C A R B O N 7 7 ( 2 0 1 4 ) 4 1 6 – 4 2 3

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Structure-mediated thermal transport of monolayergraphene allotropes nanoribbons

http://dx.doi.org/10.1016/j.carbon.2014.05.0450008-6223/� 2014 Elsevier Ltd. All rights reserved.

* Corresponding author.E-mail address: yuantong.gu@qut.edu.au (Y. Gu).

Haifei Zhan a, Yingyan Zhang b, John M. Bell a, Yiu-Wing Mai c, Yuantong Gu a,*

a School of Chemistry, Physics and Mechanical Engineering, Queensland University of Technology, 2 George St, Brisbane, QLD 4001, Australiab School of Computing, Engineering and Mathematics, University of Western Sydney, Locked Bag 1797, Sydney, NSW 2751, Australiac Centre for Advanced Materials Technology, School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Sydney,

NSW 2006, Australia

A R T I C L E I N F O

Article history:

Received 30 April 2014

Accepted 17 May 2014

Available online 24 May 2014

A B S T R A C T

We report the study of the thermal transport management of monolayer graphene allo-

trope nanoribbons (size �20 · 4 nm2) by the modulation of their structures via molecular

dynamics simulations. The thermal conductivity of graphyne (GY)-like geometries is

observed to decrease monotonously with increasing number of acetylenic linkages

between adjacent hexagons. Strikingly, by incorporating those GY or GY-like structures,

the thermal performance of graphene can be effectively engineered. The resulting het-

ero-junctions possess a sharp local temperature jump at the interface, and show a much

lower effective thermal conductivity due to the enhanced phonon–phonon scattering. More

importantly, by controlling the percentage, type and distribution pattern of the GY or

GY-like structures, the hetero-junctions are found to exhibit tunable thermal transport

properties (including the effective thermal conductivity, interfacial thermal resistance

and rectification). This study provides a heuristic guideline to manipulate the thermal

properties of 2D carbon networks, ideal for application in thermoelectric devices with

strongly suppressed thermal conductivity.

� 2014 Elsevier Ltd. All rights reserved.

which has been shown to be less stable from first principle

1. Introduction

Owing to the versatile flexibility of carbon in forming three

different hybridization states (sp3, sp2 and sp1), various carbon

allotropes can be produced such as fullerene [1], carbon nano-

tube [2] and graphene [3]. All of the carbon-based nanomate-

rials (CBNs) possess unique properties and hold enormous

potential as components in electronics, photonics, composite

materials, energy storage, sensors, and bio-applications [4–6].

Along with the surge in interests in CBNs, research efforts

have also been made to explore other 2D carbon networks

[7], such as graphyne (GY) and graphdiyne (GDY) [8,9].

Although the synthesis of GY has not been realized, GDY,

calculations [9], has been successfully synthesized recently

[10]. Similar to graphene, GY and GDY are also one-atom-thick

layers but with different atomic bonds. In addition to the sp2

carbon bonds, they contain sp hybridized bonds. Such 2D

graphene allotropes can be regarded as replacing some

@C@C@ bonds in graphene with the acetylenic linkages

AC„CA (GY) or diacetylenic linkages AC„CAC„CA (GDY)

[11]. The presence of these linkages endow the materials with

high p-junction, uniformly distributed pores, lower atom den-

sity than graphene, and tunable electronic properties [11].

The intriguing properties of these 2D graphene allotropes

have attracted increasing attention in recent years, which

Fig. 1 – Atomic configurations of c-GY. Insets show the

different number of acetylenic linkages between two

adjacent hexagons. (A color version of this figure can be

viewed online.)

C A R B O N 7 7 ( 2 0 1 4 ) 4 1 6 – 4 2 3 417

mainly concern their electronic properties [12]. It is reported

that GY and GDY are semiconductors with direct transitions

at the M and U points of the Brillouin zone, respectively

[13–16]. Specifically, Malko et al. [17] found that the 6,6,12-

GY has more remarkable electronic properties than graphene

since it does not have hexagonal symmetry and features two

self-doped non-equivalent distorted Dirac cones. Recently,

considering the exceptional electronic mobility of graphene

and the non-zero band gap of GY or GDY, researchers pro-

posed a hetero-junction structure of graphene and GY or

GDY. Such hetero-junction-based field effect transistors are

predicted to exhibit excellent switching behaviors [18],

without the severe contact resistance as in typical metal elec-

trodes [19]. Continuing efforts have also been seen in deriving

the derivatives and applications of the 2D graphene allo-

tropes, including the usage of the multilayer GY as a lithium

ion battery anode [20], GY-like membrane for water desalina-

tion [21,22], hydrogen functioned GY [23,24], calcium deco-

rated GY [25], GY nanotubes [26], and the B-N and B-C-N

analogues of the 2D carbon networks [27].

Despite their remarkable electronic properties, carbon

allotropes can also occupy a unique place in thermal manage-

ment due to their exceptional thermal property. The thermal

conductivity at room temperature ranges from �0.01 W/mK

in amorphous carbon to above 2000 W/mK in diamond or

graphene [28]. Therefore, it is of great significance to harness

the thermal properties of these carbon allotropes so as to

facilitate their applications in thermo-electronics and pho-

tonics. In contrast to the extensive studies on graphene, very

few studies have been undertaken to investigate the thermal

properties of GY. In two of these, it is reported that the ther-

mal conductivity of GY is much smaller than that of graph-

ene, and the thermal conductivity of c-GY possesses

obvious directional anisotropy [29,30].

The appealing applications along with the huge structural

flexibility of 2D carbon networks motivate us to explore the

tunability of their thermal properties through modulation of

their structures. Preliminary works have already reported

the thermal transport modification of graphene sheets and

graphene nanoribbons by their geometry size (length, width

and thickness) [31,32], and impurities/defects [33,34]. In the

present work, using large-scale molecular dynamics (MD)

simulations, the thermal transport properties of different

monolayer carbon networks are examined, including pure

GY, GDY, GY-like structures, and hetero-junctions of graphene

and GY or GDY. The obtained relation between the thermal

properties of graphene allotropes and their structure is

expected to provide guidance to the thermal management

in nano-scale devices.

2. Computational methods

Different monolayer carbon networks are constructed basing

on c-GY with an approximately identical size �20 · 4 nm2

unless otherwise stated. Fig. 1 shows the atomic configura-

tion of c-GY, which replaces some @C@C@ bonds in graphene

by acetylenic linkages AC„CA. By varying the number of

acetylenic linkages between the adjacent hexagons, different

GY-like geometries can be constructed, e.g., a GDY structure is

obtained by replacing the AC„CA linkages with diacetylenic

linkages AC„CAC„CA. The thermal transport properties of

different monolayer graphene allotropes was assessed by

the reverse non-equilibrium molecular dynamics (RNEMD)

simulations, which is based on the Muller-Plather approach

[35]. The concept of RNEMD is to inject a heat flux to the

structure and determine the resultant temperature gradient.

To impose such a heat flux, the simulation system is divided

into N slabs (with identical width) along the length direction,

and then the heat flux is generated by exchanging the velocity

of the hottest atom in the ‘‘cold’’ slab with the coldest atom in

the ‘‘hot’’ slab. The resultant heat flux J (in unit of Watt),

which is the energy transferred in a specific time through a

surface perpendicular to the heat flux direction, is then given

by

J ¼ 12tA

X

N

m2ðv2

hot � v2coldÞ ð1Þ

where t is total simulation time, A the cross-sectional area, N

the total number of exchange, m the atomic mass, and vhot

and vcold the velocities of the hot and cold atoms that are

involved in the exchange. The factor 2 in the denominator

is used because of the periodicity of the system. When the

heat flow in the structure reaches the steady state regime,

the thermal conductivity (j) is calculated by using the Fourier

law as

j ¼ �J=ð@T=@xÞ ð2Þ

where oT/ox is the temperature gradient along the heat flux

direction.

The interaction between the bonded carbon atoms is

described by the widely used reactive empirical bond order

(REBO) potential [36], which has been shown to well represent

the binding energy and elastic properties of carbon materials.

To ensure the stability of the simulations and capture the

high vibrational frequency of the triple-bonded acetylenic

linkages, a small time step of 0.1 fs was chosen. The initial

equilibrium configuration of the sample was achieved by

the conjugate gradient minimization method. The sample

was then equilibrated using Nose–Hoover thermostat [37,38]

under ambient condition for 80 ps (temperature = 300 K and

418 C A R B O N 7 7 ( 2 0 1 4 ) 4 1 6 – 4 2 3

pressure = 1 atm). Finally, the sample was relaxed in the

microcanonical ensemble (i.e., constant atom number, vol-

ume, and energy) for 800 ps. Periodic boundary conditions

were only applied along the length direction of the carbon

networks to represent a boundary condition for a nanoribbon.

All the simulations were performed using the software pack-

age LAMMPS [39].

Fig. 2 – MD results for the armchair-edged c-GY

nanoribbons. (a) A schematic view of the RNEMD model. (b)

Temperature profile of the nanoribbon along the length

direction at the simulation time of 800 ps. (c) Inverse of

thermal conductivity as a function of the inverse of length.

The error bar represents the standard deviation of the

inverse of j during the simulation time from 500 to 800 ps.

(A color version of this figure can be viewed online.)

3. Results and discussion

The temperature in the MD simulations is obtained using the

classical statistical mechanics equipartition theorem, which

is invalid when the system temperature is below the mate-

rial’s Debye temperature. Carbon-based materials have very

high Debye temperatures, for example, a graphene monolayer

is reported to have a Debye temperature of 1054 K [40]. To mit-

igate this limitation, quantum corrections [41] have been

commonly applied to the temperature T and the j predicted

by the MD simulations. However, recent research found that

quantum corrections in MD results above 200 K make mar-

ginal difference for T and j when compared with the quantum

predictions [42]. In view of this fact, quantum corrections are

neglected in the present work. It is worth mentioning that the

absolute thermal conductivity is sensitive to various factors,

such as the model size [31] and atomic potentials [34]. Herein

we focus our attention on the change of the thermal conduc-

tivity rather than its absolute magnitude with the aim to iden-

tify the effective modulation techniques for the monolayer

graphene allotrope nanoribbons. Specifically, the thickness

of the 2D carbon networks is assumed to be the CAC bond

length, i.e., 0.142 nm in this work following previous studies

[43–45]. It is noteworthy that a thickness of 0.335 nm has also

been adopted for a monolayer graphene by many researchers

[46,47]. Adopting different thickness in the calculation of j

makes no difference to the relative thermal conductivity

results that are the focus of this work.

3.1. Size-dependent thermal conductivity of c-graphynenanoribbon

Initially, we acquire the thermal conductivity of c-GY, which

has been widely discussed as a typical allotrope of graphene.

It is noted that Zhang et al. [30] or Ouyang et al. [29] have

investigated the j of different GYs earlier. However, the for-

mer study simulated GY sheets rather than GY nanoribbon

and the latter study emphasized the width effect on j. In this

section, we focus on the j of c-GY nanoribbon with armchair

edges along the length direction. Fig. 2b shows the tempera-

ture profile of the c-GY with a length of 20.06 nm. Clearly,

the temperature profile shows good symmetry with respect

to the middle of the sample. The temperature profile is non-

linear near the hot and cold regions due to the finite size

effects as observed previously [48]. To calculate the thermal

conductivity from Eq. 2, the linear middle portion (solid blue

line in Fig. 2b) is used to extract the temperature gradient

oT/ox. A value of 23.1 ± 0.9 W/mK is estimated for the c-GY

nanoribbon, which is in consistent with that reported previ-

ously [30].

Fig. 2c shows a linear dependency of j on the length of

c-GY nanoribbon, which can be understood from the kinetic

theory of phonon transport [49]. Usually, the size of the sim-

ulation cell is much smaller than the mean free path (MFP)

of phonons in the 2D carbon networks (e.g., the MFP of

phonon for graphene is �775 nm [50]), hence, besides the

phonon-phonon scattering, scattering at the heat baths (or

boundaries) of the system must be considered. Earlier works

on graphene nanoribbon [51] and other systems [48,52,53]

have confirmed that the thermal conductivity satisfies the

relation 1/j / 1/l1 + 1/lx, where l1 is the MFP in an infinite

system and lx is the length of the simulation box. Our results

agree well with this scaling relation as shown in Fig. 2c. By

extrapolating the plot in Fig. 2c to lx �1, j1 � 31.43 W/mK

is obtained for an infinite c-GY nanoribbon.

3.2. Thermal conductivity of graphyne-like nanoribbons

Through first principle calculations, researchers have investi-

gated a rich variety of GY-like geometries, which is con-

structed by varying the number of acetylenic linkages

between adjacent hexagons [8,11,16]. For simplicity, these

GY-like one-atom-thick layers are denoted as GY-n in this

Fig. 3 – Thermal conductivity of zigzag- (ZZ) and armchair-

(AC) edged GY-like networks versus the number of

acetylenic linkages. (A color version of this figure can be

viewed online.)

C A R B O N 7 7 ( 2 0 1 4 ) 4 1 6 – 4 2 3 419

work, where n represents the number of acetylenic linkages

between adjacent carbon hexagons (see Fig. 1).

Fig. 3 illustrates the thermal conductivity as a function of n

with n changing from 1 to 6. All the bond lengths are selected

following the calculations basing on density functional theory

[9]. To ensure reasonable comparisons, similar size has been

selected for all samples. In general, j is found to decrease

with increasing number of acetylenic linkages, as does their

electronic conductivity [16]. Particularly, the thermal conduc-

tivity of the armchair-edged GY-6 is �12.8 ± 0.6 W/mK, which

is nearly half of that of the c-GY (�23.1 ± 0.9 W/mK). As

expected, increasing the acetylenic linkages will induce more

phonon scattering which in turn decreases j. In addition, the

armchair-edged GY-n nanoribbons exhibit higher thermal

conductivity than their zigzag-edged counterparts, signifying

an anisotropic thermal property. It is also observed in Fig. 3

that the thermal conductivity of the armchair-edged nanorib-

Fig. 4 – Numerical results for hetero-junctions. (a) A hetero-junct

(b) The corresponding temperature profile along the length direc

GY(30%) NR with a zigzag edge. (d) The corresponding temperat

version of this figure can be viewed online.)

bon is approximately 25% higher than its zigzag-edged coun-

terpart until the number of acetylenic linkages increases to

six.

3.3. Thermal transport of monolayer carbon hetero-junctions

The previous two sections assessed the thermal conductivity

of pure GY and GY-like structures. Graphene and GY or GDY

have similar bond lengths and unit cell shapes, ensuring a

perfectly matched interface between them and construction

of graphene-GY or -GDY hetero-junctions feasible. Here, we

investigate how the thermal transport property of graphene

can be modulated by the introduction of different percent-

ages of GY or GDY.

First, we discuss the thermal transport of an armchair-

edged hetero-junction that contains �40% of c-GY and �60%

of graphene, in which the heat flux flows across graphene/

c-GY interface (see Fig. 4a). As illustrated in Fig. 4b, although

the temperature profile is still symmetrical with respect to

the middle of the sample (hot region), it is no longer mono lin-

ear (compared with that in Fig. 2b). A distinct local tempera-

ture drop is observed at the interface between graphene and

c-GY due to their different phonon properties. As demon-

strated in Fig. 4b, the temperature gradient at the c-GY region

(�37.62 K/nm) is more than five times that of the graphene

region (�6.06 K/nm), indicating that the thermal conductivity

of c-GY is more than five times lower than the graphene

region (assuming the same heat flux transferring through

the nanoribbon). This result is consistent with that of film-

like GY [30]. Consider now the hetero-junction constructed

along the zigzag direction with �30% of c-GY (see Fig. 4c).

As shown in Fig. 4d, the temperature profile is mono linear

and has a good symmetry with respect to the middle of the

nanoribbon since the heat flux flows parallel to the graph-

ene/c-GY interface. The temperature gradient of the hetero-

junction is around 13.1 K/nm, which is slightly more than

twice that of a pure graphene, but over twice smaller than

pure c-GY, signifying a j between its constituent components.

ion between graphene and c-GY(40%) with an armchair edge.

tion at 800 ps. (c) A hetero-junction between graphene and c-

ure profile along the length direction at 800 ps. (A color

Fig. 5 – (a) Effective thermal conductivity as a function of the percentage of c-GY or GDY. (b) Top figure compares the VDOS

between armchair-edged hetero-junctions with different percentages of c-GY NR, and bottom figure compares the VDOS

between zigzag-edged hetero-junctions with different percentages of c-GY NR. (A color version of this figure can be viewed

online.)

420 C A R B O N 7 7 ( 2 0 1 4 ) 4 1 6 – 4 2 3

To unveil the detailed dependency of j on the structure of

hetero-junctions, we investigate another three groups of

samples with varying percentages of c-GY or GDY, including

armchair-edged hetero-junction between graphene and GDY

(G-GDY-G-AC), armchair-edged (G-GY-G-AC) and zigzag-edged

(G-GY-G-ZZ) hetero-junctions between graphene and c-GY.

Since all armchair-edged models exhibit a local temperature

jump at the interface, a unique temperature gradient is

unavailable for the calculation of j. Herein, another tempera-

ture gradient k3 is defined by selecting two temperature

values (in the linear regions) near the cold and hot slab, as

illustrated in Fig. 4b to derive an effective thermal conductiv-

ity of those samples instead. As compared in Fig. 5a, the effec-

tive j decreases monotonously with increasing percentage of

GY or GDY in all three groups. Specifically, the effective j of

hetero-junctions experiences a sharp drop at low percentage

of GY or GDY. Thereafter, it decreases gradually and

approaches a constant. This indicates that the presence of

GY or GDY and the resultant interface in hetero-junctions

exerts significant effects on the thermal conductivity at low

percentage level but has marginal effect at high percentage

level.

To further examine the underlying transport property, we

compared the vibration density of states (VDOS) among

Fig. 6 – (a) Schematic views of different models with GY or GDY

(lower figure). (b) Atomic representation of the interface between

the graphene region. (A color version of this figure can be view

different cases (computed using the autocorrelation function

of the atomic velocities [54]). From Fig. 5b, it is readily seen

that the low and high frequency modes have been reduced

considerably due to the presence of c-GY and the degree of

reduction is proportional to the percentage of c-GY. This

reduction indicates more severe phonon–phonon scattering

and thus explains the associated degraded j.

It is evident from the above discussions that the thermal

transport properties of the monolayer graphene allotropes

can be effectively tailored by its structure. To further assess

the effects of the interface, we derive the interfacial thermal

resistance or Kapitza resistance (KR) for different hetero-junc-

tions as KR = DT/J [28,55], where DT is the temperature drop at

the interface (estimated from the two temperature values

adjacent to the interface as illustrated in Fig. 4a). Five groups

of hetero-junctions have been considered which contains

GDY and four different GYs, i.e., a-, b-, 6,6,12-, c-GY with dif-

ferent percentages of the linkages. Each group contains two

samples with graphene either locating in the middle region

or at the two ends (see Fig. 6), which enables the calculation

of KR for different directions of heat flux (i.e., KR� from graph-

ene to GY or GDY and KR+ from GY or GDY to graphene). To

ensure reasonable comparisons, these models have similar

size (length in the range of 20.0–20.6 nm and width in the

located either in the middle (upper figure) or at the two ends

graphene and GY or GDY, left of the dashed line represents

ed online.)

Fig. 7 – (a) Interfacial Kapitza resistance of different hetero-

junctions. (b) Interfacial thermal rectification factor of

different hetero-junctions. (A color version of this figure can

be viewed online.)

C A R B O N 7 7 ( 2 0 1 4 ) 4 1 6 – 4 2 3 421

range of 3.8–5.0 nm) and similar percentage of graphene

(between 59% and 62%). Fig. 7a compares the KR obtained

for different hetero-junctions. Evidently, the hetero-junction

with b-GY possess the largest KR of �20 · 10�11 m2K/W. This

can be qualitatively explained by considering the connections

in the interface between graphene and GY. For the hetero-

junction with b-GY, graphene and b-GY are sparsely and thus

weakly connected at their interface due to the compatibility

of their unit cell sizes. That is, the interface between graph-

ene and b-GY contains more vacancy defects, which leads

to an increase in the scattering of phonons and hence an

increase in KR. For all the other four groups, the KR is between

4.9 · 10�11 and 11.1 · 10�11 m2K/W, very close to those

reported for grain boundaries [51] and also isotope interface

[47] of graphene. The interfacial thermal resistance KR has a

strong adverse effect on the thermal transport in hetero-junc-

tions. Among the five groups of hetero-junctions, the one

with �41% of b-GY (see upper model in Fig. 6a) has the lowest

effective j because of the highest KR. The reduction of the

effective j is greater than 85% compared to its graphene NR

counterpart.

Fig. 7a shows that the interfacial thermal resistance

depends on the heat flux direction, implying the occurrence

of thermal rectification at the interface. Following the previ-

ous work [47], we denote the interfacial thermal rectification

(TR) factor as: TR = (KR+–KR�)/KR+ · 100%. According to

Fig. 7b, the interface between graphene and a-GY exhibits

the lowest TR (�9.5%), while the interface between graphene

and GDY has the highest TR (�25.2%). Comparable thermal

rectification effect is also reported in other systems, such as

thickness asymmetric graphene NRs [56], isotope doping

[47], and carbon nanotubes mass-loaded with heavy mole-

cules [57].

4. Conclusion

In summary, we have carried out the first comprehensive

investigation on the thermal transport properties of a range

of monolayer graphene allotropes using large-scale MD

simulations. Our results show that the presence of acetylenic

linkages in GY and GY-like nanoribbon reduces their thermal

conductivity. The degree of reduction is related to the number

of acetylenic linkages between the adjacent carbon hexagons

and the edge geometry. For the hetero-junctions combining

graphene with GY or GDY, a distinct local temperature drop is

observed at the interface perpendicular to the heat flux direc-

tion originated from their different thermal properties. The

incorporation of the weaker GY or GDY and the associated

interface in the hetero-junctions induces more severe phonon

scattering due to the reduced low and high frequency modes

as observed from the VDOS spectrum. Consequently, the effec-

tive thermal conductivity of the hetero-junctions is reduced

dramatically. Our extensive simulation results verified the

possibility of tailoring the thermal properties of hetero-junc-

tions by manipulating their structures, such as the percentage,

type and distribution pattern of GY or GDY. The present study

not only provides a fundamental understanding of the struc-

ture-mediated thermal transport of 2D carbon networks but

also indicates their potential applications in thermal manage-

ment for next generation nano-devices.

Acknowledgments

Supports from the ARC Discovery Project (DP130102120), the

High Performance Computer resources provided by the

Queensland University of Technology are gratefully

acknowledged.

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