Elastic properties of graphene

Transcription

1 Elastic properties of graphene M. I. Katsnelson P. Le Doussal B. Horowitz K. Wiese J. Gonzalez P. San-Jose V. Parente B. Amorim R. Roldan C. Gomez-Navarro J. Gomez G. Lopez-Polin F. Perez-Murano A. Morpurgo N. J. G. Couto C. Stampfer Correlations, criticality, and coherence in uantum systems Évora, Portugal, 6-10 October 014 Outline The stiffness of graphene Defects and elastic constants Electronic transport and corrugations

2 GRAPHENE S SUPERLATIVES Thinnest imaginable material largest surface area (~,700 m per gram) strongest material ever measured (theoretical limit) stiffest known material (stiffer than diamond) most stretchable crystal (up to 0% elastically) record thermal conductivity (outperforming diamond) highest current density at room T (106 times of copper) completely impermeable (even He atoms cannot sueeze through) highest intrinsic mobility (100 times more than in Si) conducts electricity in the limit of no electrons lightest charge carriers (zero rest mass) longest mean free path at room T (micron range)

3 Why are there two dimensional crystals? Thermal fluctuations: kbt L u log B d ( L) u( 0) B graphene = ev Å - = 35 N/m B diamond x d=5.4 N/m T=300K L=1Km

4 Elastic properties of graphene courtesy of M. M. Fogler

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6 Experiments C. Gomez-Navarro, J. Gomez, G. Lopez-Polin, F. Perez-Murano Load

7 Experiments A E D (N/m) 600 F Defects (%) B A Graphene Ar E 3D (TPa) Counts arxiv: D G Raman Shift (cm -1 ) A C N Fracture Force (µn) x10 13 D 4x x10 13 SiO Defects/cm Defects (%) Au 4 0 z (nm) x (µm) B 0 x10 1 4x10 1 6x10 1 3x x10 13 Defects/cm F (µ N) Counts <E D >=336±11 N/m 0 10 δ (nm) E 3D (TPa) C <E D >=538±5 N/m E D (N/m) B

8 ( ) ( ) ( ) ( ) ( ) ( )( ) = 1 h h u u h u h u r d h h u u r d h r d t h r d H y x x y y x y y y x x x y x y y x x µ λ κ ρ Two dimensional membranes Out of plane displacements lead to changes in area h L L h L Kinetic Bending Stretching Shear Two dimensional crystaline membranes are intrinsically anharmonic

9 Thermal expansion In plane strains change the freuency of out of plane modes Binding energy Flexural phonon Grüneisen parameter Thermal expansion ω = γ = κ 1 ω 4 ω u ( λ µ ) ρ = Low T u ( λ µ ) High T κ α 3 k B κ ρ log 8πκ Y Negative thermal expansion coefficient Lattice constant

10 Substrate effects Gapped flexural modes Thermal expansion

11 Out of plane fluctuations screen the in plane elastic constants 1 h c c Yu E κ 1 log κ c c Yu T T F 1 κ δ T Y u F Y = Y 10eVA T 300K 0.05eV κ 1eV l 10A YYl κ 5 1

12 Numerical results Γ Γ L T = 4 ( λ µ ) ( λ µ ) µ T = 3 4κ ρ 1 T 3 κ ρ Theory of elasticity 1

13 The self consistent screening approximation J. Physiue, 48, 1085 (1987) 4 TY d κ κ δ ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) = = = Σ Σ = p G G p d p I I b b b p G p P b d G G T π π = = = ( ) ( ) ( ) , u u η η µ λ κ η η Power law divergences Self consistent theory, valid in high dimensions Agrees well with numerical simulaions

14 Vacancies and flexural modes ( ω) G ρω 1 κ, = 4 Σ (, ω) Σ ( ω) T-matrix approximation n V n V log a localization length κρω κρω 4 κ ρω h h = 0 κ Σ 1 n V = ρ κ infinite mass vacancies Vacancies localize flexural modes Long wavelength flexural modes do not contribute to the screening of the elastic constants

15 geometric factor percolation 600 Defects (%) A Y K 1 R 1 0 n V ηu 1 1 c 0 n V E D (N/m) E 3D (TPa) intrinsic localization length 0 x x x10 13 Defects/cm nm 0 1 k F Fracture Force (µn) Defects (%) B 0 x10 1 4x10 1 6x10 1 3x x10 13 Defects/cm

16 Young modulus and induced strains a) a) b) 750 φ = 1.5 µm 750 φ = 1 µm E D (N/m) Strain % E D (N/m) c) d) Strain % Experiments Hydrostatic pressure. Formation of bubbles Theory

17 Graphene thermal expansion coefficient Prestress (N/m) 0,15 0,10 0,05 0,00 Membrane 1 Membrane Pristine Irradiated Prestress (N/m) 0,0 0,15 0,10 0,05 Pristine Irradiated Temperature (ºC) Temperature (ºC) Thermal Expansion Coefficient: Pristine: -9.4 x 10-6 K -1 Irradiated (L D ~ 5.5 nm): -1 x 10-6 K - 1 Thermal Expansion Coefficient: Pristine: -6. x 10-6 K -1 Irradiated (L D ~ 5 nm): -1.1 x 10-6 K -1 L D : Mean distance between defects as measured by Raman

18 Electron-phonon coupling Ripples induced by electrons Charge puddles induced by strains

19 GRAPHENE S SUPERLATIVES Thinnest imaginable material largest surface area (~,700 m per gram) strongest material ever measured (theoretical limit) stiffest known material (stiffer than diamond) most stretchable crystal (up to 0% elastically) record thermal conductivity (outperforming diamond) highest current density at room T (106 times of copper) completely impermeable (even He atoms cannot sueeze through) highest intrinsic mobility (100 times more than in Si) conducts electricity in the limit of no electrons lightest charge carriers (zero rest mass) longest mean free path at room T (micron range)

20 Strains and conductivity in graphene ArXiv: , Phys. Rev. X, in press DC transport on graphene on BN High mobility samples Multiterminal devices Magnetoresistance: Scattering due to intravalley processes

21 Puddles and mobility DC transport: mobility (roughly) independent of carrier concentration DC transport: correlation between mobility and puddles Weak localization: long range scatterers Mobility vs puddle density Correlation independent of sample characteristics Long range scattering mechanisms: Coulomb impurities Strains

22 Strains induce scalar and vector potentials

23 Relaxation times, mobilities, and puddles τ 1 = τ s 1 τ g 1 τ 1 s π N ε F ħ 4π τ g 1 π ħ N ε F π μ = σ nn = e h dd 1 cos θ 0 π v F k F τ nn V s V s ε =k F sin θ 4π dd 1 cos θ A A =kf sin θ 0

24 Wrinkles and transport Strains: Supress either weak localization, or weak antilocalization. Lead to long range, intravalley scattering. Induce puddles near the neutrality point. Corrugations induce scalar and gauge potentials

25 Substrate induced random forces

26 1 μn ħ e 1 4 log Λ k F 1 16α g g 1 1 λ L μ L μ L g ev

27 Bilayer graphene

28 Raman measurements: Correlations between strains and mobilities in graphene on BN

29 Anharmonic properties of graphene Anharmonic effects in membranes Negative thermal expansion coefficient Screening of the in plane stiffness The elastic response of graphene depends on the experimental setup (size, temperature, defects, pre existing strain, ) arxiv: Universal properties of transport in graphene Scattering is due to intravalley processes Interference processes (weak localization) are suppressed Puddles and transport are correlated Strains are the most likely origin of puddles and scattering ArXiv: , Phys. Rev. X, in press

30 ITN= Initial Training Network Marie Curie Program TOPIC:Spintronics in graphene Lifespan: Sep. 013 Aug 017 Groups: 9 partners (CSIC; CNRS, Manchester, Groningen, Aachen, INL, Nanogune, Graphenea, AMO) 3 associated partners #Trainees: 11 Phd students, 4 postdocs Coordinated by 30