 Nano Express
 Open Access
 Published:
Thermal Conductivity of Two Types of 2D Carbon Allotropes: a Molecular Dynamics Study
Nanoscale Research Letters volume 14, Article number: 7 (2019)
Abstract
The thermal properties of the two novel 2D carbon allotropes with fivefiveeightmembered rings are explored using molecular dynamics simulations. Our results reveal that the thermal conductivity increases monotonically with increasing size. The thermal conductivities of infinite sizes are obtained by linear relationships of the inverse length and inverse thermal conductivity. The converged thermal conductivity obtained by extrapolation in the reverse nonequilibrium molecular dynamics method is found to be in reasonable agreement with that in the equilibrium molecular dynamics method. The much lower thermal conductivity, compared with graphene, is attributed to the lower phonon group velocity and phonon mean free path. Temperature and strain effects on thermal conductivity are also explored. The thermal conductivity decreases with increasing temperature and it can also be tuned through strain engineering in a large range. The effect of strain on TC is well explained by spectra analysis of phonon vibration. This study provides physical insight into thermal properties of the two carbon allotropes under different conditions and offers design guidelines for applications of novel twodimensional carbon allotropes related devices.
Introduction
The carbon materials, e.g., diamond [1], carbon nanotubes [2,3,4,5], and graphene [6,7,8,9,10,11,12], have stimulated tremendous research interests due to their excellent thermal transport properties. Especially the lowdimensional carbon materials show outstanding properties in heat transport. As a 1D material, the high thermal conductivity (TC) of a single carbon nanotube has been observed by experiments [2, 3], and theoretical studies [4, 5]. Moreover, as a singleatomthick flat twodimensional (2D) carbon material, graphene is considered as a revolutionary material for the future generation of thermal conductive reinforced composites due to its high TC [6,7,8,9,10,11,12]. It is also reported that the TC of graphyne can reach 40% of graphene and it has potential applications in thermal management [13,14,15].
Inspired by the fascinating characteristics of these carbon allotropes, researchers have made intensive efforts to study the carbon allotropes and their derivatives in recent years. The experimental and theoretical approaches have been adopted to investigate the novel 2D carbon allotropes, such as the sp^{2}like carbon layer with five, six, and sevenmembered rings [16]; 2D amorphous carbon with fourmembered rings [17]; planar carbon pentaheptite [18]; 2D carbon semiconductor with patterned defects [19]; several 2D flat carbon networks [20]; octagraphene [21]; Tgraphene [22]; and Hnet [23]. Identifications of the unique properties of these 2D carbon allotropes are significant for future generations of nanomaterials in electronic, photonic, and thermal fields [16,17,18,19,20,21,22,23].
With growing interest in exploring new structures of the 2D carbon allotropes, Su et al. [24] proposed two novel energetically competitive and kinetically stable 2D carbon allotropes composed of octagons and pentagons via the firstprinciple calculation. The kinetic stability of these two carbon sheets was confirmed by calculating their phonon dispersion curves. Due to the fact that structures of these two carbon allotropes can be viewed as copying the fivefiveeightmembered rings (558) ribbon along a straight line path and along a zigzag path, these two carbon allotropes are thus named as octagon and pentagon grapheneline (OPGL) and octagon and pentagon graphenezigzag (OPGZ), respectively. The formation energy of these two carbon allotropes are 0.31 eV/atom and 0.34 eV/atom, respectively. The values are much lower than the formation energy of previously synthesized graphyne, i.e., 0.76 eV/atom [25]. It is noted that the OPGZ possesses remarkable anisotropy of electronic structure which has potential applications in electronic devices [24]. Consequently, to meet the requirements of electronic applications of OPGL and OPGZ, it is inevitable and necessary to research the thermal dissipation properties of the two novel structures. Till now, the thermal properties of these two structures are still not clear.
In this work, we investigate the thermal properties of the two novel 2D carbon allotropes using molecular dynamics simulations. Size, strain, and temperature effects on TC are explored. The results are analyzed by calculating the vibration density of states (VDOS) of phonons. Our research of the thermal properties of these two carbon allotropes indicates their potential applications in thermal management devices.
Model and Methods
The structures of OPGL (Fig. 1a) and OPGZ (Fig. 1b) contain representative cells composed of octagons and pentagons [24]. In order to distinguish the edge types of the structures, we define the chirality of armchair and zigzag just like graphene (see Fig. 1). These two structures can be formed by the representative 558 ribbon indicated by the red atoms using translational symmetry along the green rows.
All MD simulations are performed using the largescale atomic/molecular massively parallel simulator (LAMMPS) package [26]. We use the optimized Tersoff potential by Lindsay and Broido [27], with small modifications, i.e., modified optimized Tersoff potential, to describe the interactions among the carbon atoms. Lindsay and Brodio optimized two parameters compared to the original Tersoff potential [28], one for the equilibrium bond angle and one for the attractive interaction strength. According to this optimized Tersoff potential [27], the equilibrium bond length in graphene is 1.4388 Å, which is larger than the experimental value of 1.42 Å [29]. Because the only lengthrelated parameters in the Tersoff potential are λ_{1} in the repulsive function (f^{R} = A exp.(λ_{1}r)) and λ_{2} in the attractive function (f^{A} = B exp(λ_{2}r)), we can obtain the correct bond length by multiplying these two parameters by a factor of 1.4388/1.42. That is, we change λ_{1} from 3.4879 Å^{−1} to 3.5333 Å^{−1} and change λ_{2} from 2.2119 Å^{−1} to 2.2407 Å^{− 1}. These modifications only change the length scale of the potential in a global way. Based on this modified optimized Tersoff potential, the corresponding equilibrium lattice parameters in MD simulation are as follows: OA = 3.63 Å, OB = 9.38 Å in OPGL and OA = 6.78 Å, OB = 5.04 Å in OPGZ, which are in good agreement with the previous study of Su et al. [24], i.e., OA = 3.68 Å, OB = 9.12 Å in OPGL and OA = 6.90 Å, OB = 4.87 Å in OPGZ.
Reverse nonequilibrium molecular dynamics (rNEMD) [30] simulations are performed to calculate the TC. The periodic boundary conditions are adopted in x and y dimensions. The structures of OPGL and OPGZ are initially optimized via the PolakRibiered version of conjugated gradient algorithm [31], and a 0.25ns NoséHoover thermal bath [32, 33] is employed later to ensure the system reaches the equilibrium state at 300 K (with a time step of 0.25 fs). After approaching the equilibrium state, the model is divided into 50 slabs along the heat transfer direction. As shown in Fig. 2a, the 1st slab is assigned to be the heat sink while the 26th (middle slab of the sample) is the heat source, and the heat flux transfers from the heat source (hot region) to the heat sink (cold region). The heat flux transport direction is defined as the length direction (L) while the transverse direction is the width (W) direction. The heat flux J is released/injected between these two slabs by exchanging the kinetic energies between the hottest atom, which has the highest kinetic energy, in the heat sink slab and the coldest atom, which has the lowest kinetic energy, in the heat source slab. The heat flux J can be obtained by calculating the exchanging amount of the kinetic energy between the heat sink and the heat source slab according to the following equations.
where t_{swap} is the total time of exchanging kinetic energy, N_{swap} denotes the amount of exchanging atoms pairs, m is the mass of atom, and v_{h} and v_{c} represent the velocity of exchanging atoms (the hottest atom with the highest kinetic energy in the heat sink slab and the coldest atom with the lowest kinetic energy in the heat source slab), respectively. The temperature of each slab is collected and averaged over 3.0 ns to obtain temperature distribution when system reaches nonequilibrium steady state (after 1.5 ns). The value of TC (κ) is then calculated by using the Fourier’s law as
where A is the crosssectional area of heat transfer (A is obtained by multiplying the width and thickness of the model), and ∂T/∂L denotes the temperature gradient after the system reaches nonequilibrium steady state (see Fig. 2b). The factor 2 represents the fact that the heat flux transports in two directions away from the heat source. The thickness of model is assumed to be the interlayer equilibrium spacing of graphene (0.34 nm) [8, 10, 34, 35].
Results and Discussions
We first examine the system size effect on the TC of the two carbon allotropes. Simulation samples are generated with the same width of 3 nm but different length varying from 50 to 1000 nm. It should be noted that all of the values of the sample length mentioned in this work are the effective length (L_{eff}) of heat transfer. That is, the effective sample length is half of the sample length (L), i.e., L_{eff} = L/2, which is attributed to the heat flux transferring from the middle (the heat source) to the both ends (the heat sink) of the sample in the rNEMD method. Particularly, we have confirmed that the TC does not depend on the sample width by calculating the thermal conductivities of samples with fixed length of 50 nm but different width of 3 nm, 6 nm, 9 nm, and 12 nm, respectively, as shown in Fig. 3. The TC of OPGL along the zigzag and the armchair directions are named as κ_{OPGLZ} and κ_{OPGLA}, respectively. Similarly, κ_{OPGZZ} and κ_{OPGZA} are used to represent the TC of OPGZ along the zigzag and the armchair directions. The simulation results show that the TC of OPGL and OPGZ in the two chiral directions increases monotonically with sample length varying from 50 to 1000 nm. It is attributed to that in the long sample, the acoustic phonons with longer wavelength are involved to heat transfer [9, 36]. Respectively, the TC of 50nm and 1000nmlong OPGL and OPGZ along the zigzag direction are κ_{OPGLZ50} = 125 W/mK, κ_{OPGLZ1000} = 296 W/mK, κ_{OPGZZ50} = 94 W/mK, and κ_{OPGZZ1000} = 236 W/mK. Along the armchair direction, the TC of OPGL and OPGZ are κ_{OPGLA50} = 105 W/mK, κ_{OPGLA1000} = 316 W/mK, κ_{OPGZA50} = 93 W/mK, and κ_{OPGZA1000} = 214 W/mK.
In order to extract the TC of infinitely long samples, an inverse fitting procedure is employed. The relationship between the inverse length and inverse TC is expressed as [37,38,39]:
where κ_{∞} is the extrapolated TC of an infinite sample, l is the phonon mean free path, and L_{eff} is the effective length of heat transfer. Equation (3) suggests that the relationship between the inverse length and inverse TC should be linear. As shown in Fig. 4, a linear relationship between the inverse length and inverse TC is observed. By extrapolating to L^{−1} = 0, the TC of infinite samples, i.e., κ_{OPGLZ} = 310 W/mK, κ_{OPGLA} = 332 W/mK, κ_{OPGZZ} = 247 W/mK, and κ_{OPGZA} = 228 W/mK, are obtained.
In addition, we also express the running TC in the equilibrium molecular dynamics (EMD) method by establishing the sample with the same length and width of 20 nm (this simulation sample size has been tested to be large enough to eliminate finitesize effects). According to the work by Fan et al. [39, 40], the TC calculations in the EMD method is based on the GreenKubo formula [41, 42], in which the running TC along the x direction can be expressed as follows:
where κ_{B} is the Boltzmann’s constant, V is the volume of the system, T is the absolute temperature of the system, 〈J_{x}(0)J_{x}(t^{'})〉 is the heat flux autocorrelation function, t is the correlation time, and J_{x} is the heat flux in the x direction. The symbol 〈〉 represents the time average in EMD simulations. The maximum correlation time is 2 ns, which has been tested to be large enough. As shown in Fig. 5, the running TC for OPGL and OPGZ at two chiral directions at 300 K are expressed by averaging the results of 100 independent simulations with different initial velocity. We can further obtain the TC of an infinite sample by averaging the running TC in correlation time from 1.0 to 2.0 ns. That is, the converged TC of OPGLZ, OPGLA, OPGZZ, and OPGZA are 313 W/mK, 344 W/mK, 261 W/mK, and 233 W/mK, respectively, which are in reasonable agreement with the results by extrapolation in the rNEMD method.
It is found that the TC of these two carbon allotropes is much lower than that of graphene (3000–5000 W/mK) [7, 43]. To explain this phenomenon and explore physical insight, we calculate three important parameters, i.e., C_{v}, v_{g}, and l, based on the classical lattice thermal transport equation:
where C_{v} is heat capacity, v_{g} is effective phonon group velocity, and l is phonon mean free path.
The sample with both length and width of 20 nm is adopted to investigate the heat capacity at 300 K. The heat capacity is computed following the approach of McGaughey and Kaviany [44], which has been used in the approachtoequilibrium molecular dynamics simulations [45]. We calculate the total energy E at temperature of T = 290 K, 295 K, 300 K, 305 K, 310 K in the canonical ensemble, and the results are averaged over 60 ps of ten independent simulations with different initial velocity. As shown in Fig. 6, the slope in the linear fitting of energytemperature curve is the heat capacity.
It should be noted that the phonon group velocity we calculate here is the effective phonon group velocity v_{g} rather than average phonon group velocity v. As shown in Fig. 7, the effective phonon group velocity can be obtained by comparing the results of the rNEMD and the EMD simulations. That is, an effective system length L_{eff} can be defined in the EMD method by multiplying the upper limit of the correlation time t in the GreenKubo formula Eq. (4) by an effective phonon group velocity v_{g}, L_{eff} ≈ v_{g}t. The running TC κ(t) of the EMD method can also be regarded as a function of the system length κ(L_{eff}). In comparison with the average phonon group velocity, the effective phonon group velocity is rough estimate, but it has been extensively used in studying thermal transport in lowdimensional lattice models [46] and has also been used for graphene [40] and allotropes of Si [39].
Based on Eq. (3), the phonon mean free path can be obtained by extrapolation in the rNEMD method. To compare the TC of these two carbon allotropes with that of graphene, we also present these three parameters of graphene. The heat capacity of graphene is calculated through the above method while the effective phonon group velocity and phonon mean free path are obtained in other works [7, 40]. It can be found that the heat capacities of these two carbon allotropes are close to that of graphene; however, the effective phonon group velocity and phonon mean free path are much lower than that of graphene, which leads to the lower TC of the two materials (see Table 1).
Furthermore, we explore the dependence of TC on the temperature, as shown in Fig. 8. The temperature region of 200 K to 300 K is the major range that we focus on. Simulation samples are generated with the same width of 3 nm but different length of 50 nm, 75 nm, 100 nm, 150 nm, and 200 nm, respectively. As shown in Fig. 8a, b, we give the inverse TC of OPGLZ and OPGLA at various temperatures as a function of the inverse sample length. Similar to the extrapolation in size effect at 300 K, the thermal conductivities of an infinite sample at various temperatures are extracted by doing extrapolation procedure. As shown in Fig. 8c, d, all the converged thermal conductivities are normalized by the TC at 300 K (κ_{0}).
Figure 8 indicates that along both the zigzag and the armchair directions, the TC decreases with increasing temperature for both OPGL and OPGZ. The trend of TC varies with temperature (from 200 to 500 K) is in good agreement with those of previous TC studies of graphene [8, 36, 47]. This phenomenon is derived from the enhancement of Umklapp scattering processes which play a critical role in heat transport [8, 36, 47]. Additionally, when the temperature varies from 300 to 500 K, the κ_{OPGLZ}, κ_{OPGLA}, κ_{OPGZZ}, and κ_{OPGZA} drops by 42%, 40%, 36%, and 37%, respectively. The dependence of TC of these two carbon allotropes on temperature shows that it is necessary to consider the temp effects for their practical applications.
The thermal properties of the twodimensional materials, e.g., graphene [48, 49], silicene [34, 50, 51], and phosphorene [37], are sensitive to strain engineering. It has been reported that the TC of graphene with small size decreases with increasing tensile strain [48], and TC also can be enhanced by increasing strain when the sample is larger than 500 μm [49]. The unusual dependences of TC on sample size and strain is attributed to the competition between the boundary scattering and phononphonon scattering. In addition, the TC of silicene is found to increase at small tensile strain but decrease at large strain due to the competition between the phonon softening in the inplane modes and phonon stiffening in the outofplane modes [34, 50, 51]. Therefore, it is significant and necessary to investigate the relationships between TC behavior and tensile strain for both OPGL and OPGZ structures.
We first investigate the mechanical properties of these two carbon allotropes. The sample size is about 5 nm long and 5 nm wide. To avoid any spurious high bond forces and nonphysical strain hardening [52, 53], the cutoff distance is fixed at (R = S = 1.95 Å). This cutoff distance in the modified optimized Tersoff potential is also consistent with that in previous Tersoff potentials (1.8–2.1 Å) [28, 53,54,55] that are being used to simulate CC bond. All the simulations are initiated by relaxing atomistic configuration of structure to a minimum potential energy state. Uniaxial tensile strain is applied with the strain rate of 0.0002 ps^{−1}. It should be noted that the interlayer equilibrium spacing of graphene (3.4 Å) is used to represent the interlayer equilibrium distance of the two structures. The mechanical properties of these two carbon allotropes are listed in Table 2, with comparison of graphyne and graphene [56]. The superscript characteristics of z and a represent zigzag and armchair sheets, respectively.
It is seen from Table 2 that along the zigzag direction, the Young’s modulus of the OPGL and OPGZ are 538 GPa and 492 GPa, and along the armchair direction, the Young’s modulus are 648 GPa and 550 GPa, respectively. It indicates that the Young’s modulus of the OPGL and OPGZ are close to that of graphyne (503.1^{z} and 525.0^{a}) but lower than that of graphene (856.4^{z} and 964.0^{a}). Stressstrain relationships of the two carbon allotropes along the zigzag and the armchair directions are shown in Fig. 9. According to the fracture behaviors of these two carbon allotropes, we further obtain the ultimate strain (tension) of these two carbon allotropes. Respectively, along the zigzag direction, the ultimate strain (tension) of the OPGL and the OPGZ are 17.2% and 10.9%, and along the armchair direction, the ultimate strain (tension) are 8.7% and 7.9%. We found that the structure of OPGL has higher strength under tensile strain in the zigzag direction. However, compared with graphyne and graphene, the ultimate strains (tension) of the two carbon allotropes are lower.
We then study the strain effect on TC of these two carbon allotropes by applying uniaxial tensile strain along the heat transfer direction. Simulation samples have the same width of 3 nm but different length of 50 nm, 75 nm, 100 nm, 150 nm, and 200 nm, respectively. The thermal conductivities of an infinite sample at various strains are extracted by doing extrapolation procedure (see Fig. 10a, b). As illustrated in Fig. 10c, d, all the converged thermal conductivities are normalized by the TC of stress free at 300 K (κ_{0}), we further give the relative TC (κ/κ_{0}) of the two carbon allotropes as a function of various uniaxial strains. Figure 10 clearly shows that the TC of both OPGL and OPGZ decreases monotonically with increasing tensile strain, which is consistent with previous studies in graphene [34, 48] but in sharp contrast to silicene [34, 50, 51] and phosphorene [37]. As shown in Fig. 10, the maximum reduction of κ_{OPGLZ}, κ_{OPGLA}, and κ_{OPGZZ}, κ_{OPGZA} are 49%, 44%, 37%, and 31%, respectively. Particularly, the TC of OPGL along the zigzag direction can be tuned through strain in a large range.
In order to further elucidate the strain effect on thermal transport properties of OPGL and OPGZ, we calculate the VDOS of phonons of OPGLZ at typical strain. The VDOS are calculated by a Fourier transform of the autocorrelation function of atomic velocity. The function is defined as follows:
As illustrated in Fig. 11, the phonon softening (red shift) in inplane and outofplane directions is observed. This phenomenon is in good agreement with previous studies in graphene under tensile strain [34, 48]. Particularly, compared with the VDOS in outofplane direction, the phonon softening in inplane direction is obvious. It indicates that the decline of TC of OPGL and OPGZ is mainly owing to the straininduced phonon softening in inplane direction.
Conclusions
In summary, both EMD and rNEMD simulations have been performed to investigate the thermal properties of the two novel 2D carbon allotropes composed of octagons and pentagons. The size, temperature, and strain effects on TC are obtained. Our results reveal that the TC increases monotonically with increasing size. The thermal conductivities of infinite sizes are obtained by linear relationships of the inverse length and inverse TC. The converged TC obtained by extrapolation in the reverse nonequilibrium molecular dynamics method is found to be in reasonable agreement with that in the equilibrium molecular dynamics method. The much lower TC, compared with graphene, is attributed to the lower phonon group velocity and phonon mean free path. Our findings provide important insights for the effects of size, temperature, and strain on thermal transport properties of OPGL and OPGZ, and indicate potential applications in thermal management devices in micro/nanoelectronics fields.
Abbreviations
 558:

Fivefiveeightmembered rings
 OPGL:

Octagon and pentagon grapheneline
 OPGZ:

Octagon and pentagon graphenezigzag
 rNEMD:

Reverse nonequilibrium molecular dynamics
 TC:

Thermal conductivity
 VDOS:

Vibrational density of states
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Acknowledgements
We thank Dr. Zheyong Fan (Aalto University) for his helpful discussions.
Funding
We gratefully acknowledge support by the National Natural Science Foundation of China (Grant Nos. 11502217, 11572251, and 11872309), China Postdoctoral Science Foundation (Grant Nos. 2015 M570854 and 2016 T90949), HPC of NWAFU, the Youth Training Project of Northwest A&F University (Grant No. Z109021600), National Key Research and Development Plan, China (Grant No. 2017YFC0405102), the Fundamental Research Funds for the Central Universities (Grant Nos. 3102017jc1003 and 3102017jc11001) of China.
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The original idea was conceived by N. W.; the simulation design and data analysis were performed by S. L., N. W., and C. L. H. R., Y. Z., X. X., and K. C. performed the molecular dynamic simulation studies. The manuscript was drafted by S. L., N. W., and C. L. All authors have approved to the final version of the manuscript.
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Keywords
 Twodimensional material
 Molecular dynamics
 Thermal conductivity
 Carbon allotrope