APPLIED PHYSICS LETTERS 96, 203112 共2010兲
Thermal conductivity of graphene ribbons from equilibrium molecular dynamics: Effect of ribbon width, edge roughness, and hydrogen termination William J. Evans,1 Lin Hu,2 and Pawel Keblinski2,a兲 1
Rensselaer Nanotechnology Center, Rensselaer Polytechnic Institute, Troy, New York 12180, USA Department of Materials Science and Engineering and Rensselaer Nanotechnology Center, Rensselaer Polytechnic Institute, Troy, New York 12180, USA
2
共Received 27 January 2010; accepted 4 May 2010; published online 21 May 2010兲 We use equilibrium molecular dynamic simulations to compute thermal conductivity of graphene nanoribbons with smooth and rough edges. We also study effects of hydrogen termination. We find that conductivity is the highest for smooth edges and is essentially the same for zigzag and armchair edges. In the case of rough edges, the thermal conductivity is a strong function of the ribbon width indicating the important effect of phonon scattering from the edge. Hydrogen termination also reduces conductivity by a significant amount. © 2010 American Institute of Physics. 关doi:10.1063/1.3435465兴 Graphene, a monolayer sheet of hexagonal lattices of sp2 bonded carbon atoms, has been the subject of much investigation over the past decade due to its exceptional electronic and thermal properties.1,2 Graphene nanoribbons 共GNRs兲, which are narrow 共typically⬍ 20 nm兲 strips of graphene, also became the subject of significant research because of equally extraordinary electrical,3 thermal,1,4 and mechanical5 properties with significant application potential to future nanoelectronic/mechanical devices. While many of the GNRs properties are similar to those exhibited by carbon nanotubes, a major advantage of GNRs is a more straightforward fabrication process.6 Recent measurements of the thermal conductivity, , of a partially suspended graphene sheet7 found to be as high as 5300 W/m K at room temperature. First principle calculations by Nika et al.8 and Kong et al.9 determined values in the range of 2000–6000 W/m K. Several research groups using the Brenner potential 共rather than a Tersoff potential which we use兲 and nonequilibrium molecular dynamics 共NEMD兲 simulations4,10,11 found significantly lower values of in the range of several hundred Watt per meter Kelvin depending on the width, edge type 共armchair or zigzag兲, and roughness. We use equilibrium molecular dynamics 共EMD兲 simulations to investigate the effect of GNR width on thermal conductivity for GNRs with both smooth and rough edges. Such simulations have much smaller size effects by comparison with NEMD, in which heat sources and sinks are employed that scatter heat-carrying phonons. The geometry of a typical ribbon is shown in Fig. 1. Armchair and zigzag, smooth and rough edges are considered. We also mimic hydrogen 共H兲-termination of the edges using a united atom approach by increasing the mass of edge carbon atoms to 13 amu. This way we do not explicitly model high frequency motion of the H atom. However, this high frequency motion should not affect thermal conductivity since it is dominated by much lower frequency phonons. Also, in real ribbons the high frequency modes are not exa兲
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cited due to quantum mechanical statistics. We employ periodic boundary conditions along the ribbon length direction. To describe interatomic interactions we use the Tersoff potential,12 which was used in numerous studies of carbon nanotube thermal conductivity.13,14 We first equilibrate each structure for 100000 time steps 共we use a time step of 0.182 fs兲 at constant volume and temperature of 300 K. Following the equilibration, we continue with a constant energy run 共NVE兲 for up to ⬃2 ⫻ 107 time steps. Thermal conductivity is determined via the Green-Kubo expression15
yy =
1 ⍀kBT2
冕
m
具Jy共兲Jy共0兲典d ,
共1兲
0
where ⍀ is the system volume, kB, is the Boltzmann constant, T is system temperature, and Jy is the y component of
FIG. 1. Atomic structure of rough-edged 共left兲 and smooth-edged 共right兲 GNR models.
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FIG. 2. Typical Green–Kubo results for 67 Å wide GNR with smooth zigzag edges computed from independent ensembles of heat currents 共dashed curves兲. The solid curve is the average of the independent ensembles. The inset shows a corresponding HFACF which rapidly decreases to zero.
the heat current. Angle brackets denote autocorrelation so that the bracketed term is the heat flux autocorrelation function 共HFACF兲. ⍀ is calculated as the product of ribbon planar area 共after equilibration兲 and nominal graphite interplanar distance 共3.35 Å兲. In the case of GNRs, y denotes the Cartesian direction parallel to ribbon length direction. In the case of graphene sheet calculations, the reported data are the average over xx and yy. To assess the convergence of the Green–Kubo integral, we computed its value for successive 1 ns segments of heat current data for each case. Figure 2 shows a typical result, indicating good convergence allowing us to extract averages, standard deviations, and errors, which are typically less than 10% 共see Figs. 3 and 4兲. We first calculate the thermal conductivity of approximately square graphene sheets ranging in size from 17 to 140 Å. Since these sheets are fully periodic in both in-plane directions, 共H兲-termination is not applicable. The computed conductivity 共Fig. 3兲 is very high at small sheet sizes, but for sizes larger than ⬃50 Å the conductivity becomes relatively size independent and converges to the value of 8000–10 000 W/m K. Larger conductivity values at very small sizes might
FIG. 3. Thermal conductivity as functions of edge size for square graphene sheets, with periodic boundary applied for both in-plane directions. The dashed lines bound the range of values for large sheet sizes.
Appl. Phys. Lett. 96, 203112 共2010兲
FIG. 4. 共Color online兲 Comparison of computed thermal conductivity for armchair and zigzag GNRs with smooth, rough, and H-terminated edges as a function of ribbon width. Armchair edge data is displaced by +5 Å for clarity. Chart symbols are as follows: 䊏 smooth zigzag, 䉱 rough zigzag, 䊐 smooth armchair, 䉭 rough armchair, ⽧ smooth+ C13 zigzag, 䉲 rough+ C13 zigzag, 〫 smooth+ C13 armchair, 䉮 rough+ C13 armchair.
be associated with the fact that a limited number of phonons are present in the system; thus, due to lack of phononphonon combinations that satisfy the energy and momentum conservation rules for scattering, diminished phonon-phonon scattering occurs.16 Based on the results of graphene sheet calculations we select the length for all simulated GNRs to be 100 Å. We vary the width of the ribbons from 10 to 100 Å. All conductivity data is compiled in Fig. 4. For smooth edges without 共H兲-termination the conductivity is very high 共⬃3000 W / m兲, quite strikingly, even for 10 Å width, and increases with increasing width. At a ribbon width of 100 Å for either type of edge the thermal conductivity reaches values of ⬃6000 to 7000 W/m K with no 共H兲-termination; only about 25% less than the value calculated for graphene sheets. The conductivity for both edges is quite similar for all ribbon widths. Recent NEMD simulations of GRNs 共Ref. 10兲 reported that zigzag GNRs have consistently larger thermal conductivity. We note, however, that our simulations are equilibrium and use the Tersoff potential while NEMD results reported by Hu10 used the Brenner potential, and result in much lower thermal conductivity values. Figure 4 also shows the thermal conductivity of roughedged GNRs and H-terminated smooth and rough edges as a function of ribbon width. It is immediately evident that the rough-edged GNRs with no 共H兲-termination at narrowest widths have a much reduced conductivity, on the order of 500 W/m K, compared to smooth edged GNRs values on the order of 3000 W/m K. With rough edges, at small ribbon widths the conductivity is proportional to the width, which suggests that edge-phonon scattering is a dominant scattering mechanism. Equally important, H-terminated atoms also have a significant effect on lowering thermal conductivity. It is noteworthy that even with rough edges and/or H-termination, conductivity reaches relatively large values of ⬃3000– 4000 W / m K for ribbon widths ⬃100 Å. Results on a range of ribbon structures, with various edge defects and widths allow us to analyze contributions of various scattering mechanisms to the overall thermal resistivity, = 1 / . Assuming independent scattering mechanisms we can write the overall scattering rate,17 S, as 1 / total ⬀ SU
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Appl. Phys. Lett. 96, 203112 共2010兲
Evans, Hu, and Keblinski
+ SBS + SBR + SHT, where the subscripts U, BS, BR, and HT refer to Umklapp, smooth boundary, rough boundary, and H-termination related scattering, respectively. Using the value for graphene 共Fig. 3兲 we obtain SU = 1 / 10 000. Using as an example results for 50 Å zigzag edge ribbon we can further evaluate, SBS = 1 / 5400− SU = 1 / 11 810, SBR = 1 / 2500 − SU − SBS = 1 / 4500, and SHT = 1 / 3200− SBS − SU = 1 / 8000. This can be used to predict the total scattering rate for the rough edge H terminated ribbon as SU + SBS + SHT + SBR = 1 / BR+HT, resulting in conductivity Rough+H-terminated = 1880 W / m K. This value is very close to the one obtained by direct simulation 共1870 W/m K, see Fig. 4兲. As demonstrated by the analysis presented above the boundary scattering is dominant for narrow ribbons. When this is the case, conductivity should be proportional to the ribbon width which is indeed true as demonstrated by the data presented 共Fig. 4兲. For smooth edges, however, the limit of conductivity is not zero when ribbon width→ 0; on the contrary, conductivity in this limit is quite high, ⬃2000 W / m K 关no 共H兲-termination兴. The obvious explanation is that the scattering is much less a factor for smooth edges than for rough edges. However, the fact that thermal conductivity remains high even when width→ 0 for smooth edges is inconsistent with the boundary scattering premise. This suggests that for narrow ribbons with smooth edges, a picture of phonons scattering from edges might not be appropriate. In fact one might consider narrow ribbons as one dimensional, rather that two dimensional, objects akin to small diameter carbon nanotubes. In this context, the diameter dependence of thermal conductivity is caused by size dependent phonon spectra, rather than by scattering by the edges. Another relevant example is high thermal conductivity observed in individual straight polyethylene chains.18 In summary, we have used EMD to compute the thermal conductivity of graphene nanoribbons in sizes ranging from 10 to 100 Å. For smooth edges conductivity is very high, even in the limit of very narrow ribbons. With rough edges
the conductivity is much lower and controlled by edgephonon scattering mechanism. Similarly, H-termination of edge atoms has a very significant effect on conductivity for both smooth and rough edges. The authors acknowledge support from the AFOSR/ MURI program under Grant No. FA9550-08-1-0407 and by NRI-NIST Institute for Nanoelectronics Dicovery and Exploration 共INDEX兲. 1
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