Thermal Transport Around Tears in Graphene

Thermal transport around tears in graphene

G. C. Loh, E. H. T. Teo, and B. K. Taya)

School of Electrical and Electronic Engineering, Nanyang Technological University, Singapore 639798

(Received 29 October 2010; accepted 18 December 2010; published online 24 February 2011)

Tears in any material act as barriers to phonon transport. In this study, molecular dynamics

simulations are employed to investigate thermal transport around tears in graphene. Specifically,

thermal boundary conductance across different tear orientations and lengths is computed. Analysis

of vibrational density of states suggests that long-wavelength acoustic phonons within the spectrum

range 0–700 cm�1 are vital to thermal transport across the tears. Different phonon scattering

phenomena are observed for both tear orientations. It is proposed that the dissimilitude of the

scattering processes encountered by phonons carrying energy around the tears to the opposite end

explains why thermal transport is generally more efficient for longitudinal tears in our simulations.VC 2011 American Institute of Physics. [doi:10.1063/1.3549735]

I. INTRODUCTION

Thermal energy is transported predominantly by elec-

trons and phonons. In semiconductors, the primary heat car-

riers are wavelike lattice vibrations, or phonons. Through

recent academic and industrial efforts, it has been discovered

that materials such as carbon nanotubes and graphene1–7

have excellent intrinsic thermal conductance. However it is

imperative to note that utilizing these materials is only a par-

tial step to ensure the efficient removal of heat. At interfaces

between materials, phonon transmission is often not perfect.

As the efficiency of interfacial transport may dictate the

effective thermal conductance, especially in complex nano-

structures with numerous interfaces, it is increasingly perti-

nent that this physical phenomenon is explored further.

Thermal boundary resistance, or Kapitza resistance, is

an interfacial resistance that exists due to a mismatch of the

lattice-vibrational spectra of the two media. Not all phonons

transmit through the interface, and the probability of trans-

mission is governed essentially by the matching of the vibra-

tional spectra and the phonon velocities. Two well known

models provide theoretical and numerical interpretations to

this phenomenon. The first tool is the acoustic mismatch

model (AMM)8,9 which defines the interfacial thermal resist-

ance to be a function of phonon density on both media.10

The transmission coefficient from medium 1 to medium 2 is

given by 11

sr1!2 ¼4Z2Z1

ðZ1 þ Z2Þ2; (1)

where Zi refers to the acoustic impedance, and is the product

between the density q and phonon velocity c of medium i.As a manifestation of the phonon dynamics, and an extension

of Fourier’s law, thermal boundary resistances (TBRs) can

also be calculated by quantifying the temperature drop at the

interface, and the thermal flux across it. The following rela-

tionship applies

RK ¼DT

J; (2)

where DT is the temperature drop, and J is the thermal flux

through a cross-sectional area per unit time.11

The second tool is the diffuse mismatch model (DMM)12

which postulates that scattering at the interface is completely

diffusive in nature. In other words, the scattered phonons lose

all memory of polarization and incident angle.13 Essentially,

only energy is conserved through the interface. In addition,

within this model, elastic scattering is commonly assumed,

suggesting that a phonon of frequency x will only scatter

with another phonon of the same frequency. Therefore, pho-

non transmission is entirely determined by the phonon popu-

lation of the lower Debye frequency material. Basically,

there is more species mixing at elevated temperatures. The

interface roughness is greater, magnifying the probability of

diffusive scattering. Consequently, at high temperatures,

DMM generally predicts a more accurate thermal boundary

resistance than AMM. However, as a matter of fact, DMM

results can still disagree with experimental data by more than

an order of magnitude. It is suggested that the discrepancies

may be due to reasons including multiple elastic phonon scat-

tering at the interface,14,15 the presence of inelastic scatter-

ing,13 electron-phonon resistances,16 and different definitions

of interface roughness.14,17

Other than interfaces, thermal boundary resistances also

exist in defected materials. Although defects, especially

point defects, are strictly not “boundaries”, the scattering of

phonons at these defects begets a temperature discontinuity,

similarly observed at boundaries. This thermal barrier abates

the efficiency of heat sinks to direct away thermal energy,

effectively reducing the overall thermal conductivity.

In the work by Terrones,18 graphene-related defects are

presented, including structural, topological, doping-induced,

nonsp2-carbon, and high-strain folding defects. Structural

defects distort the lattice structure; topological defects or

Stone–Thrower–Wales (STW) defects are created when a

carbon–carbon bond is rotated 90� within four neighboring

hexagons, inducing the transformation of two pentagons and

two heptagons; doping-induced defects refer to the presence

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

[email protected].

0021-8979/2011/109(3)/043508/6/$30.00 VC 2011 American Institute of Physics109, 043508-1

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of substituted atoms; non-sp2-carbon defects comprise

vacancies, adatoms, edge sites, and interstitials; high-strain

folding defects are formed when adjacent graphene layers

are annealed to form loops. In this work, we emphasize on

the non-sp2-carbon defect of tears. A vital difference

between tears and normal interfaces is the presence of cova-

lent bonds. Here we assume that covalent bonds are broken

at a tear, and the only interactions between the affected

atoms are noncovalent bonds such as van der Waals. At

interfaces, both covalent and noncovalent bonds may exist.

Graphene is a type of carbon allotrope in which the

atoms are arranged in a two-dimensional honeycomb lat-

tice.19 It is the fundamental building block for all common

types of graphitic allotropes: zero-dimensional fullerene,

one-dimensional carbon nanotube, and three-dimensional

bulk graphite. Its superior electrical conductivity19 and unri-

valled thermal conductivity have sparked a lot of interest in

various scientific niches. Recently attempts to reduce gra-

phene to desired sizes and geometries include20 (1) subse-

quent steps of electron beam lithography and plasma

etching,21,22 (2) chemically exfoliating graphene nanoribbons

from graphite,23 (3) using scanning tunneling microscopy

(STM) to electrochemical-etch graphite,24 (4) catalytic reac-

tion of Fe and Ni particles on graphene,25,26 (5) unzipping

carbon nanotubes by chemical oxidation,27,28 and (6) using

scanning probe microscopic manipulation to cut oxidized

graphene sheets.20 Nevertheless, there is a strong likelihood

that defects will undesirably exist in the resultant nanosized

graphene. Any of the aforementioned reducing procedures

may create partial tears at random locations on the material,

altering the intrinsic material properties of graphene.

This study aims to delineate the effects of tearing on

thermal transport in graphene. Specifically, factors such as

the length and orientation may well be crucial in the under-

standing of phonon transport around tears. Due to the resem-

blance of graphene and carbon nanotubes, since they are

both built from the honeycomb lattice, we can draw parallel-

ism between the two materials.

II. SIMULATION DETAILS

Molecular dynamics simulations were carried out to

probe the mechanism of phonon scattering in the presence of

tears. 480 atoms were contained in a 13.3 A by 114.9 A by

13.3 A simulation cell and arranged in hexagonal rings to

represent a graphene nanoribbon. Tears were introduced by

positioning some of the atoms further away from each other,

such that the distance between them exceeds the defined cut-

off distance of 2.0 A. This removed some of the covalent

bonds and created tears. Iterations of the simulations were

done for zigzag and armchair graphene nanoribbons, and the

tears were oriented both perpendicular (transverse) and par-

allel (longitudinal) to the thermal flow (Fig. 1). The carbon–

carbon interactions were expressed by the Brenner poten-

tial.29 The velocity Verlet algorithm was adopted to predict

the next positions and velocities, with a time step of 0.5 fs.

Nose–Hoover30,31 thermostats were used at the extreme ends

of the system to create a temperature difference. Noncova-

lent bonds were not included in the simulations, because of

two reasons: (1) they are generally weak, and do not serve as

a major thermal pathway, (2) simplicity of simulations.

III. RESULTS AND DISCUSSION

A. Normalized thermal boundary conductance

The thermal boundary conductance (TBC) across tears

(transverse and longitudinal) is calculated for different tear

lengths, both for zigzag and armchair graphene nanoribbons,

by dividing the thermal flux by the temperature drop. Since

the tear length is varied, TBC is normalized to the lattice

constant of graphene (1.452 A) for the purpose of compari-

son. In Fig. 2, the unconnected symbols represent the nor-

malized TBCs, while the connected symbols are the

calculated means of the normalized TBCs for zigzag and

armchair ribbons. There is a palpable downward trend in

both lines; thermal boundary conductance decreases as the

tear length increases. Furthermore, thermal conductance

across a longitudinal tear is generally greater than that across

a transverse tear. The TBCs are reduced by approximately

FIG. 1. (Color online) Schematic diagrams of (a) transverse and longitudi-

nal tears in grapheme (top panel), and (b) the sites of investigation (bottom

panel).

043508-2 Loh, Teo, and Tay J. Appl. Phys. 109, 043508 (2011)

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40% with the increase of tear length from 1 to 6 units. We

will explore and explain the rationale behind these observa-

tions. We have to accentuate that it is misleading to compare

across orientations for the same tear length—although both

orientations may have 1 unit of tear, the absolute length is

not exactly identical. What is more important is to reveal the

trend when tear length is varied.

B. Vibrational density of states

The vibrational density of states (VDOS) is calculated

by performing a Fourier-transform on the velocity autocorre-

lation function. The local density of states (LDOS) around

the tear provides crucial information on the phonon modes

that are scattered at the particular locations, and the degree

of scattering. Although the exact mode conversions are

unknown, some speculation can be made. In addition, the

strengthening or weakening of the different modes at various

locations discloses essential information on the behavior of

phonons when they approach line defects such as tears. Criti-

cal locations where local density of states is computed are

deemed to be at opposite ends of the tear, parallel to the flow

of thermal energy [Fig. 1(b)], i.e., Zigzag/Transverse/6 tear

units/Site B (abbreviated as ZT6-B), ZT6-E, ZT6-F, ZL6-A,

ZL6-D. The figure also shows that LDOS are obtained at first

and second nearest neighbors of the torn site, with the

numerals 1 and 2 in the figure referring to the first and sec-

ond nearest neighbor, respectively. It will be presented later

that most of the scattering occurs within the distance 2.51 A

(distance of second nearest neighbor from atom). Minute dif-

ferences do exist between second nearest neighbor LDOS at

different sites, indicating the presence of scattering at least

beyond the second nearest neighbors, but the extent of scat-

tering is relatively nondescript.

To identify the modes, the LDOS for the three dimen-

sions are calculated separately. The relative strength of the

three sets of LDOS allows us to speculate if a certain mode

is in-plane transverse (iT), longitudinal (L), or out-of-plane

transverse (oT) in orientation. It is vital to note that this oper-

ation is only a prediction. By comparing these one-

dimensional LDOS and Raman spectroscopic data,32 the ori-

entation and type (acoustic/optical) of the modes are ascer-

tained (Fig. 3). In general, the in-plane modes are the main

carriers of thermal energy, while the out-of-plane modes

have low group velocities.33

Figure 4(b) shows the LDOS at first and second nearest

neighbors for ZT6-E. It is evident that the first nearest neigh-

bor G-band at around 1600 cm�1 is in-plane transverse opti-

cal–dominant (iTO) (Fig. 3). The G-band appears due to the

doubly degenerate zone center E2g mode.34,35 Dresselhaus32

reported that unlike single-walled carbon nanotubes

(SWNTs), the G-band for graphite does not consist of two

features. It is further described in the work by Jorio36 that

confinement and curvature of these degenerate modes in

SWNTs induce them to split into the two features.

The percentage change in VDOS from second to first

nearest neighbor is computed for each case. For a transverse

tear (Fig. 4), in each case (ZT6-B, ZT6-E, ZT6-F), the out-

of-plane acoustic (oTA), longitudinal acoustic (LA) modes,

and G-band are enhanced. The G-band is slightly red-shifted

near the tear. Although there are differences for sites B, E,

and F, these differences are minute, and the general observa-

tions remain. Furthermore, similar features of all the second

nearest neighbor LDOS (Figs. 4 and 5) suggest that phonon

scattering occurs mainly within an approximate distance of

2.51 A from the tear.

For a longitudinal tear (Fig. 5), oTA and LA modes are

strengthened nearer the tear (comparison of VDOS intensity

at first and second nearest neighbor positions), but an un-

identified mode at around 1800 cm�1 is created. The G-

band reduces in intensity. It is hypothesized that the fall in

the G-band intensity, (G-band comprises both in-plane

transverse and longitudinal modes, from the one-dimen-

sional VDOS) suggests that some of the modes in G-band

are involved in anharmonic three-phonon interaction, to cre-

ate the modes in the 1800 cm�1 band. Some of the G-band

modes may be annihilated, and some acoustic modes are

created as a result. The mode nature prediction indicates

that the 1800 cm�1 band is both longitudinal and in-plane

transverse in orientation.

FIG. 2. (Color online) Relationship between normalized thermal boundary

conductance and tear length unit.

FIG. 3. (Color online) Prediction of mode orientation of ZT6-E.

043508-3 Loh, Teo, and Tay J. Appl. Phys. 109, 043508 (2011)

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From these results, it is clear that very different pho-

nonic scattering phenomena takes place around transverse

and longitudinal tears. To further support the previous state-

ments regarding the predominantly scattered modes, a com-

parative analysis (Fig. 6) of the VDOS for both tear

orientations and different tear lengths are performed. The

standard deviation of VDOS at two different tear lengths is

also derived to detect the region of the phonon spectra which

is most affected. It is mentioned previously that we suggest

the presence of anharmonic three-phonon scattering at the

tear, which involves the modes of G-band. It is uncertain

which are the exact modes that collide, but with the appear-

ance of the 1800 cm�1 band and a weakening of the G-band

in the longitudinal case, it seems that this new band

“evolves” from the modes in G-band. According to Dressel-

haus’ work,32 this band is close to the iTOLA band—a

FIG. 4. (Color online) Local density of states at different sites near trans-

verse tear; (a) ZT6-B, (b) ZT6-E, and (c) ZT6-F; The curves in the top panel

refer to the percentage change in VDOS from second to first nearest neigh-

bor at each site.

FIG. 5. (Color online) Local density of state at different sites near longitudi-

nal tear; (a) ZT6-A and (b) ZT6-D; The curves in the top panel refer to the

percentage change in VDOS from second to first nearest neighbor at each

site.

FIG. 6. (Color online) Comparative analysis of LDOS (transverse and longi-

tudinal tears) of different tear lengths.

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combination mode of iTO and LA. It is close but not the

iTOLA band since the peak of LA mode at around 400–450

cm�1 increases, implying that there are now more LA

modes, and not annihilated to from the iTOLA band. For the

transverse case, the red-shift effect is stronger with a longer

tear. Closer observation also shows that there are mainly two

regions of the spectrum that undergo scattering – 0–700

cm�1 and 1400–2000 cm�1. Those modes in the central

region of the spectrum are not affected significantly.

Succinctly, optical phonons and high frequency acoustic

phonons are more strongly scattered by “line defects” in the

direction of thermal flow, in the case of a longitudinal tear.

This is in stark contrast to the degree of scattering encoun-

tered by the long-wavelength acoustic phonons. It is apparent

that they are much weakly scattered in the vicinity of the

tear, regardless of the tear orientation. It is known that inter-

faces are more efficient at scattering short-wavelength pho-

nons, and the transmission coefficient of long-wavelength

phonons is relatively high.33 Our results show and propose

that a similar scattering mechanism may exist around tears.

Moreover, acoustic phonons are the dominant carriers of

heat in semiconductors and insulators, since they have high

group velocities. By inspection and comparison, the trans-

verse tear long-wavelength acoustic phonons are scattered to

a larger extent than the longitudinal tear ones, mainly within

the frequency range 300–600 cm�1. This might explain the

greater thermal conductance across the longitudinal tear.

The 1D LDOS standard deviation (3 and 6 units trans-

versely) plot indicates that within the frequency range of

300–600 cm�1 (shaded), the in-plane transverse and out-of-

plane modes are essentially most affected by the increase in

tear length [Fig. 7(b)]. The magnitude of standard deviation

is relatively smaller for the longitudinal case [Fig. 7(a)], for

all polarizations and frequencies in the long-wavelength

acoustic range.

C. Proposed mechanism

Here we propose another possible reason for the lower

thermal conductance across the transverse tear. A tear acts as

a thermal barrier. When phonons reach the tear, some of

these phonons are scattered to different frequencies and ori-

entations. Essentially, the primary path for thermal energy to

be transmitted to the opposite end of the tear is around it. For

illustration, refer back to Fig. 1(b). Phonons reaching site F

of the transverse tear are obstructed by the barrier, and to

transfer the energy to the opposite end—site B, the major

energy transmission path is around the tear. The noncovalent

interactions between atoms at the tear are relatively weak to

play a significant role in thermal transport. For convenience

of explanation, sites A and D are named the lateral sites,

since they are lateral to the direction of energy flux, while

sites alongside B, E, and F are called the obstruction sites.

As the obstruction sites occupy a greater total area of scatter-

ing than the lateral sites, iTA mode (at around 288 cm�1

according to Dresselhaus)32 on the obstruction sites (eg., site

F in Fig. 4(c) is the dominant carrier around the tear. Fig.

4(c) shows that the VDOS of iTA mode does not increase

significantly when scattered. On the other hand, by the same

rationale, the LA mode on the obstruction sites is the primary

heat carrier around the longitudinal tear. Figure 5(b) shows

that LA mode strengthens by around two times in the vicin-

ity. It is suggested that it is due to this slight difference in the

type of principal heat carriers and the degree of strengthen-

ing, together with the extent of scattering of the long-wave-

length acoustic phonons at the obstruction sites that

distinguish the probability of transmission across both orien-

tations of tear. Based on this conjecture, the tear length is an

important factor. The larger the tear length is, the longer the

energy transmission path is around the tear. Phonons travel-

ing around the tear along this path now experience more

scattering (as observed in Fig. 6). The mean free path of pho-

nons is further reduced. Here an analogy is drawn between

thermal boundary conductance and intrinsic thermal conduc-

tivity. From the classical kinetic model of thermal

conductivity,

j ¼ 13

cvvl; (3)

where cv is the heat capacity of the material, v is the phonon

mean group velocity, and l is the phonon mean free path. A

decrease in the mean free path implies that the phonons col-

lide more frequently, and the overall thermal conductivity is

reduced. Similarly, thermal boundary conductance drops with

more phonon scattering. Figure 8 completes the picture by fur-

ther exhibiting the difference between the scattering processes

at the lateral sites of both transverse and longitudinal tears.

IV. CONCLUSION

In this novel study, we examine the effect of structural

tears on thermal conductance in graphene, using classical mo-

lecular dynamics simulations. It is learnt that the extent of scat-

tering is dependent on two factors: (1) tear orientation and (2)

tear length. In our analysis of one-dimensional and three-

dimensional vibrational density of states, we expound that pho-

nons are scattered differently around transverse and longitudi-

nal tears; at a transverse tear, there is a red-shift of the G-band,

while it is suggested that a 1800 cm�1 band evolves from the

G-band at a longitudinal tear. Long-wavelength acousticFIG. 7. (Color online) Standard deviation of long-wavelength acoustic

VDOS at a (a) longitudinal tear and (b) transverse tear.

043508-5 Loh, Teo, and Tay J. Appl. Phys. 109, 043508 (2011)

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phonons are scattered more strongly at a transverse tear. It is

proponed that it is due to the disparity in phonon scattering

processes along the energy transmission path around the tear

that causes the difference in thermal conductance across the

tear. Using the same explanation, longer tears reduce thermal

conductance. In concise terms, thermal transport across trans-

verse tears is more efficient than across longitudinal ones.

ACKNOWLEDGMENTS

The codes used in this work are developed from a code

provided by the Maruyama Group, the University of Tokyo.

The main author was supported by the Nanyang Research

Scholarship, and the work funded by the MOE Tier 2 Fund.

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FIG. 8. (Color online) Local density of states at lateral sites near (a) trans-

verse and (b) longitudinal tear; the curves in the top panel refer to the percent-

age change in VDOS from the second to first nearest neighbor at each site.

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