Rippling and Ripping Graphene

Rippling and Ripping Graphene Michael Marder Professor of Physics Center for Nonlinear Dynamics and Department of Physics The University of Texas at Austin Clifton Symposium, June 2012, Symi Marder (UT Austin) Graphene Symi 1 / 25

Outline Rippling 1 Rippling Experimental Motivation Theoretical Background Impurity Mechanism 2 Ripping Experimental Motivation Fracture Mechanics Framework Numerics and Continuum Theory 3 Conclusions Marder (UT Austin) Graphene Symi 2 / 25

Graphene Ideal Rippling Experimental Motivation 500 nm Regions of perfect twodimensional crystal (A) (B) 1.42 Å Marder (UT Austin) Graphene Symi 3 / 25

Rippling Experimental Motivation Graphene Ripples: Meyer, Geim, Katnelson, Novoselov, Booth, and Roth, Nature(2007). Marder (UT Austin) Graphene Symi 4 / 25

Rippling Theoretical Background First theoretical ideas Thermal Fluctuations, Fasolino, Los, and Katsnelson (2007) and Abedpour et al. (2007) Response: thermal fluctuations are too fast to have been seen! [ 1 E = d 2 r 2 ρḣ2 + κ ( 2 h ) ] 2, (1) 2 where ρ is the mass per area. Thus thermally excited ripples of wavenumber k should be oscillating at a frequency ω given by ω 2 = κ ρ k4. (2) f 10 10 Hz (3) Marder (UT Austin) Graphene Symi 5 / 25

Rippling First theoretical ideas Mermin-Wagner Theorem, Meyer et al (2007) Theoretical Background Mermin-Wagner theorem forbids two-dimensional crystals Response: This refers to in-plane motion on enormous length scales l a exp[ga 2 /k B T], (4) where a is a lattice spacing and G is the shear modulus of graphene. Breakdown in long-range order due to in-plane motions at distances of order l > 10 30 m Marder (UT Austin) Graphene Symi 6 / 25

Rippling Rippling from metric distortions Impurity Mechanism y a x b Marder (UT Austin) Graphene Symi 7 / 25

Rippling Impurity Mechanism Edge effects alone Ripples are 10 Åhigh and 30 Åwide. Decay within 3Åas they leave the edge of the sheet. Marder (UT Austin) Graphene Symi 8 / 25

Rippling Impurity Mechanism Impurity Mechanism Correspondence with Geim Dear Mike, I mulled this over.... It is 100% likely that there are a plenty of defects like OH... OH increases C-C bond length by 10% ( Xu et al. J. Phys. Chem. 2007) Molecular dynamics simulations, MEAM carbon potentials, 200 200Å systems Marder (UT Austin) Graphene Symi 9 / 25

Rippling Impurity Mechanism Impurity Mechanism Thompson-Flagg, Moura, and Marder, EPL (2009) 85 46002 Average ripple wavelength (Å) 60 40 20 0 2.0 rms ripple amplitude (Å) 1.5 1.0 0.5 0 0.0 0.1 0.2 0.3 0.4 OH concentration Note peak-to-peak amplitude is 6 times rms amplitude: low concentrations constent with experiment... but even low concentrations of free radicals are problematic. Marder (UT Austin) Graphene Symi 10 / 25

Outline Ripping 1 Rippling Experimental Motivation Theoretical Background Impurity Mechanism 2 Ripping Experimental Motivation Fracture Mechanics Framework Numerics and Continuum Theory 3 Conclusions Marder (UT Austin) Graphene Symi 11 / 25

Experimental Motivation Work with Maria Moura Ripping Experimental Motivation H Conley, Vanderbilt van der Zande, Nano Lett 2010 Marder (UT Austin) Graphene Symi 12 / 25

Ripping Fracture Mechanics Framework Previous work on fracture of graphene In-plane Lu, Gao and Huang, Modelling and Simulation in Mat Sci, & Eng (2011) Terdalkar, Chem Phys Lett 2010 Out-of plane Kawai, PRB (2009) Sen and Small (2010) Kim, Nano Lett. (2011) Marder (UT Austin) Graphene Symi 13 / 25

Ripping Fracture Mechanics Framework Back-Gate Voltage Configuration Suspend graphene between two shelves Uniform downward force due to voltage difference with substrate Analyze fracture in this geometry Marder (UT Austin) Graphene Symi 14 / 25

Geometry of failure Ripping Fracture Mechanics Framework U = dr ds [ k 2 ( ) dθ 2 ] s + f a ds sin(θ(s ) ds 0 It turns out that radius of curvature of bend depends upon downward force f a. Marder (UT Austin) Graphene Symi 15 / 25

Geometry of failure Ripping Fracture Mechanics Framework U 1 6 f m 2 l 2 a n + C kf a nml; C = 4 3 (2 2) Adding in fracture energy Γl provides criterion for onset of failure. Marder (UT Austin) Graphene Symi 16 / 25

Ripping Numerics and Continuum Theory Simulations with MEAM Support Support L Force f Systems on the order of 100 100Å. Marder (UT Austin) Graphene Symi 17 / 25

Simulations with MEAM Movie 1 Ripping Numerics and Continuum Theory Marder (UT Austin) Graphene Symi 18 / 25

Simulations with MEAM Movie 2 Ripping Numerics and Continuum Theory Marder (UT Austin) Graphene Symi 19 / 25

Simulations with MEAM Movie 3 Ripping Numerics and Continuum Theory Marder (UT Austin) Graphene Symi 20 / 25

Simulations with MEAM Movie 4 Ripping Numerics and Continuum Theory Marder (UT Austin) Graphene Symi 21 / 25

Numerics vs Analytics Green, 100 100, Orange, 100 200Å Ripping Numerics and Continuum Theory 4. 10 11 3. 10 11 F N atom 2. 10 11 1. 10 11 0 0 2. 10 9 4. 10 9 6. 10 9 8. 10 9 1. 10 8 l m Marder (UT Austin) Graphene Symi 22 / 25

Ripping Numerics and Continuum Theory Connection to Experiment Electric field, estimate E 20 10 6 V/m Force, estimate 10 16 N/atom Numerical fracture energy Γ 3.8 10 9 J/m These values do not quite work; critical crack length of 1 µm in 2 µm sheet. Fracture energy therefore probably must be less than value extracted numerically from MEAM. Marder (UT Austin) Graphene Symi 23 / 25

Outline Conclusions 1 Rippling Experimental Motivation Theoretical Background Impurity Mechanism 2 Ripping Experimental Motivation Fracture Mechanics Framework Numerics and Continuum Theory 3 Conclusions Marder (UT Austin) Graphene Symi 24 / 25

Conclusions Conclusions Mechanics and fracture mechanics provides a valuable framework to analyze graphene, although it is atomically thin. Support Support L Force f Marder (UT Austin) Graphene Symi 25 / 25