Graphene With Defects

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Mechanical and thermal transport properties of graphene with defects

Feng Hao,1 Daining Fang,1,2 and Zhiping Xu1,3,a)1Department of Engineering Mechanics, Tsinghua University, Beijing 100084, China2College of Engineering, Peking University, Beijing 100871, China3Center for Nano and Micro Mechanics, Tsinghua University, Beijing 100084, China

(Received 18 May 2011; accepted 1 July 2011; published online 25 July 2011)

The roles of defects including monatomic vacancies and Stone-Wales dislocations in the

mechanical and thermal properties of graphene are investigated here through molecular dynamics

(MD) simulations. The results show that Youngs modulus of a defected graphene sheet has a

gentle dependence with the concentration of defects, while the thermal conductivity is much more

sensitive. Analysis based on the effective medium theory (EMT) indicates that this sensitivity

originates from the scattering of phonons by defects and delocalized interaction between

them, which leads to a transition from propagating to diffusive mode as the concentration

increases.VC 2011 American Institute of Physics. [doi:10.1063/1.3615290]

Graphene, the extremely monolayer material, attracts in-

tensive research interests recently owing to its outstanding

mechanical, thermal, and electronic properties, which are,

more interestingly, very sensitive to its microstructures.1 For

example, the tensile stiffness and strength of graphene sheet

are on the order of 1 TPa and 100 GPa, respectively2 and the

thermal conductivity of monolayer graphene sheet is

observed to be 5500 W/mK,35 that is one order higher than

common engineering materials with high thermal conductiv-

ities, such as copper that has a thermal conductivity of 400

W/mK. In contrast to metals where electrons carry most of

the heat and define the heat transport processes, the high

thermal conductivity of graphene is attributed to phonons.

Apparently, the key of these features relies on the structural

perfection of the hexagonal graphene lattice and strong in-

plane sp

2

bond between carbon atoms. However, defectsunavoidably present in the graphene materials from various

synthesis methods and their effects on the mechanical and

thermal properties need to be clarified.68

The experimental techniques to produce graphene sheets

can be classified into two categories. In physical methods

including both mechanical peeling and chemical vapor depo-

sition, defects such as vacancies and dislocations are widely

observed,68 for example, dislocation concentration as high as

3.6% is observed by atomic resolution transmission electron

microscopy image.8 In chemical methods such as reduced gra-

phene oxide approach, epoxy groups reside on the graphene

basal-plane.911 Furthermore, under electron irradiation or

severe Joule-heating condition, sublimation, evaporation, anddoping can lead to considerable defects concentration.1214

Open edges and edge roughness also scatter the heat flux but

are only critical for narrow graphene nanoribbons.15 In order

to evaluate their effects, here we pursue both equilibrium and

nonequilibrium molecular dynamics (MD) simulations for

some insights of the underlying mechanisms, which will

advise fabrications of graphene materials towards specified

mechanical and thermal applications.

In our MD simulations, we use the LAMMPS package.16

Periodic boundary condition is applied to a square graphene

sheet with length L 6 nm. The adaptive intermolecular reac-tive empirical bond-order potential functions and correspond-

ing parameters are used for the interatomic interactions

between carbon atoms.17 This method is widely used to pre-

dict the mechanical and thermal transport properties of gra-

phene materials and their derivatives. In this work, only

monatomic vacancies and Stone-Wales (SW) dislocations

with varying concentration are considered. The mechanical

properties under tensile loads and in-plane thermal conductiv-

ities are calculated at a concentration up to the critical value

(4%) that leads to structural failure at 300 K, for monatomic

vacancies or saturation of Stone-Wales dislocations.

The structure with a certain defect concentration is

firstly equilibrated at ambient condition (temperatureT 300 K where the quantum correction is negligible, andpressure P 1 atm) under a Nose-Hoover thermostat for 500ps. In the tensile simulation, a uniaxial load is applied in ei-

ther the armchair and zigzag direction and the other direction

is relaxed to obtain its Youngs modulus. In the equilibrium

Green-Kubo simulation, the system is subsequently switched

to a microcanonical ensemble. The atomic positions and

velocities are collected to calculate the thermal flux and its

correlation functions. The thermal conductivity is thus

obtained by following the Green-Kubo formula and averag-

ing on three samples and two runs for each. In the nonequili-

brium Muller-Plathe simulations, the graphene sheet with

certain defects concentrations are sandwiched between twopristine graphene sheets of the same sizes. Elastic collisions

between the hottest atom in the left and the right contacts

are forced every 50 fs to produce a thermal flux. To utilize

the periodic boundary condition, the sandwich structure is

mirrored.18,19 This simulation lasts for 5 ns to obtain the spa-

tial pattern of directed thermal flux.

We define the concentration f of monatomic vacancies

as the number density of atoms removed from the pristine

graphene sheet. For a Stone-Wales dislocation, f is defined

by considering two defected atoms. Youngs modulus and

thermal conductivity of the monolayer graphene sheets are

a)Author to whom correspondence should be addressed. Electronic mail:

[email protected].

0003-6951/2011/99(4)/041901/3/$30.00 VC 2011 American Institute of Physics99, 041901-1

APPLIED PHYSICS LETTERS 99, 041901 (2011)

Author complimentary copy. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp
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plotted in Figs. 1(a) and 1(b), where the results at various

concentrations are summarized.

For the mechanical properties, it can be seen that Youngs

moduli Yof defected graphene sheets feature a linear depend-

ence on the defect concentration. Youngs modulus Y0 of a

defect-free graphene sheet is 1.1 TPa when a thickness of 3.2

nm is used. We plot the relative Youngs modulus of defect

graphene sheets Ymv/Y0 and YSW/Y0 in Fig. 1(a). The curve

can be fitted into a linear function for monatomic vacancies as

Ymv/Y0 0.996 0.028fand a much more smoothly decreas-ing for Stone-Wales dislocations, which can be explained as

that it contains two heptagons and two pentagons, which pre-

serve interatomic sp2 bonding, while monatomic vacancy

breaks the integrity of pristine sheet that results in a higher

formation energy about Emv 14 eV in comparison with thenucleation energy ESW 6 eV for the Stone-Wales disloca-tion. A linear fitting for the Stone-Wales dislocations fails

here as is hard to define the concentration.

In contrast to the gentle dependence of Youngs modu-

lus, the thermal conductivity j, however, is much more sen-

sitive to the presence of these two defects. We can consider

the defected graphene sheet as a composite with the pristinegraphene lattice as the matrix and defects as inclusions. The

overall conductivity of a composite can be estimated by the

thermal conductivities of the inclusion and matrix as

jcomp1 jinc

1 jmat1. Fitting the simulation results in

Fig. 1(b) gives jmv/j0 (1.008 5.718 f)1 for monatomic

vacancies and jSW/j0 (1.001 3.330 f)1, where j0 is the

thermal conductivity of pristine graphene at 300 K. As a

result, the thermal conductivity of a pristine monolayer gra-

phene sheet is reduced to its half by introducing monatomic

vacancies at a concentration f1/2 0.175% or Stone-Walesdefects at a concentration of 0.3%. Using a characteristic

size a 0.14 nm that is the diameter of one carbon atom in

the basal plane of graphene, the prediction from the Max-well-Garnett formula based effective medium theory (EMT)

jEMT/j0 (1 f)/(1 0.5f) is also plotted in Fig. 1(b),which shows distinct difference with the simulation results.

The contradiction suggests that at this length scale of defect,

the compatible boundary condition is broken by the strong

scattering at the interfaces between pristine graphene lattice

and defects. Thus, a question is raised for practical applica-

tions that how Maxwell-Garnett formula could be improved.

The thermal conductivity of a solid can be approxi-

mately estimated by j Cvl/3, where C is the specific heat,and v is the group speed of sound wave in solid in the spirit

of Debye. For graphene v (Y/q)1/2 21.3 km/s for longitu-

dinal acoustic (LA) phonons. l is the mean phonon free path

that is reported as 775 nm for graphene sheets from experi-

mental measurements that is the origin of their ultrahigh

thermal conductivities in combination with the high

stiffness.5

We calculate the phonon spectrum based on the Fourier

transformation of the velocity auto-correlation functions

hv0vti from molecular dynamics simulation.18 Theresults show that even at a monatomic vacancy concentration

as high as 1%, the shape and peaks are preserved well, which

demonstrates that the specific heat C kB(hx/kBT)2exp(hx/

kBT)/[exp(hx/kBT) 1]2 and group velocities v dx/dk for

a phonon mode with a frequency x and wave vector k have

negligible change. A quantitative estimation based on the

shift of peaks shows that it only leads to less than 5% reduc-

tion of thermal conductivity if the reduction of l is not taken

into account.

This result suggests that the presence of defects of low

concentration has less effects on the group speed v and spe-

cific heat C for the phonons in graphene that is consistent

with the gentle reduction of Youngs modulus as we observe

before. However, the impacts of defects on the thermal con-

ductivity ldefectphonon is dominantly accounted by the meanfree path l1 l1defect-phonon l1

phonon-phonon in addition to

the phonon-phonon scattering mechanism. Moreover, Fig.

1(b) also shows that when the defect concentration is high

enough, l becomes less sensitive in comparison with the sit-

uation of low concentration, suggesting a transition from

propagating to diffusive mechanism.20 This is also reflected

in our further calculations for the temperature dependence of

thermal conductivities when 2% monatomic vacancies are

introduced. In Fig. 2, it is shown that not only the thermal

conductivity j and the temperature dependence is much

reduced but also there exists a peak at T 200 K, and thelow-temperature reduction ofj is attributed to scattering of

phonons with small wave vectors.15

Moreover, as fincreases,the sensitivity ofj on f is much reduced due to the disorder

nature of phonon transport processes.

To obtain more insights into the sensitivities of mechan-

ical and thermal properties on the defect concentration, we

plot the stress distribution around a monatomic vacancy in

Fig. 3(a) and the heat flux in Fig. 3(b). These plots show

distinct difference between the stress and heat flux distribu-

tion around the defect, i.e., the influence region of stress dis-

tribution is much more localized in comparison with the heat

flux and significant scattering occurs around the defects.

To predict the effective thermal properties of a nano-

composite, Nan and his collaborators apply the Maxwell-

Garnett effective medium theory by introducing a so-called

FIG. 1. (Color online) (a) Youngs modulus and (b)

thermal conductivity of a monolayer graphene sheet

with monatomic vacancies or Stone-Wales disloca-

tions. Inset in (a): a monatomic vacancy and Stone-

Wales defect in graphene as usually encountered in

experiments. Numerical fitting and predictions from

the Maxwell-Garnett EMT are also plotted for the

thermal conductivities in subplot (b). Point defects

scatter phonons at a larger scale than the defect size

and scatter centers have strong correlation at elevated

concentration. These effects are not included in EMT

and lead to large deviation from the simulation results.

041901-2 Hao, Fang, and Xu Appl. Phys. Lett. 99, 041901 (2011)



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Kapitza radius for the interfacial thermal resistance.21 This

theory is further modified for nanocomposites with spheroi-

dal inclusions.22,23 However, here it is difficult to define an

interface between the defected atoms and others in a pristine

hexagonal lattice. To quantitatively characterize this differ-

ence, we introduce an influence coefficient R in the

Maxwell-Garnett formula instead, i.e.,

jeff jm 1 dRfb

1 Rfb

; b

j jmj d 1jm

; (1)

where d 3 is dimension of the problem, and jeff, j, and jmare the thermal conductivities of the whole defected gra-

phene sheet, defects, and the pristine graphene sheet, respec-

tively. Our calculations show that R is strongly correlated to

the defect concentration f, which can be fitted into

R (0.002 0.011f)1. At low defect concentration, thepoint defects serve as local scattering centers to the heat flux

through them. While at elevated concentrations, the fast

decaying of R with respect to increasing f indicates that dif-

ferent scattering centers interact with each other, which

results in a delocalized scattering to the propagating modes

phonons and the overall scattering cross-section is reduced

in comparison with discrete and non-interacting defects.

When this delocalization is established as the defect concen-

tration is high enough, both R and l become less sensitive

with respect to that of low concentration.

In summary, we performed molecular dynamics simula-

tions for defected graphene sheets. It is found that Youngs

modulus is reduced with a linear dependence for vacancies and

a much more smooth decrease for Stone-Wales dislocations.

On the other hand, thermal conductivity relies dramatically onthe defect concentration, especially at low concentration. The

shortening of phonon mean free path l is responsible for this

reduction. At higher defect concentrations, the scattering cen-

ters percolates throughout the whole material and the thermal

conductivity of defected graphene sheet behaves similarly as

in disordered materials, where diffusive modes dominates the

thermal transfer process and the temperature dependence is

much reduced. These understadings could be used to evaluate

the quality of graphene for related applications. Similar

phenomena are expected for functionalized or doped graphene

sheets, e.g., hydrogenated, oxidized, or fluorinated ones where

functional groups behave as the scattering centers as defects do

here, that could inspire nanoengineering approaches to tune the

mechanical and thermal properties of graphene.11,24

This work is supported by Tsinghua University through

the Key Talent Support Program and the National Science

Foundation of China through Young Scholar Grant 11002079

(Z.X.). This work is also supported by the Shanghai Super-

computer Center of China.

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FIG. 2. (Color online) Temperature dependence of the thermal conductiv-

ities j/j0 of pristine graphene and defected graphene with monatomic

vacancies.

FIG. 3. (Color online) (a) Stress and (b) heat flux distribution in a pristine

graphene sheet and around a monatomic vacancy defect.

041901-3 Hao, Fang, and Xu Appl. Phys. Lett. 99, 041901 (2011)

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Graphene With Defects