoms global
wave distortions, 10001)
? 0.85 ? 1- ?/2
In agreement with phenom. ? 0.8.
(J. Los et al, 2009)
24
Chemical bonds I
25
Chemical bonds II
RT tendency to formation of single and double
bonds instead of equivalent conjugated
bonds Bending for chemical reasons
26
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?
27
Midgap states due to ripples
Guinea, MIK Vozmediano, PR B 77, 075422 (2008)
Periodic pseudomagnetic field due to
structure modulation
28
(No Transcript)
29
Zero-energy LL is not broadened, in contrast with
the others
In agreement with experiment (A.Giesbers,
U.Zeitler, MIK et.al., PRL 2007)
30
Midgap states Ab initio I
Wehling, Balatsky, Tsvelik, MIK Lichtenstein,
EPL 84, 17003 (2008)
DFT (GGA), VASP
31
Midgap states Ab initio II
32
Electronic structure of point defects
Impurity potential ? T-matrix ? Greens function
? local DOS
Greens function in the presence of defects
Equation for T-matrix
U is scattering potential
33
Dirac spectrum
Greens function for massless Dirac case
E ?vk
Greens function for ideal case (continuum model)

Contains logarithmic divergence at small energy
34
Results TB model, singleand double impurity
(Wehling et al, PR B 75, 125425 (2007))
35
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

36
Electronic structure results
NO2

N2O4
Single molecule is paramagnetic, dimer is
diamagnetic
37
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

38
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
39
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
40
Water or graphene role of substrate II
Just water no resonances near the Dirac point

41
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
42
Charge-carrier scattering mechanisms in graphene
Conductivity is approx. proportional to
charge- carrier concentration n (concentration-ind
ependent mobility). Standard explanation (Nomura
MacDonald 2006) charge impurities
Novoselov et al, Nature 2005
43
Scattering by point defectsContribution to
transport properties
Contribution of point defects to resistivity ?


Justification of standard Boltzmann equation
except very small doping n gt exp(-psh/e2), or
EFt gtgt 1/ln(kFa) (M.Auslender and MIK, PRB
2007)
44
Radial Dirac equation


45
Scattering cross section
Wave functions beyond the range of action of
potential


Scattering cross section
46
Scattering cross section II
Exact symmetry for massless fermions

As a consequence

47
Cylindrical potential well


A generic short-range potential scattering is
very weak
48
Resonant scattering case

Much larger resistivity
Nonrelativistic case

The same result as for resonant scattering for
massless Dirac fermions!
49
Charge impurities
Coulomb potential

Scattering phases are energy independent. Scatteri
ng cross section s 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!!!

50
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/hvF ß lt ½
-irrelevant, if ß gt ½ - up to ß ½ Cannot
explain a strong suppression of scattering

51
Experimental situation II
Ponomarenko et al, PRL 102, 206603 (2009)

Almost no sensitivity to screened
medium (ethanol, ? 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)
52
Clusterization
For some charge impurities (e.g., Na, K)
barriers are low (lt 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!
53
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!
54
Main scattering mechanismscattering 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
55
Main screening mechanism II
Harmonic ripples _at_ 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!!!)
56
Hydrogen from single atomto 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
57
Hydrogen from single atomto 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))
58
Towards graphane experiment



59
Role of ripples
Hydrogenation of flat surface is not favorable
with respect to H2
(Boukhvalov MIK 2009)

1. Create ripple as a hemisphere (2) put pair of H
   atoms (3) optimize the structure

60
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?!



61
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?)