Grain Boundary Energy and Grain Size Dependences of Thermal

Oct 2, 2014 - For further discussion, we introduce the integrated pDOS (Figure 6(b)) and the following equation for determining thermal conductivity b...
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Grain Boundary Energy and Grain Size Dependences of Thermal Conductivity of Polycrystalline Graphene H. K. Liu,†,‡ Y. Lin,*,† and S. N. Luo*,‡ †

State Key Laboratory of Electronic Thin Films and Integrated Devices, University of Electronic Science and Technology of China, Chengdu, Sichuan 610054, P. R. China ‡ The Peac Institute of Multiscale Sciences, Chengdu, Sichuan 610207, P. R. China ABSTRACT: We investigate with molecular dynamics simulations the dependences of thermal conductivity (κ) of polycrystalline graphene on grain boundary (GB) energy and grain size. Hexagonal grains and grains with random shapes and sizes are explored, and their thermal properties and phonon densities of states are characterized. It is found that κ decreases exponentially with increasing GB energy, and decreasing grain size reduces κ. GB-induced phonon softening and scattering, as well as reduction in the number of heat conducting phonons, contribute to the decrease in thermal conductivity.

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INTRODUCTION Graphene is a two-dimensional (2D) carbon film with unique thermal, mechanical, and electronic properties bearing promise in many applications.1−4 Defect-free graphene’s extraordinary properties such as ultrahigh electronic mobility,5,6 superior thermal conductivity,7,8 and mechanical strength with exceptional stretchability,9−11 have been demonstrated in experimental12−15 and theoretical16−19 investigations. Recently, a chemical vapor deposition (CVD) technique was developed to grow large-scale single-layer graphene films20,21 consisting of randomly oriented graphene grains. It is shown that the grain boundaries (GBs) result in not only the deterioration of electronic22,23 and mechanical properties,24−26 but also thermal transport properties.27−29 Yazyev and Louie introduced a general approach for constructing dislocations in graphene characterized by arbitrary Burgers vectors as well as grain boundaries,30 covering a whole range of possible misorientation angles. Bagri et al. studied thermal transport across twin grain boundaries using nonequilibrium molecular dynamics (MD) simulations.27 Previous studies focused mostly on parallel, regularstructured (e.g., repeating pentagon−heptagon pairs) GBs,31−33 which largely represent bicrystals in a sense. However, grains can be of different shapes and sizes and may form GB triple junctions and other random, complex patterns in reality. A recent independent study examined heat conduction in a special, idealized polycrystalline graphene with small hexagonal grains and repeating pentagon−heptagon dislocations.34 Since GB structure and thus GB energy, grain size, and grain shape may all contribute to thermal conductivity, here we perform MD simulations of more realistic polycrystalline structures containing hexagonal grains or grains of random © XXXX American Chemical Society

grain shapes and sizes and explore the effects of the average GB energy and grain size (in terms of GB fraction) on thermal conductivity of polycrystalline graphene. GBs induce reduction in thermal conductivity, and this reduction is quantified and can be attributed to phonon softening and scattering.

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METHODOLOGY Different grain patterns have been observed in experiments, including uniform hexagonal graphene flakes grown on the surface of liquid copper.35 We thus construct polycrystalline graphene structures with the widely used Voronoi tessellation method,36−38 where a set of grain centers is first specified, and for each center there is a corresponding region consisting of all atoms closer to that center than to any other centers. The formal definition is Gi = A ∈ S|d(A , Ci) ≤ d(A , Cj) for all j ≠ i

(1)

Here S is a space created with a distance function d. A, Gi, and Ci denote atoms, grain i, and grain center i, respectively. GBs may form polygons of different shapes due to different grain centers and different numbers of grains chosen. A unit polycrystalline configuration with hexagonal grains and tilt GBs is shown in Figure 1(a). Type-1 grain in Figure 1(a) is the reference grain for rotation. The rotation angle is 30° for type 2; for types 3 and 4, the rotation angles are ϕ and −ϕ, respectively. All the neighboring grains are of different grain types. This construction leads to random GBs (in order to mimic reality) and shapes, so the GB structures are with Received: August 8, 2014 Revised: September 25, 2014

A

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characteristics of pristine and polycrystalline configurations explored in this work are summarized in Table 1. The total numbers of defects in different hexagonal-grained configurations are nearly the same owing to their same GB fractions. The random-grained polycrystalline graphene is more defective given its larger GB fraction. The pentagon and heptagon defect types are dominant among different defects; the fractions of these two defect types in the hexagonal-grained structures are similar, indicating that the difference in their GB energies is mostly due to other defects. GB energy is a most useful and widely used method in characterizing GBs. We also characterize polycrystalline graphene with selected area electron diffraction (SAED) simulation.39 The scattered intensity I(k) is F(k)F *(k) N with the structure factor I(k) =

Figure 1. Atomic configurations of polycrystalline graphene with hexagonal grains (a) and with grains of random shapes and sizes (b). Some structure details at the GB triple junctions in (a) are shown in (c) and (d). Color-coding refers to different grains (e.g., 1−4).

N

F(k) =

E̅GB (J/m2)

Npent

Nhept

Noct

Nother

Ntotal

pristine 5° 10° 15° 20° 25° random

160000 158422 158416 158429 158425 158412 158183

0.00 4.60 1.42 2.11 2.94 3.85 6.52

0 109 122 123 118 112 156

0 111 131 129 123 117 167

0 61 38 41 49 59 112

0 44 21 21 23 29 94

0 325 312 314 313 317 529

(4)

Here k is the reciprocal space vector; r is the position of an atom in the real space; f is the atomic scattering factor for electrons;40 and N is the number of atoms in the selected area/ region. For a given wavelength λ, the diffraction angle θ and k are related via Bragg’s law41 2 sin θ 1 = = |k| λ dhkl

(5)

where d represents the d-spacing between (hkl) planes. We use λ = 0.025 Å (200 keV)40 in SAED simulations. Figure 2 shows

Table 1. Structural and Defect Characteristics of the Pristine and Polycrystalline Graphene Configurationsa system size

∑ f j exp(2πi k·rj) j=1

randomly occurring defects of different types including vacancies and octagons (Figure 1(c) and Figure 1(d)). All the grains in polycrystalline graphene are structurally the same except that neighboring grains are rotated relatively by an angle. The dimensions of the polycrystalline graphene sheet are about 60 nm × 70 nm, containing about 160 000 carbon atoms. We explore five ϕ values for hexagonal configurations (Table 1). In

ϕ

(3)

a

Here N denotes defect number; subscripts pent, hept, oct, and other refer to pentagon defects, heptagon defects, octagon defects, and other types of defects, respectively.

addition, a fully random, 10 grain, polycrystalline structure is generated with random grain centers and rotation angles (Figure 1(b)). To characterize the bulk GB characteristics of a polycrystalline graphene specimen, one parameter is the average GB energy of polycrystalline graphene, E̅ GB, defined as EGB ̅ =

Figure 2. Calculated SAED patterns of the pristine graphene, the hexagonal-grained polycrystalline graphene, and the random-grained polycrystalline graphene simulated in our work. The scale bar refers to the relative electron intensity. The electron energy is 200 keV.

Epl − Epr hL

(2)

the SAED patterns: with increasing GB energy, the diffraction spots become broadened, and the Debye−Sherrer rings becomes less sparse showing increased randomness and “disordering”. In addition, the grain size effects are explored by scaling down the grain sizes (or the area of a graphene sheet) listed in Table 1, while keeping the GB configurations and thus, E̅GB, the same. A convenient parameter to characterize the average grain size is GB fraction, ρGB = L/S, where S is the area of the graphene sheet. For hexagonal grains, the grain size is reduced

where Epl is the total energy of the polycrystalline graphene; Epr is the total energy of the corresponding pristine graphene with the same number of atoms; h is the nominal graphite interplanar distance (3.35 Å); and L is the total length of GBs. Free surface contributions to the total energy are considered in the calculation. For comparing bulk thermal conductivities, the average GB energy is more appropriate than individual GB energies since there are numerous different GBs in a polycrystalline specimen. The structural and defect B

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from about 30 to 10 and 5 nm; these values correspond to ρGB = 0.03, 0.09, and 0.18, respectively. For different polycrystalline graphene specimens, their bulk thermal conductivities are of direct interest, so it is more suitable to use the parameters for characterizing their bulk properties, E̅ GB and ρGB, than those for individual GBs. The large-scale atomic/molecular massively parallel simulator (LAMMPS) is used in our MD simulations.42 The atomic interactions are defined by a widely used, optimized Tersoff potential,43 which has been shown to improve the accuracy of thermal calculations.28,44−46 Thermal conductivity is calculated with the Green−Kubo method47,48 κ=

1 ΩkBT 2

∫0

∞

⟨Ji (0)Ji (t )⟩dt

Figure 3. Thermal conductivity of pristine graphene as a function of time during consecutive, 100, 120 ps periods (dashed curves). The red curve represents the average of the 100 runs. Inset: integration of HFACF over time.

(6)

where Ω = Sh is the system volume; kB is the Boltzmann constant; T is temperature; Ji is the ith component of the heat flux (i = x, y); and t is time. The term within the angle brackets denotes the heat flux autocorrelation function (HFACF). The polycrystalline graphene configurations are relaxed with the conjugate gradient method and then equilibrated with the constant−volume−temperature ensemble and a Nose-Hoover thermostat at 300 K for 1 ns, followed by 0.5 ns thermalization with the microcanonical (NVE) ensemble. Periodic boundary conditions are applied in the in-plane directions, and the out-ofplane dimension of the supercell is sufficiently large.27 Then heat flux is recorded for autocorrelation during long NVE runs. Since the simulations are performed at discrete time steps of Δt = 0.5 fs, eq 6 can be rewritten as49 κ=

Δt ΩkBT 2

M

N−m

∑ (N − m)−1 ∑ m=1

n=1

Ji (n)Ji (n + m)

Figure 4. Normalized thermal conductivity, κ̂ = κ/κ0, of hexagonalgrained graphene as a function of GB fraction. κ0 refers to pristine graphene. Numbers in the legend denote E̅ GB.

(7)

where Ji(m + n) is the ith component of the heat flux at time step n + m. The total number of integration steps M is set to be smaller than the total number of MD steps, N. The Green− Kubo method requires long time simulations to obtain converged κ,49 and our simulation durations are 12 ns. Each HFACF is calculated every 120 ps to obtain a converged thermal conductivity value. There are 100 conductivity values calculated for a 12 ns run, and the final κ value is obtained by averaging the last 70 values.

or GB fraction ρGB, κ decreases rapidly with increasing E̅ GB, e.g., for E̅GB < 3.1 J/m2, and it becomes “saturated” when the GB energy is bigger than E̅ GB = 6 J/m2 (Figure 5). However, thermal conductivity decreases with decreaing grain size less rapidly (Figure 4). Thus, at sufficient high GB energies, thermal conductivity is more sensitive to grain size than GB energy. GBs impede thermal transport, consistent with a previous study

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RESULTS AND DISCUSSION Figure 3 shows a typical HFACF (inset), thermal conductivities of consecutive, 100, 120 ps periods, and the average over the last 70 periods, for pristine graphene containing 160 000 atoms. κ is about 2430 ± 237 W/mK, consistent with previously reported values for pristine graphene predicted with the Tersoff potential23,27 and with other calculations and experiments (2000−5000 W/mK).8,50 Thermal conductivities for pristine graphene and hexagonal- (5 configurations) and randomgrained (1 configuration) polycrystalline graphene, with three different grain sizes or GB fractions, are obtained from the Green−Kubo method. Defects affect thermal transport at the atomistic scale, and the ensemble thermal transport properties are expected to be significantly influenced by GBs in polycrystalline graphene.51 For given GB configurations or E̅GB, κ decreases with increasing GB fractions or decreasing grain sizes, partly owing to increasing phonon scattering at GBs (Figure 4). Mortazavi reported similar results in randomgrained polycrystalline graphene with grain size less than 5 nm (grain size is up to 25 nm in this work).52 For a given grain size

Figure 5. κ̂ as a function of E̅ GB, for pristine graphene and polycrystalline graphene with hexagonal grains and grains of random shapes and sizes. The solid curves denote exponential fittings, and numbers in the legend denote ρGB. C

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suggesting that the phonon transmission is weakened by increased GB energy in a twin boundary system.28 The simulation results in Figure 5 can be fitted with an exponential function ⎧ E̅ ⎫ ̂ + aexp⎨− GB ⎬ κ ̂ = κ∞ ⎩ ε ⎭

(8)

where the normalized thermal conductivity κ̂ = κ/κ0; κ̂∞, a, and ε are fitting parameters. κ̂∞ is the asymptotic value of thermal conductivity, and ε is the characteristic energy reflecting the overall effect of different defects at GBs. ε can be regarded as phonon mobility, analogous to the Matthiessen’s rule53 1 1 1 1 1 = + + + + ··· ε εvacancies εpentagons εheptagons εoctagons (9) Figure 6. (a) Phonon DOS of the pristine and polycrystalline graphene. The inset shows the details of the first pDOS peak. (b) The integrated pDOS. The inset shows integration of the first pDOS peak.

Here subscripts denote contributions of various defects to the inverse mobility. In other words, defects at GBs reduce the effective mobility and thus thermal conductivity. κ̂∞ decreases with increasing GB fraction (or decreasing grain size), and it is the opposite for ε. κ̂∞ is 0.24, 0.20, and 0.17, and ε is 0.60, 0.87, and 1.17 J/m2, for ρGB = 0.03, 0.09, and 0.18, respectively. The values suggest that phonon scattering at GBs increases with increasing GB fraction or decreasing grain sizes. Similarly, Khitun et al. reported that the in-plane thermal conductivity of a quantum-dot superlattice54,55 decreases with increasing fraction of quantum dots. They described the phonon transport inside a quantum dot within a continuum approximation and attributed the reduction to acoustic phonon scattering. The quantum dots were of different sizes and shapes, analogous to GBs in our cases. Compared to the hexagonalgrained graphene, κ for the random-grained graphene with the same system sizes is lower (below the fitted curves, Figure 5) due to the higher GB fractions. Thus, increasing GB energy and GB fraction both reduce thermal transport capability. However, for high E̅GB, the former becomes less effective, and GB fraction plays a more important role since the reduction in κ is more rapid and becomes saturated with increasing E̅GB, compared to decreasing grain size (cf. Figures 3 and 4). Thermal conductivity is closely related to phonons, in particular acoustic phonons. We calculate phonon density of states (pDOS) for pristine and polycrystalline graphene (Figure 6(a)) from additional NVE simulations, where the atomic velocities are recorded every 2.5 fs for 100 ps, by the Fourier transformation of time-dependent velocity autocorrelation function (VCAF) g (ω) =

1 2π

∫0

∞

⟨υ(0)υ(t )⟩eiωt dt

κ=

1 3

∫0

∞

C(T , ω)υg λ(T , ω)dω

(11)

Thus, GB-induced reduction in υg contributes to the decrease in κ compared to pristine graphene. However, the difference in red shifts of the pDOS is not pronounced among all the polycrystalline graphene. For further discussion, we introduce the integrated pDOS (Figure 6(b)) and the following equation for determining thermal conductivity by polarization modes (longitudinal acoustic, LA, and transverse acoustic, TA, modes)28 κ=

1 2

∫0

ω

⎛ dω ⎞2 df (ω) τ(ω)g (ω)⎜ ℏωdω ⎟ ⎝ dq ⎠ dT

(12)

where q is wave vector, τ is relaxation time, and ℏ is the reduced Planck constant; the 1/2 factor is due to the 2D nature. For scattering caused by defects, the relaxation time is related to the scattering cross-section σ and defect density η as58 τ=

1 ησυg

(13)

The integrated pDOS is shown in Figure 6(b). There are ∼10% more phonons in the pristine graphene than the polycrystalline graphene over the whole frequency range and 3% more phonons in the hexagonal-grained graphene than the random-grained graphene. The variations in the number of phonons are consistent with those in κ. Meanwhile, increasing GB energy or GB fraction in polycrystalline graphene yields more defects and grain boundaries (Table 1). From eq 10, the larger scattering cross section σ and defect density η lead to the decrease in phonon relaxation time τ and thus κ according to eq 9. Thus, an increase in η and σ in polycrystalline graphene leads to a decrease in τ and thus κ, even though there is no obvious difference in υg. Furthermore, the phonon number reduction of the g(ω) peaks, in particular the acoustic phonons, decreases the number of heat-conducting phonons and thus heat conduction. In our simulations of polycrystalline graphene, the main contribution of the in-plane pDOS comes from the LA and TA modes around 24 THz (Figure 6(a)). The inset to

(10)

Here ω is angular frequency; υ is velocity; and the term with the angle brackets represents VACF. Since graphene is 2D, only in-plane, longitudinal/transverse phonons are considered.56 Figure 6(a) shows two main peaks in pDOS located around 24 and 53 THz (the latter is also referred to as the G-band). Our results on pristine graphene are in accord with previous calculations.51,57 The G-band of pristine graphene is located at 52.2 THz and that of polycrystalline graphene at 50.9 THz. Therefore, the polycrystalline graphene shows a 2% red shift in pDOS compared to the pristine graphene, indicating phonon softening and a reduction of phonon group velocities (υg). Thermal conductivity at a given temperature T is related to υg, heat capacity C, and phonon mean free path λ as53 D

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Figure 6(a) (Figure 6(b)) shows that a higher pDOS peak (more phonon number) corresponds to a lower GB energy or a smaller GB fraction at the same frequency and, along with an increased relaxation time τ, leads to higher conductivity as seen from eq 9.

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CONCLUSIONS We have investigated the thermal conductivity of polycrystalline graphene, including its GB energy and grain size dependences. The configurations are generated by Voronoi tessellation with hexagonal and random-shaped grains, mimicking those in real applications. The ensemble thermal conductivity is calculated with the Green−Kubo method, and its relations with GB size (GB fraction) and GB energy are quantified. Our results indicate that the thermal conductivity decreases with increasing GB fraction or GB energy. At sufficient high GB energies, thermal conductivity is more sensitive to grain size than GB energy. Phonon densities of states are computed for pristine and polycrystalline graphene. Phonon softening and scattering and reduction in the number of heat-conducting phonons contribute to decreasing heat conductivity with increasing GB energy or decreasing GB size.

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AUTHOR INFORMATION

Corresponding Authors

*E-mail: [email protected] *E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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REFERENCES

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The Journal of Physical Chemistry C

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dx.doi.org/10.1021/jp508035b | J. Phys. Chem. C XXXX, XXX, XXX−XXX