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?