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  ic* 


 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 =ck
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
(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?)
