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: [email protected] (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. IntroductionOwing 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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