PHYSICAL REVIEW B 75, 153303 共2007兲

Thermal conductivity of GaN nanotubes simulated by nonequilibrium molecular dynamics Zhiguo Wang,1,* Fei Gao,2 Jean-Paul Crocombette,3 X. T. Zu,1 L. Yang,1 and William J. Weber2 1Department

of Applied Physics, University of Electronic Science and Technology of China, Chengdu 610054, People’s Republic of China 2 Pacific Northwest National Laboratory, P.O. Box 999, Richland, Washington 99352, USA 3CEA-Saclay, DEN/DMS/SRMP, 91191 Gif-Sur-Yvette, France 共Received 27 November 2006; published 12 April 2007兲

Thermal conductivity of GaN nanotubes along the tube axis is investigated over the temperature range from 600 to 2300 K using homogeneous nonequilibrium molecular dynamics. In general, the thermal conductivity of nanotubes is smaller than that for the bulk GaN single crystal. The thermal conductivity is also found to decrease with temperature and increase with increasing wall thickness of the nanotubes. The change of phonon spectrum and surface inelastic scattering may account for the reduction of thermal conductivity in the nanotubes, while thermal softening and high-frequency phonon interactions at high temperatures may provide an explanation for its decrease with increasing temperature. DOI: 10.1103/PhysRevB.75.153303

PACS number共s兲: 68.65.⫺k, 44.10.⫹i, 65.80.⫹n

As the sizes of electronic, optical, and mechanical devices are being decreased to the nanometer level, and the speed of their operation is being steadily increased, the thermal transport of the low-dimensional system is becoming an important issue since it plays an important role in controlling the performance and stability of these nanodevices. Although one-dimensional nanostructures show unique properties unattainable in bulk due to high surface-to-volume ratios at the nanoscale, some drawbacks of these nanostructures are inevitable. One of the challenges is that the thermal conductivity of nanostructures is greatly reduced, as compared with that in the bulk.1–4 Recently, GaN nanotubes have been successfully produced,5–8 with proposed applications in nanoscale electronics, optoelectronics, and biochemical-sensing field. In this Brief Report, the thermal conductivity of GaN nanotubes is investigated using homogeneous nonequilibrium molecular dynamics. The thermal conductivity J ␭ is related to the heat flux and the applied temperature gradient by Fourier’s law Jជ q = ជ T. The energy flux expression is derived from the enJ·ⵜ −␭ ជ · Jជ 共rជ , t兲 = 0. In a ergy balance equation 共1 / V兲关⳵E共rជ , t兲 / ⳵t兴 + ⵜ q classical approach, the heat flux can be given by the following expression in a solid:9,10

N

1 d 1 Jជ q共t兲 = 兺 r iE i = V V dt i

冉兺 N i

N

vជ iEi +

冊

the Green-Kubo approach. In addition, it has been established that finite-size effects do play a role in applying this method.16 The steady-state NEMD method relies on imposing a temperature gradient across the simulation cell and is therefore analogous to the experimental conditions. This method requires large simulation boxes in order to partition the system into a number of slabs where equilibrium can take place. In addition, instabilities may appear in the neighborhood of hot and cool slabs.17 These drawbacks may be overcome with another nonequilibrium molecular-dynamics method, the homogeneous field method.18–22 In this method, a “heat field” ជf ext is introduced in Newton’s equations so as ជ +D J · ជf ext共t兲, where D J to produce a desired heat flow maជ i = F i i i is defined as J ab = D i

冊

1 兺 共Fជ ij · vជ i兲rជij . 2 i,j⫽i

␭ = lim

The thermal conductivity can be obtained from molecular dynamics using either equilibrium simulations,11–13 based on Green-Kubo equations, steady-state nonequilibrium simulations,14,15 or nonequilibrium molecular–dynamics 共NEMD兲 simulations. The Green-Kubo approach is an equilibrium MD method that uses current fluctuations to compute the thermal conductivity via the fluctuation-dissipation theorem. However, very long simulation times are needed to sufficiently converge the current autocorrelation function with

N

which represents the coupling between the perturbation and the system. The N particle system is coupled to the heat field ជf ext. The coupling is defined in such a way that the energy dissipation is proportional to Jជ qជf ext. The thermal conductivity can then be obtained from extrapolation to zero-field amplitude:

冉

lim

ជf ext→0 t→⬁

共1兲

1098-0121/2007/75共15兲/153303共4兲

冉

N

N

pជ 2i 1 1 + ␾i ␦ab − 兺 raij f bij + 兺 兺 ra f b , 共2兲 2 j=1 2m 2N k=1 j=1 kj kj

冊

具Jq共t兲典 . VTfជext

共3兲

As pointed out in the literature,22 the method works only for ជf ext not too large and not too small. If the ជf ext is too large, a solitary wave will travel in the direction of heat flow, and heat is transported in the form of a highly localized energy pulse carried by a soliton. In this case, the average value of the heat flux is nearly independent of ជf ext. However, when the heat field is too small, the noise-signal ratio would become too large, and the accuracy of this method would be drastically reduced.

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PHYSICAL REVIEW B 75, 153303 共2007兲

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FIG. 1. 共Color online兲 Top view of GaN nanotubes that are enclosed by 兵100其 side planes.

The empirical interatomic interaction used in this work is the Stilling-Weber potential23 that has been parametrized to reproduce bulk structures and mechanical properties. The potentials can handle dangling bonds, wrong bonds, and excess bonds in bulk GaN very well. In addition, they have been successfully employed to evaluate the Young modulus of defect-free and defected single-crystal GaN nanotubes.24,25 In all the MD simulations, a 0.5 fs time step is used. The fictitious force was set along the tube length direction, and the force f ext was set to 共5 – 8兲 ⫻ 106 m−1. The averaging of the heat flux in Eq. 共3兲 was calculated over the last 10 ps of a 50 ps simulation. To maintain a constant temperature within the box, the following scaling method is adopted:26 new 冑 is the velocity of particle i after correcvnew i = v i T D / T R. v i tion, and TD and TR are the desired and actual temperatures of the system, respectively. As the simulation temperatures are higher than the Debye temperature 共600 K for wurtzite GaN兲, the quantum correction of the temperature is not considered in this Brief Report. Nanotubes with 关100兴-lateral oriented facets are simulated, with lengths varying from 2.08 to 10.4 nm. The inner radius of the nanotube is 0.92 nm, and the outer radius ranges from 1.29 to 2.67 nm, i.e., the thickness of the nanotubes changes from 0.37 to 1.75 nm, and the corresponding number of atoms changes from 3840 to 21 840 for the nanotubes with a length of 10.4 nm. Figure 1 shows the cross sections of GaN nanotubes which are enclosed by 兵100其 side planes. The stability of the structure in Fig. 1 has been examined for 100 ps at room temperature, and we do not observe any bond breaking. The hexagonal shape at the cross section remains similar to the original configuration. The melting temperature of the GaN nanotubes has been determined to increase with the thickness of the nanotubes to a saturation value that is close to the melting temperature of bulk GaN.27 For comparison, the thermal conductivity of bulk GaN was initially calculated. The force f ext was set along the 关001兴 direction. In performing this calculation, we have carefully checked the size effect on the thermal conductivity by varying the dimension of the MD cell, and the numbers of atoms used in the present calculations are 768, 1600, 2880, 4704, 9720, and 12 800. The calculated thermal conductivity of bulk GaN at 600 K as a function of the number of atoms in the cell is shown in Fig. 2共a兲. The results suggest that the

FIG. 2. 共a兲 Thermal conductivity of bulk GaN at 600 K as a function of number of atoms and 共b兲 temperature dependence of thermal conductivity of bulk GaN, where the number of atoms is 2880.

thermal conductivity initially increases with increasing numbers of atoms and saturates at a value of 152 W / mK, when the number of atoms is greater than 2880. Consequently, 2880 atoms in a MD cell were used for the calculations of thermal conductivity of bulk GaN over the temperature range from 300 to 2300 K, and the results are shown in Fig. 2共b兲. It can be seen that the thermal conductivity of bulk GaN is 215 W / mK at 300 K, which is consistent with experimental values. Jeżowski et al.28 reported that the thermal conductivity of bulk GaN is 220 W / mK at 300 K, while Florescu et al.29 obtained the thermal conductivities of fully and partially coalesced lateral epitaxial overgrown GaN/sapphire to be 186–205 and 200– 210 W / mK, respectively. In general, the thermal conductivity decreases with increasing temperature, as shown in Fig. 2共b兲. Following the initial rapid reduction of thermal conductivity, it saturates at a value of 30 W / mK at temperatures higher than 1800 K. In the case of GaN nanotubes, one main concern of using MD to calculate the thermal conductivity is the size effect of the simulation box due to periodic boundary conditions. To verify that the tube length does not affect the MD results, the sensitivity of the thermal conductivity to the simulation tube length has been carefully examined. Figure 3 shows the dependence of thermal conductivity on the length of the GaN nanotubes with thicknesses ranging from 0.37 to 1.75 nm at a simulation temperature of 600 K. The error bars shown in the figure correspond to ±5 W / mK. In small simulation systems, the calculated thermal conductivity is between 19 and 34 W / mK, depending on the thickness of the nanotubes. Because of the overestimation of phonon scattering in a small cell, these values may be not accurate. As shown in Fig. 3,

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FIG. 3. Dependence of thermal conductivity on the tube length of GaN nanotubes with different thicknesses at a simulation temperature of 600 K.

the thermal conductivity increases with increasing length of the nanotube and converges to a constant when the nanotube length is larger than 6.0 nm. In order to eliminate possible phonon scattering, a tube length of 10.4 nm is used in the following simulations. As described above, the fictitious force must be carefully set to obtain accurate results. In the present calculations, we have tested the fictitious force and found that the force f ext in the range of 共5 – 8兲 ⫻ 106 m−1 provides reliable results. The relationship between the thermal conductivity and f ext obtained by the nonequilibrium molecular-dynamics calculation is shown in Fig. 4, where the simulation temperature is 600 K. The thermal conductivities of the nanotubes can be obtained from extrapolation to zero-field amplitude and are found to be 28, 31, 35, and 41 W / mK for nanotubes with wall thicknesses of 0.37, 0.64, 1.20, and 1.75 nm, respectively, at 600 K. A similar approach has been applied to calculate the thermal conductivity of various GaN nanotubes at other temperatures considered at the present study. The temperature dependence of thermal conductivity for GaN nanotubes is shown in Fig. 5 and the results indicate that the thermal conductivities decrease with increasing temperature for these nanotubes. The thermal conductivities also increase with increasing nanotube wall thickness, but the difference becomes small at higher temperatures. Furthermore, the thermal conductivities of the nanotubes are smaller than the corresponding conductivity of bulk GaN. A similar reduction of thermal conductivity for nanowires has also been previously observed theoretically1–3 and experimentally.4

FIG. 4. Dependence of thermal conductivity on the applied force f ext at a simulation temperature of 600 K for the GaN nanotubes with different thicknesses at a given length of 10.4 nm.

FIG. 5. Temperature dependence of thermal conductivity for GaN nanotubes with different thicknesses at a given length of 10.4 nm.

The thermal conductivity of Si nanowires has been found to be 1–2 orders of magnitude smaller than that for bulk Si.2 In general, the thermal conductivity of a nanotube can be written as30 ␭共T兲 = 31 兺 Cvl over all phonon states, where C, v, and l are the specific heat, group velocity, and phonon mean free path, respectively. The reduction of the nanotube thermal conductivity could be attributed to 共i兲 the change of phonon spectrum in one-dimensional structures, which modifies the phonon group velocity and the scattering mechanisms,31,32 and 共ii兲 the boundary inelastic scattering, which increases diffuse reflections on the surfaces. As the thickness of the nanotubes increases, so does the thermal conductivity, mainly because the boundary scattering rate decreases.33 The phonon-phonon interaction increases with size reduction due to the confinement, which causes an increase of thermal resistance and a decrease of heat conduction.34 The slope of the ␭共T兲 curve of the nanotubes is smaller than that of bulk GaN, which is an indication of the strong phonon-phonon scattering in GaN nanotubes. The decrease of thermal conductivity with increasing temperature should be associated with thermal softening and higherfrequency phonon interactions at high temperatures. The lower lattice stiffness at high temperatures results in low average phonon group velocities. Also at high temperatures, high-frequency acoustic- and optical-phonon interactions become appreciable, lowering the mean free path.3 In conclusion, we have calculated the thermal conductivity of GaN nanotubes using homogeneous nonequilibrium molecular dynamics, and the results show that the thermal conductivities of GaN nanotubes are 28, 31, 35, and 41 W / mK for nanotubes with thicknesses of 0.37, 0.64, 1.20, and 1.75 nm at 600 K, respectively, which is smaller than the value of 150 W / mK in bulk GaN at the same temperature. The thermal conductivity decreases with increasing temperature and increases with increasing nanotube wall thickness. The change of phonon spectrum, surface inelastic scattering, thermal softening, and higher-frequency phonon interactions at high temperatures may account for the observed phenomena. This work was supported by the Program for Innovative Research Team in UESTC and by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, U.S. Department of Energy under Contract No. DE-AC0576RL01830.

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