electronic structure and magnetismelectronic structure crystal structure of graphene: two...
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Gauge fields in corrugated Gauge fields in corrugated graphenegraphene
Mikhail KatsnelsonTheory of Condensed Matter
Institute for Molecules and MaterialsRadboud University Nijmegen
CollaborationCollaboration
A.Geim
and K.Novoselov, Manchester
A.Fasolino, J.Los, Nijmegen -
Theory
U.Zeitler, J.C.Maan, Nijmegen –
High FieldMagnet Laboratory
T.Wehling
and A.Lichtenstein, Hamburg
F.Guinea
and M.Vozmediano, Madrid
OutlineOutline1. Introduction
2. Intrinsic ripples in 2D: Application to graphene
3. Dirac fermions in curved space: Pseudomagnetic fields
4. Anomalous QHE and pseudo-Landau levels
5. Scattering by ripples
Allotropes of CarbonAllotropes of Carbon
GrapheneGraphene: prototype : prototype truly 2D crystaltruly 2D crystal
NanotubesNanotubes FullerenesFullerenes
Diamond, Graphite
TightTight--binding description of the binding description of the electronic structureelectronic structure
Crystal structureCrystal structureof of graphenegraphene::Two Two sublatticessublattices
MasslessMassless Dirac fermionsDirac fermions
Spectrum near K (K’) points is linear. Conical cross-points: provided by symmetry and thus robust property
UndopedUndoped ElectronElectron HoleHole
MasslessMassless Dirac fermions IIDirac fermions II
ac
yi
x
yi
xciH 0**
23
0
0γ=
⎟⎟⎟⎟
⎠
⎞
⎜⎜⎜⎜
⎝
⎛
∂∂
+∂∂
∂∂
−∂∂
−= hh
If Umklapp-processes K-K’ are neglected:2D Dirac massless fermions with the Hamiltonian
“Spin indices’’ label sublattices A and Brather than real spinrather than real spin
Ripples on Ripples on graphenegraphene: Dirac : Dirac fermions in curved spacefermions in curved space
Freely suspendedFreely suspendedgraphenegraphene membranemembraneis partially crumpledis partially crumpled
J. C. Meyer et al,J. C. Meyer et al,Nature 446, 60 (2007)Nature 446, 60 (2007)
2D crystals in 3D space 2D crystals in 3D space cannot be flat, due to cannot be flat, due to bending instabilitybending instability
Statistical Mechanics of FlexibleStatistical Mechanics of Flexible MembranesMembranes
D. R. Nelson, T. Piran
& S. Weinberg (Editors), Statistical Mechanics of membranes and SurfacesWorld Sci., 2004
Continuum medium theory
Statistical Mechanics of FlexibleStatistical Mechanics of Flexible Membranes IIMembranes II
Elastic energy
Deformation tensor
Statistical Mechanics of FlexibleStatistical Mechanics of Flexible Membranes III Membranes III
Harmonic approximation: membrane cannot beflat
Anharmonic
coupling (bending-stretching) isessential; bending fluctuations grow with thesample size L
as Lς, ς ≈ 0.6
Ripples with
various size, broad distribution,power-law correlation functions of normals
Harmonic ApproximationHarmonic Approximation
Correlation
function of height fluctuations
Correlation
function of normals
In-plane components:
AnharmonicAnharmonic effectseffectsIn 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)
AnharmonicAnharmonic effects IIeffects II
B is 2D bulk modulus
Typical height fluctuations scales as
More or less flat phase
Computer simulations 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., Nov.2007)
A snapshotA snapshot for room temperaturefor room temperature
Broad distribution of ripple sizes + some typicallength due to intrinsic tendency of carbon to be
bonded
Chemical bonds IChemical bonds I
Chemical bonds IIChemical bonds II
RT: tendencyto formation of single and double bonds instead ofequivalent conjugated bonds
Bending for “chemical”
reasons
PseudomagneticPseudomagnetic fields due to ripplesfields due to ripples
Deformation tensor in the plane
coordinates in the plane
displacement vector
displacements normal to the plane
PseudomagneticPseudomagnetic fields IIfields IINearest-neighbour approximation: changes ofhopping integrals
“Vector potentials” K
and K’
points are shiftedin opposite directions;Umklapp processes restore time-reversal symmetry
Suppression of weaklocalization?
E =0N =0
N =2N =1
N =4N =3
EN
=[2ehc∗2B(N + ½
±
½)]1/2E =hc∗
k
E =0
pseudospin
The lowest Landau level is at ZERO energyand shared equally by electrons and holes
hωC
Anomalous Quantum Hall Effect
Anomalous QHE in singleAnomalous QHE in single-- andand bilayerbilayer graphenegraphene
Single-layer: half-integerquantization since zero-energy Landau level has twice smaller degeneracy
Bilayer: integer quantizationbut no zero-ν
plateau
(chiral fermions withparabolic gapless spectrum)
HalfHalf--integer quantum Hall effect integer quantum Hall effect and and ““index theoremindex theorem””
0/φφ=− −+ NN
AtiyahAtiyah--Singer index theorem: number of Singer index theorem: number of chiralchiralmodes with zero energy for modes with zero energy for masslessmassless Dirac Dirac
fermions with gauge fieldsfermions with gauge fieldsSimplest case: 2D, electromagnetic field
(magnetic flux in units of the flux quantum)
Consequence: ripples should not broadenzero-energy Landau level
Experiment: activation energy in Experiment: activation energy in QHEQHE
(A.Giesbers, U.Zeitler, MIK et.al., PRL 2007)
Schematic DOS with broadening due todisorder
Experiment: activation energy in Experiment: activation energy in QHE IIQHE IIQHE is observed @ RT
All data between 100 Kand 300 K are fitted by singleArrhenius exponent
Low-temperatures: morecomplicated, sometimessplitting (QHE ferromagnet?)
Experiment: activation energy in Experiment: activation energy in QHE IIIQHE IIIActivation energies for ν=2 (red) and ν=6 (blue). Dashed lines –
theoretical
Landau level positions
ν=2: follows LL at high field; ν=6: never
(broadening)
Low fields –
level mixing etc.
States with zero energyStates with zero energy
Pseudomagnetic fields from the ripples cannot broaden the LL: topological protection
Scalar potential fluctuations broadenzero-enery level in more or less the same way as for other LL’s.
MidgapMidgap states due to ripples states due to ripples Guinea, MIK & Vozmediano, 2007
Periodic pseudomagnetic
field due to structuremodulation
Zero-energy LLis not broadened,in contrast with the others
Ripples with circular symmetryRipples with circular symmetry
A=0.4l/a=50E = 0
MidgapMidgap states: states: AbAb initio Iinitio IWehling, Balatsky, MIK & Lichtenstein 2007
DFT (GGA)
VASP
MidgapMidgap states: states: AbAb initio IIinitio II
MidgapMidgap states: states: AbAb initio IIIinitio III
MidgapMidgap states: states: AbAb initio IVinitio IV
Mechanism of charge Mechanism of charge inhomogeneityinhomogeneity??
Midgap
states (pseudo-Landau levels): infinitecompressibility due to δ-functional DOS peak
Charge inhomogeneity
opens the gap due tomodulation of electrostatic potential
Modulation of NNN hopping
(A.Castro
Neto):
similar effect but probablytoo small: t’/t
≈
1/30.
ChargeCharge--carrier scattering carrier scattering mechanisms in mechanisms in graphenegraphene
Novoselov
et al, Nature 2005
Conductivity is approx.proportional to charge-carrier concentration n(concentration-independentmobility).
Standard explanation(Nomura & MacDonald 2006):charge impurities
Scattering by point defects:Scattering by point defects: Contribution to transport propertiesContribution to transport properties
Contribution of point defects to resistivity ρ
Justification of standard Boltzmann equation except very small doping: n > exp(-πσh/e2)(Auslender
& MIK, PR B 2007)
Radial Dirac equationRadial Dirac equation
Scattering cross sectionScattering cross sectionWave functions beyond the range of actionof potential
Scatteringcross section:
Scattering cross section IIScattering cross section IIExact symmetry for massless fermions:
As a consequence
Cylindrical potential wellCylindrical potential well
Resonant scattering caseResonant scattering case
Much larger resistivity
Nonrelativistic case:
The same result as for resonant scatteringfor massless Dirac fermions!
Charge impuritiesCharge impuritiesCoulomb 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)
Experimental situationExperimental situationSchedin
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 magnitude
Nonlinear screening (Shytov, MIK, Levitov, PRL2007): if Ze2/ħvF = β
< ½
-irrelevant, if β
> ½
- up
to β
= ½.
Explanation (?) –
screening by layer of water between graphene
and quartz?
Main scattering mechanism:Main scattering mechanism: scattering by ripples?scattering by ripples?
Scattering by random vector potential:
Random potential due to surface curvature
Assumption: intrinsic ripples due to thermalfluctuations
MIK & Geim, Phil. Trans. R. Soc. A 366, 195 (2008)
Main screening mechanism IIMain screening mechanism II“Harmonic” ripples @ RT
For the case
The same concentrationdependence as for charge impurities
The problem: quenching mechanism?!
(Disorder, Coulomb forces…)
Conclusions and final remarksConclusions and final remarksSpecific of 2D systems: ripplesCorrugations result in pseudomagnetic fields “Index theorem” and mid-gap pseudo-LL: relevacnce for QHE andcharge inhomogeneity?Scattering by ripples as limiting factor for charge-carrier mobility in graphene?