Graphene: Continuum Mechanics and Atomistic Modeling

Rui Huang

Center for Mechanics of Solids, Structures and Materials

Department of Aerospace Engineering and Engineering Mechanics

The University of Texas at Austin

Acknowledgments

� Qiang Lu (postdoc)

� Wei Gao (graduate student)

� Zachery Aitken (undergraduate student, now at Caltech)

� Prof. Marino Arroyo (Spain)

� Prof. Li Shi (UT/ME)

� Funding: NSF (ARRA)

From Wikipedia:

Graphene is a one-atom-thick planar sheet of sp2-bonded carbon

atoms that are densely packed in a honeycomb crystal lattice. The term

Graphene was coined as a combination of graphite and the suffix -ene

by Hanns-Peter Boehm, who described single-layer carbon foils in

1962. Graphene is most easily visualized as an atomic-scale chicken

wire made of carbon atoms and their bonds.

Graphene Based Devices

Garcia-Sanchez, et al., 2008.

Suspended nanoribbons for NEMS devicesPatterned graphene on oxide for bipolar

p-n-p junctions

Ozyilmaz et al., 2007.

Mechanical properties of monolayer graphene

• High in-plane stiffness

• High tensile strength

• High bending stiffness (relative to its own weight)

� Isotropic, linearly elastic under infinitesimal deformation

� Anisotropic, nonlinear under finite deformation

� Graphene nanoribbons: edge effects? size dependent?

� Substrate-supported graphene: adhesive interactions?

Nonlinear Continuum Mechanics of 2D Sheets

2D-to-3D deformation gradient:

In-plane deformation: 2D Green-Lagrange strain tensor

Bending: 2D curvature tensor (strain gradient)

d d=x F X

X1

X2

x1

x3x2

J

iiJ

X

xF

∂

∂=

( )JKiKiJJK FFE δ−=2

1

2

iI iIJ i i

J I J

F xn n

X X X

∂ ∂Κ = =

∂ ∂ ∂

Strain energy (hyperelasticity): ( )∫ Φ=A

dAU ΚE,

Lu and Huang, Int. J. Applied Mechanics 1, 443-467 (2009).

2D Stresses and Moments

2nd Piola-Kirchoff stress and moment

(work conjugates)

Tangent moduli:

IJ

IJE

S∂

Φ∂=

IJ

IJMΚ∂

Φ∂=

KLIJKL

IJIJKL

EEE

SC

∂∂

Φ∂=

∂

∂=

2

KLIJKL

IJIJKL

MD

Κ∂Κ∂

Φ∂=

Κ∂

∂=

2

KLIJIJ

KL

KL

IJIJKL

EE

MS

Κ∂∂

Φ∂=

∂

∂=

Κ∂

∂=Λ

2Intrinsic coupling between

tension and bending

Κ

Κ

Κ

ΛΛΛ

ΛΛΛ

ΛΛΛ

+

=

12

22

11

333231

232221

131211

12

22

11

333231

232221

131211

12

22

11

22 d

d

d

dE

dE

dE

CCC

CCC

CCC

dS

dS

dS

ΛΛΛ

ΛΛΛ

ΛΛΛ

+

Κ

Κ

Κ

=

12

22

11

332313

322212

312111

12

22

11

333231

232221

131211

12

22

11

22 dE

dE

dE

d

d

d

DDD

DDD

DDD

dM

dM

dM

An incremental form of

the generally nonlinear

and anisotropic behavior:

Lu and Huang, Int. J. Applied Mechanics 1, 443-467 (2009).

Units for 2D quantities

( )ΚE,Φ strain energy density function: J/m2

IJ

IJE

S∂

Φ∂= 2D stress: N/m

KLIJKL

IJIJKL

EEE

SC

∂∂

Φ∂=

∂

∂=

2

2D in-plane modulus: N/m

IJ

IJMΚ∂

Φ∂= moment intensity: (N-m)/m

KLIJKL

IJIJKL

MD

Κ∂Κ∂

Φ∂=

Κ∂

∂=

2

bending modulus: N-m

KLIJIJ

KL

KL

IJIJKL

EE

MS

Κ∂∂

Φ∂=

∂

∂=

Κ∂

∂=Λ

2

coupling modulus: N

�Analogous to the 2D plate/shell theories (but no thickness).

(n,n): armchair

(n,0): zigzag

(2n,n)

αααα

Atomistic Modeling of Graphene

• 2nd-generation REBO potential (Brenner et al., 2002)

– Bond angle effect (second-nearest neighbors)

– Dihedral angle effect (third-nearest neighbors)

– Radical energetics (defects and edges)

• Molecular Mechanics: energy minimization for static equilibrium states.

• Stress and moment calculations

– Energy derivation

– Virial stress calculations

– Direct force evaluation

0 0.05 0.1 0.15 0.2 0.25 0.3−7.5

−7

−6.5

−6

Nominal strain ε

En

erg

y (

eV

)

A

B

C

D

Uniaxial Stretch of Monolayer GrapheneE

ner

gy p

er a

tom

(eV

)

Lu and Huang, Int. J. Applied Mechanics 1, 443-467 (2009).

0 0.05 0.1 0.15 0.2 0.25 0.30

5

10

15

20

25

30

35

40

Nominal strain ε

No

min

al str

ess P

11 (

N/m

)

α = 0 (zigzag)

α = 10.89o

α = 19.11o

α = 30o (armchair)

A

C D

B

εε

d

dP

Φ=)(11

Nominal stress-strain

relation:

Anisotropic tangent moduli

0 0.05 0.1 0.15 0.2 0.25 0.3−30

−20

−10

0

10

20

30

Green−Lagrange strain E11

C3

1 (

N/m

)

(n,n): armchair

(n,0): zigzag

(2n,n)

αααα

Graphene is linear and isotropic under

infinitesimal deformation, but becomes nonlinear

and anisotropic under finite deformation.

Coupling between

tension and shear

0 0.05 0.1 0.15 0.2 0.25 0.3−50

0

50

100

150

200

250

300

350

C11 (

N/m

)

α = 0 (zigzag)

α = 10.89o

α = 19.11o

α = 30o (armchair)

0 0.05 0.1 0.15 0.2 0.25 0.3−50

0

50

100

150

C21 (

N/m

)

=

12

22

11

333231

232221

131211

12

22

11

2dE

dE

dE

CCC

CCC

CCC

dS

dS

dS

Lu and Huang, Int. J. Applied Mechanics 1, 443-467 (2009).

Fracture strength under uniaxial stretch

0 0.05 0.1 0.15 0.2 0.25 0.30

5

10

15

20

25

30

35

40

Nominal strain ε

Nom

inal str

ess P

11 (

N/m

)

α = 0 (zigzag)

α = 10.89o

α = 19.11o

α = 30o (armchair)

Nom

inal fr

actu

re s

train

0 5 10 15 20 25 300.1

0.2

0.3

0.4

0 5 10 15 20 25 3030

32

34

36

Chiral angle (degree)

Nom

inal fr

actu

re s

tress (

N/m

)

02

2

=∂

Φ∂=

∂

∂

εε

PFracture occurs as a result of

intrinsic instability of the

homogeneous deformation:

� The nominal stress and strain to fracture depend on the direction

of uniaxial stretch.Lu and Huang, Int. J. Applied Mechanics 1, 443-467 (2009).

Monolayer graphene under uniaxial tension

0 0.05 0.1 0.15 0.2 0.25 0.30

10

20

30

40

Nominal strain

Nom

inal 2D

str

ess (

N/m

)

zigzag (MM/REBO)

armchair (MM/REBO)

zigzag (Wei et al., 2009)

armchair (Wei et al., 2009)

0 0.05 0.1 0.15 0.2 0.25 0.30

100

200

300

400

Nominal strain

2D

Young's

modulu

s (

N/m

)

zigzag (MM/REBO)

armchair (MM/REBO)

zigzag (Wei et al., 2009)

armchair (Wei et al., 2009)

• Disagreement: the REBO potential underestimates the initial

Young’s modulus.

• Agreement: the fracture stress/strain is higher in the zigzag

direction than in the armchair direction.

Bending Modulus of Monolayer Graphene

0 0.5 1 1.5 2 2.5 30

0.05

0.1

0.15

Bending curvature (1/nm)

Str

ain

en

erg

y p

er

ato

m (

eV

/ato

m)

B1, armchair

B1, zigzag

B2, armchair

B2, zigzag

B2*, armchair

B2*, zigzag

0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

Bending curvature (1/nm)

Bendin

g m

odulu

s (

nN

−nm

)

B1, armchair

B1, zigzag

B2, armchair

B2, zigzag

B2*, armchair

B2*, zigzag

• Bending moment-curvature is nearly linear, with slight

anisotropy.

• Including the dihedral angle effect leads to higher

bending energy and bending modulus.

Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 42, 102002 (2009).

0 0.5 1 1.5 2 2.50

0.05

0.1

0.15

0.2

0.25

Bending curvature (1/nm)

Bendin

g m

odulu

s (

nN

−nm

)

B1, armchair

B1, zigzag

B2, armchair

B2, zigzag

B2*, armchair

B2*, zigzag

Effect of dihedral angle

i

j

k

l

θθθθijk

θθθθjil

θθθθq4

θθθθq2θθθθq1

θθθθq3

ΘΘΘΘq3

ΘΘΘΘq4ΘΘΘΘq1

ΘΘΘΘq2

Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 42, 102002 (2009).

)()( ijAijijRij rVbrVV −=

RC

ij

DH

ijjiijij bbbb Π+++= −− )(21 πσπσ

( )[ ]∑≠

Θ−=),(,

20 )()(cos12 jilk

jlcikcijkl

DH

ij rfrfT

b ijljikijkl nn ⋅=Θcos

Physical Origin of Bending Modulus

Bending modulus of a thin elastic plate:

For monolayer graphene, bending moment and

bending stiffness result from multibody interatomic

interactions (second and third nearest neighbors).

• D = 0.83 eV (0.133 nN-nm) by REBO-1

• D = 1.4 eV (0.225 nN-nm) by REBO-2

• D = 1.5 eV (0.238 nN-nm) by first principle

0 0( ) 14

2 3

ijA

ijk

bV r TD

σ π

θ

− ∂= − ∂

3~ Ehd

dMD

κ= h

Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 42, 102002 (2009).

Coupling between bending and stretching

0 0.005 0.01 0.015 0.020.113

0.114

0.115

0.116

0.117

0.118

Strain ε = R/R0 − 1

Str

ain

energ

y p

er

ato

m (

eV

/ato

m)

R

Lu, Arroyo, and Huang, J. Phys. D: Appl. Phys. 42, 102002 (2009).

( ) ( ) ( ) 2200

20

2

1

2

1

2

1, εκκσεκκκεκ CDMDW +−++−+≈

� The tube radius increases upon

relaxation, leading to simultaneous

bending and stretching.

nm 397.00 =R

AIREBO vs REBO

AIREBO: Adaptive intermolecular REBO potential (Stuart et al.

2000); implemented in LAMMPS.

• Add a LJ term for long-range weak interactions

• Add a torsional term to improve the dihedral angle effect of

single bond (C-C and C-H)

• Modified parameters in the spline function of REBO

Effects on mechanical properties of graphene:

• The equilibrium bond length is slightly shorter (1.397 Ǻ vs

1.420 Ǻ)

• Increase the in-plane stiffness (good)

• Screw up the bending modulus (bad)

Excess Edge Energy and Edge Force

1.391

1.425

1.420

1.420

1.367

1.412

1.425

1.398

1.425

1.4201.421

1.420

Edge energy (eV/nm ) Edge force (eV/nm) r0

Armchair Zigzag Armchair Zigzag (nm)

DFT [17]

(GPAW)9.8 13.2 - - 0.142

DFT [18]

(VASP)10 12 -14.5 -5 0.142

DFT [22]

(SIESTA)12.43 15.33 -26.40 -22.48 0.142

MM [20]

(AIREBO)- - -10.5 -20.5 0.140

MD [21] - - -20.4 -16.4 0.146

MM

(REBO)10.91 10.41 -8.53 -16.22 0.142

Zigzag edge:

Armchair edge:

fZfZ

fAfA

0 0.2 0.4 0.6 0.8 1−7.4

−7.3

−7.2

−7.1

−7

−6.9

−6.8

1/W (nm−1

)

Energ

y p

er

ato

m (

eV

)

armchair, un−relaxed

armchair, 1−D relaxation

zigzag, un−relaxed

zigzag, 1−D relaxation

Eq. (6), γ = 11.1 eV/nm

Eq. (6), γ = 10.6 eV/nm

Lu and Huang, Phys. Rev. B 81, 155410 (2010).

γγ

W

SU

NUWU 0

00

2)( +=+=

Edge buckling of GNRs

2 4 6 8 10 1210.27

10.275

10.28

10.285

10.29

Buckle wavelength λ (nm)

Excess e

nerg

y (

eV

/nm

)

y = 2e−05*x4 − 0.00071*x

3 + 0.0093*x

2 − 0.054*x + 10

2 4 6 8 10 12 14 1610.87

10.875

10.88

10.885

10.89

Buckle wavelength λ (nm)

Excess e

nerg

y (

eV

/nm

)

y = 2.2e−06*x4 − 0.00011*x

3 + 0.0019*x

2 − 0.014*x + 11

Zigzag GNR

Armchair GNR

Intrinsic wavelength ~ 6.2 nm

Intrinsic wavelength ~ 8.0 nm

�The wavelengths for edge buckling do not scale with D/f.

Lu and Huang, Phys. Rev. B 81, 155410 (2010).

Zigzag GNRs Armchair GNRs

GNRs under Uniaxial Tension

FF

( ) ( ) ( )LWLU εγεε 2+Φ=

δεδ FLU =

ε

γ

εεσ

d

d

Wd

d

d

dU

WLW

F 21+

Φ===

GNRs under Uniaxial Tension

( )ε

γ

εεσ

d

d

Wd

d

W

F 2+

Φ==

FF

Zigzag GNRs Armchair GNRs

Lu and Huang, arXiv:1007.3298 (2010).

2D Young’s Moduli of GNRs

( )ε

γ

εεσ

d

d

Wd

d 2+

Φ=

( )2

2

2

2 2

ε

γ

εε

d

d

Wd

dE +

Φ=

Young’s modulus under

infinitesimal strain:

edgebulkE

WEE 000

2+=

The nonlinear dependence of the excess edge energy on strain leads

to anisotropic, width-dependent Young’s modulus for GNRs.

Lu and Huang, arXiv:1007.3298 (2010).

Fracture of graphene nanoribbons

Zigzag edge

Armchair edge

Lu and Huang, arXiv:1007.3298 (2010).

Graphene on Oxide Substrates

Ozyilmaz et al., 2007.

HR-STM image (Stolyarova et al., 2007)

The 3D morphology is important for the transport properties of graphene-based devices.

Ishigami et al., 2007.

RMS2

Correlation

length

Van der Waals Interaction

Lennard-Jones potential for

particle-particle interactions:

Monolayer-substrate interaction

(energy per unit area):

−

Γ−=

9

0

3

00

2

1

2

3)(

h

h

h

hhU

hr h

U

0h0

-Γ0

Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).

1 2

6 12( )

LJ

C CW r

r r= − +

Van der Waals Thickness and Energy

Gupta et al., 2006.

� Interlayer spacing in graphite ~ 0.34 nm;

� AFM measurements of h0 for graphene on oxide range from 0.4 to 0.9 nm;

� The adhesion energy (Γ0) has not been measured directly;

� Theoretically estimated values for Γ0 range from 0.6 to 0.8 eV/nm2.

Sonde et al., 2009.

Flat Graphene on Flat Surface

( )3 9

0 00

3 1

2 2vdW

h hU z

z z

= −Γ −

max 230 MPaσ =

0 0.6h nm=

Representative values:

( )

−

Γ=

10

0

4

0

0

0

2

9

z

h

z

h

hzvdWσ

maxσ

z

Interfacial

strength:0

0max 466.1

h

Γ=σ

Initial

stiffness: 2

0

00

27

hk

Γ=

2eV/nm 6.00 =Γ

MPa/nm 72000 =k

Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).

Strain-Induced Instability

� The competition between the elastic strain energy of graphene and the van der Waals

interaction energy sets a critical strain for instability as well as the equilibrium

corrugation wavelength beyond the critical strain.

44344214342143421PlaneIn

gg

BendingPlaneIn

g

CDCU

−−

+

+

≈ 4

4

2

422

64

32

4

2

4

~δ

λ

πδ

λ

π

λ

πε

+

+−Γ≈

4

0

2

0

08

675

4

271

~

hhU

gg

vdW

δδ

Elastic strain energy

of graphene:

Van der Waals

Interaction Energy:

h0

substrate

z

x

ελ

+=

λ

πδ

xhz gg

2sin0

vdWg UUU~~~

+=Total free energy:

Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).

Strain-Induced Corrugation of Graphene

41

0

2

0

272

Γ=

Dhc πλ

0

036

Ch

Dc

Γ−=ε

0074.0−=cε

nmc 68.2=λ

Representative values:

N/m353=C eV5.1=D

0 0.6h nm=2

eV/nm 6.00 =Γ

� On a flat surface, a graphene monolayer is flat and stable below a critical

compressive strain;

� Beyond the critical strain, the monolayer corrugates with increasing amplitude

and decreasing wavelength.

Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).

Graphene on a Corrugated Surface

( ) ( ) ( ) ( )2

0

2

2

0

2

0

1,~~

hhU

hhhUhUhU

sgsg

vdWgvdW

δδδδδ +

+

+≈

( ) ( )hkh

h

h

h

hhU vdW

42

5

2

9 2

0

11

0

5

001 =

+

−

Γ=

( )

−

Γ=

λ

π

λ

π

λ

π

λπ

hK

h

hhK

h

hhU

2

24

29 656

11

0

3

323

5

00

3

2

substrate

z

x

λ

h

h

s

and find to

energy total the minimize

and Given

gδ

λδ ,

=

λ

πδ

xz ss

2sin

+=

λ

πδ

xhz gg

2sin

van der Waals

interaction energy:

vdWg UUU~~~~~

+=

Total free energy:

Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).

Substrate Induced Corrugation of Graphene

ΓΓ=

0

2

0000

,;,εδλ

δ

δ C

h

D

hhf s

s

g

� Conformal at long wavelengths.

� Non-conformal at short wavelengths.

� Transition between the two states depends on the amplitude of substrate

surface corrugation, becoming more abrupt with increasing amplitude.

ΓΓ=

0

2

00000

,;,εδλ C

h

D

hhg

h

h s

Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).

Effect on Adhesion Energy

� The adhesion energy decreases as the corrugation wavelength decreases,

approaching a plateau at the short‐wavelength limit.

� The adhesion energy decreases with increasing amplitude of the substrate

surface corrugation.

� Better adhesion at the conformal state than the non-conformal state.

Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).

Effects of Mismatch Strain

� A tensile strain tends to flatten the supported graphene, while a

compressive strain tends to increase the corrugation amplitude.

� A “snap-through” instability occurs at a critical tensile strain.

� A “buckling” instability occurs at a critical compressive strain.

Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).

Critical Strain for Buckling Instability

• With increasing amplitude of the substrate surface corrugation:

– The critical compressive strain for buckling decreases.

– The corresponding buckling wavelength increases.

Aitken and Huang, J. Appl. Phys. 107, 123531 (2010).

Summary

� Nonlinear continuum mechanics for 2D graphene monolayer

� Atomistic modeling of graphene under bending and stretching

� Excess edge energy, edge forces, and induced edge buckling

� Graphene nanoribbons under uniaxial tension: edge effects on elastic modulus and fracture

� Graphene on substrates: van der Waals interaction and corrugation; strain-induced corrugation