Atomistic simulation of the size and orientation dependences of

mal conductivity of Si nanowires is about two orders of mag- nitude smaller than that of bulk Si,2 and the thermal conduc- tivity of ZnO nanobelts is an order of ...
205KB taille 6 téléchargements 325 vues
APPLIED PHYSICS LETTERS 90, 161923 共2007兲

Atomistic simulation of the size and orientation dependences of thermal conductivity in GaN nanowires Zhiguo Wanga兲 and Xiaotao Zub兲 Department of Applied Physics, University of Electronic Science and Technology of China, Chengdu, 610054, People’s Republic of China

Fei Gaoc兲 and William J. Weber Pacific Northwest National Laboratory, P.O. Box 999, Richland, Washington, 99352

Jean-Paul Crocombette CEA-Saclay, DEN/DMN/SRMP, 91991 Gif-Sur-Yvette, France

共Received 12 January 2007; accepted 24 March 2007; published online 20 April 2007兲 The thermal conductivity of GaN nanowires has been determined computationally by applying nonequilibrium atomistic simulation methods using the Stillinger-Weber 关Phys. Rev. B 31, 5262 共1985兲兴 potentials. The simulation results show that the thermal conductivity of the GaN nanowires is smaller than that of a bulk crystal and increases with increasing diameter. Surface scattering of phonons and the high surface to volume ratios of the nanowires are primarily responsible for the reduced thermal conductivity and its size dependence behavior. The thermal conductivity is also found to decrease with increasing temperature and exhibits a dependence on axial orientation of the nanowires. © 2007 American Institute of Physics. 关DOI: 10.1063/1.2730747兴 The thermal conductivity of one dimensional nanostructures has been of great interest in recent years because of the application of nanoarchitectures in microelectronics1 and the very high heat density that will be transported in these nanostructures. Unfortunately, very few materials are known to exhibit high thermal conductivity at reduced dimensions. Traditional semiconductors show a dramatic reduction of thermal conductivity at the nanoscale. For example, the thermal conductivity of Si nanowires is about two orders of magnitude smaller than that of bulk Si,2 and the thermal conductivity of ZnO nanobelts is an order of magnitude lower than that for bulk ZnO crystals.3 Thermal conductivity studies of nanowires using the Boltzmann transport equation and the equations of phonon radiative transfer show that surface boundary and interface scattering are the main mechanisms for the decrease of the thermal conductivity.4,5 Gallium nitride is a high temperature semiconductor material that emits brilliant light and is considered for the next generation of high frequency and high power transistors that are capable of operating at high temperatures. Single-crystal gallium nitride nanowires have already shown promise for realizing photonic and biological nanodevices based on blue light emitting diodes6 and short-wavelength ultraviolet nanolasers.7,8 Due to its anisotropic and polar nature, GaN exhibits directional-dependent properties,9 and the growth direction of GaN nanowires can been controlled by using heteroepitaxy on different single-crystal templates, mediated by catalytic clusters.10–14 Nanowires have been grown along the 关001兴 crystallographic axis with hexagonal cross ¯ 0兴 and 关110兴 exsections,10–12 while those grown along 关11 hibit triangular cross sections.13,14 Such low dimensional GaN-based structures represent important nanometer-scale building blocks for potential optoelectronic, high temperature/high power, and spintronic devices.15 The a兲

Author to whom correspondence should be addressed. Electronic mail: zgwangគ[email protected] Electronic mail: [email protected] c兲 Electronic mail: [email protected] b兲

growth direction is critical in determining the nanowire’s thermal, optical, electrical, and mechanical properties. Quantitative understanding of the size and orientation dependences of thermal conductivity could provide design and fabrication criteria for nanoscale devices. In this letter, we used homogeneous nonequilibrium molecular dynamics with a well established empirical potential to investigate the thermal conductivity of GaN single-crystal nanowires. The thermal conductivity determines the heat current due to a temperature gradient via Fourier’s law, which is given as J␮ = −兺␯␭␮␯⳵T / ⳵x␯. Experimentally, ␭ is obtained by measuring the temperature gradient that results from the application of a heat current. The energy flux expression is derived from the energy balance equation 1 ⳵E共r,t兲 + ⵱ · Jq共r,t兲 = 0, V ⳵t

共1兲

where E共r , t兲 is the instantaneous local energy and Jq共r , t兲 is the instantaneous local heat flux. The integration of Eq. 共1兲, combined with the definition of the total instantaneous heat flux in the statistical ensemble of constant energy, leads to the following expression:16,17 N

Jq共t兲V =

N

N

1 d 兺 riEi = 兺i viEi + 2 i,j⫽i 兺 共Fij · vi兲rij . dt i

共2兲

In molecular dynamics 共MD兲 simulations, the thermal conductivity can be computed either using equilibrium MD based on Green-Kubo equations, steady-state nonequilibrium MD, or nonequilibrium molecular dynamics 共NEMD兲 simulations. In the equilibrium methods, the autocorrelation function, which decreases slowly and nonmonotonically towards zero, must be evaluated and integrated over a long time. The slow convergence of the integral makes equilibrium MD simulations much more expensive in terms of computation time. In addition, it has been established that finite-size effects do play a role in applying this method.18 In steady-state nonequilibrium MD, two heat reservoirs with high and low

0003-6951/2007/90共16兲/161923/3/$23.00 90, 161923-1 © 2007 American Institute of Physics Downloaded 20 Apr 2007 to 130.20.227.172. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

161923-2

Appl. Phys. Lett. 90, 161923 共2007兲

Wang et al.

temperatures are attached on opposite sides of the MD cell. By measuring the average heat flux and temperature gradient, the thermal conductivity can be calculated as the ratio of the two quantities. Also there are some inherent disadvantages in this approach.19 These drawbacks may be overcome with another nonequilibrium molecular dynamics method, the homogeneous field method.20–24 In this method, a “heat field” fext is introduced in Newton’s equations so as to proJ is duce a desired heat flow: mai = Fi + Di · fext共t兲, where D i defined as J ab = D i





N

N

N

p2i 1 1 + ␾i ␦ab − 兺 raij f bij + 兺 兺 ra f b , 2 2m 2N k=1 j=1 kj kj j=1

共3兲

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



␭ = lim lim

fext→0 t→⬁



具Jq共t兲典 . VTfext

共4兲

As pointed out in the literature,24 this method works only for fext that is not too large and not too small. If fext 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 which case, the average value of the heat flux is nearly independent of fext. 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. In the present calculations, we have tested the fictitious force and found that a force f ext in the range of 共5 – 8兲 ⫻ 106 m−1 provides reliable results. The empirical interatomic interaction used in this work is the Stillinger-Weber potential25 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, it has been employed to evaluate the Young modulus of defect-free and defective single-crystal GaN nanotubes.26,27 In all the MD simulations, a 0.5 fs time step is used. The fictitious force was set along the wire length direction. The averaging of the heat flux in Eq. 共4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: = vi冑TD / TR, where vnew is the velocity of particle i after vnew i i correction, and TD and TR are the desired and actual temperatures of the system, respectively. Based on reported experimental observations,10–14 we constructed 关001兴-oriented GaN nanowires with hexagonal ¯ 0兴- and 关110兴-oriented GaN nanowires cross sections and 关11 with triangular cross sections directly from bulk GaN by removing atoms outside a hexagon or a triangle and replacing with vacuum space. The top views of these GaN nanowires have been published in Refs. 28 and 29. The thermal conductivity of bulk GaN calculated using the NEMD without quantum correction is 215 W / m K at 300 K, which is consistent with experimental values. Jezowski et al.30 reported that the thermal conductivity of bulk GaN is 220 W / m K at 300 K. The thermal conductivities of bulk GaN were 151, 167, and 157 W / m K as the

FIG. 1. Dependence of thermal conductivity on wire length for 关001兴oriented GaN nanowires with 兵100其 side planes at a simulation temperature of 600 K.

¯ 0兴, and 关110兴 direcforce f ext was set along the 关001兴, 关11 tions, respectively, at 600 K, which indicates less directional dependence. Figure 1 shows the dependence of thermal conductivity on wire length for 关001兴-oriented nanowires with diameters ranging from 2.02 to 6.44 nm at a simulation temperature of 600 K. The results show that at larger cell sizes thermal conductivity becomes independent of the length of the nanowires. Because of the overestimation of phonon scattering in a small cell, these values may not be accurate in small simulation systems. For all the nanowires considered in the present study, the wire length dependence of thermal conductivity showed the same characteristics. Wire lengths of 10.4, 11.08, and 11.2 nm are used in the following simulations for ¯ 0兴-, and 关110兴-oriented nanowires. the 关001兴-, 关11 In Fig. 2, the thermal conductivity of 关001兴-oriented GaN nanowire with a diameter of 6.44 nm is shown as a function of external field f ext at simulation temperatures from 600 to 2100 K. A linear dependence of thermal conductivity on f ext is observed, and the thermal conductivity of the nanowires can be obtained from extrapolation to zero field amplitude. For example, the thermal conductivities are 46.67, 18.05, and 3.11 W / m K for the nanowire at 600, 1200, and 2100 K. A similar approach has been applied to calculate the thermal conductivity of all the GaN nanowires in the present study. ¯ 0兴-, and The thermal conductivities of the 关001兴-, 关11 关110兴-oriented GaN nanowires with different diameters were evaluated with the NEMD simulation in the temperature range between 600 and 2100 K. As the classical method is valid only for temperature well above the Debye temperature, only variations above this temperature are reported here. Figure 3 shows the size and temperature dependences ¯ 0兴-, and 关110兴of thermal conductivity for the 关001兴-, 关11 oriented GaN nanowires. As shown in Fig. 3, the calculated

FIG. 2. Dependence of thermal conductivity on the applied force f ext at simulation temperatures from 600 to 2100 K for 关001兴-oriented GaN nanowires with a diameter of 6.44 nm. Downloaded 20 Apr 2007 to 130.20.227.172. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

161923-3

Appl. Phys. Lett. 90, 161923 共2007兲

Wang et al.

thermal conductivity increases with increasing diameter and decreases with increasing temperature. The MD results also reveal a reduction in the thermal conductivity of GaN nanowires relative to those of the corresponding bulk crystal. The thermal conductivity also exhibits a strong orientational de¯ 0兴- and 关110兴pendence. The thermal conductivity of 关11 oriented GaN nanowires is substantially lower that that of the 关001兴-oriented ones.

FIG. 3. Size and temperature dependences of thermal conductivity for 共a兲 关001兴-oriented nanowires with 兵100其 side planes, 共b兲 关001兴-oriented nano¯ 0兴-oriented nanowires, and 共d兲 关110兴wires with 兵100其 side planes, 共c兲 关11 oriented nanowires.

Two of the authors 共Z.W. and X.Z.兲 are grateful to the Program for Innovative Research Team in UESTC, and another two of the authors 共F.G. and W.J.W.兲 were supported by the Division of Materials Sciences and Engineering, Office of Basic Energy Sciences, U.S. Department of Energy under Contract No. DE-AC05-76RL01830. 1

R. C. Liu, C. S. Pai, and E. Martinez, Solid-State Electron. 43, 1003 共1999兲. 2 S. G. Volz and G. Chen, Appl. Phys. Lett. 75, 2056 共1999兲. 3 A. J. Kulkarni and M. Zhou, Appl. Phys. Lett. 88, 141921 共2006兲. 4 X. Lü, W. Z. Shen, and J. H. Chu, J. Appl. Phys. 91, 1542 共2002兲. 5 D. Stewart and P. Norris, Microscale Thermophys. Eng. 4, 89 共2000兲. 6 Y. Huang, X. F. Duan, Y. Cui, and C. M. Lieber, Nano Lett. 2, 101 共2002兲. 7 J. C. Johnson, H. J. Choi, K. P. Knutsen, R. D. Schaller, P. D. Yang, and R. J. Saykally, Nat. Mater. 1, 106 共2002兲. 8 S. Gradečak, F. Qian, Y. Li, H. G. Park, and C. M. Lieber, Appl. Phys. Lett. 87, 173111 共2005兲. 9 P. Waltereit, O. Brandt, A. Trampert, H. T. Grahn, J. Menniger, M. Ramsteiner, M. Reiche, and K. H. Ploog, Nature 共London兲 406, 865 共2000兲. 10 B. D. Liu, Y. Bando, C. C. Tang, F. F. Xu, and D. Golberg, Appl. Phys. Lett. 87, 073106 共2005兲. 11 T. Kuykendall, P. J. Pauzauskie, Y. F. Zhang, J. Goldberger, D. Sirbuly, J. Denlinger, and P. D. Yang, Nat. Mater. 3, 524 共2004兲. 12 J. Zhang, L. D. Zhang, X. F. Wang, C. H. Liang, X. S. Peng, and Y. W. Wang, J. Chem. Phys. 115, 5714 共2001兲. 13 S. Y. Bae, H. W. Seo, J. Park, H. Yang, H. Kim, and S. Kim, Appl. Phys. Lett. 82, 4564 共2003兲. 14 T. Kuykendall, P. Pauzauskie, S. Lee, Y. F. Zhang, J. Goldberger, and P. D. Yang, Nano Lett. 3, 1063 共2003兲. 15 D. Vashaee, A. Shakouri, J. Goldberger, T. Kuykendall, P. Pauzauskie, and P. Yang, J. Appl. Phys. 99, 054310 共2006兲. 16 P. B. Allen and J. L. Feldman, Phys. Rev. B 48, 12581 共1993兲. 17 R. J. Hardy, Phys. Rev. 132, 168 共1963兲. 18 P. K. Schelling, S. R. Phillpot, and P. Keblinski, Phys. Rev. B 65, 144306 共2002兲. 19 A. Maeda and T. Munakata, Phys. Rev. E 52, 234 共1995兲. 20 D. Evans and G. Morris, Statistical Mechanics of Non Equilibrium Liquids 共Academic, London, 1990兲, Chap. 6, p. 33. Available online at http:// rsc.anu.edu.au/~evans/evansmorrissbook.php 21 A. Maeda and T. Munakata, Phys. Rev. E 52, 234 共1995兲. 22 S. Motoyama, Y. Ichikawa, Y. Hiwatari, and A. Oe, Phys. Rev. B 60, 292 共1999兲. 23 S. Berber, Y. K. Kwon, and D. Tamanek, Phys. Rev. Lett. 84, 4613 共2000兲. 24 F. Zhang, D. J. Isbister, and D. J. Evans, Phys. Rev. E 61, 3541 共2000兲. 25 J. Kioseoglou, H. M. Polatoglou, L. Lymperakis, G. Nouet, and Ph. Komninou, Comput. Mater. Sci. 27, 43 共2003兲. 26 B. Xu, A. J. Lu, B. C. Pan, and Q. X. Yu, Phys. Rev. B 71, 125434 共2005兲. 27 B. Xu and B. C. Pan, J. Appl. Phys. 99, 104314 共2006兲. 28 Z. G. Wang, X. T. Zu, F. Gao, and W. J. Weber, J. Mater. Res. 22, 742 共2007兲. 29 Z. G. Wang, X. T. Zu, F. Gao, and W. J. Weber, J. Appl. Phys. 101, 043511 共2007兲. 30 A. Jeżowski, B. A. Danilchenko, M. Boćkowski, I. Grzegory, S. Krukowski, T. Suski, and T. Paszkievicz, Solid State Commun. 128, 69 共2003兲. 31 M. A. Stroscio, Y. M. Sirenko, S. Yu, and K. W. Kim, J. Phys.: Condens. Matter 8, 2143 共1996兲. 32 A. Balandin and K. L. Wang, Phys. Rev. B 58, 1544 共1998兲. 33 L. Shi, Q. Hao, C. Yu, N. Mingo, X. Y. Kong, and Z. L. Wang, Appl. Phys. Lett. 84, 2638 共2004兲. 34 L. H. Liang and B. W. Li, Phys. Rev. B 73, 153303 共2006兲. Downloaded 20 Apr 2007 to 130.20.227.172. Redistribution subject to AIP license or copyright, see http://apl.aip.org/apl/copyright.jsp

thermal conductivity decreases with temperature; this is due to umklapp process 共phonon-phonon interactions兲. The thermal conductivity decreases from 46.67 to 3.11 W / m K as the temperature increases from 600 to 2100 K for the 关001兴oriented GaN nanowires with a diameter of 6.44 nm. At high temperatures, higher frequency acoustic and optical phonon interactions become appreciable, lowering the mean free path and conductivity. The thermal conductivity of the nanowires increases with increasing diameter and is clearly lower than bulk GaN. However, the size dependence is small at low temperatures and disappears at elevated temperatures. The orientational dependence of nanowire thermal conductivity is clearly apparent from Fig. 3. The thermal conductivity of ¯ 0兴- and 关110兴-oriented GaN nanowires is lower than that 关11 of the 关001兴-oriented ones. At 600 K, the calculated thermal conductivity increases from 36.86 to 46.67 W / m K with increasing diameter for the 关001兴-oriented nanowires. For the ¯ 0兴-oriented GaN nanowires, the thermal conductivity is 关11 nearly independent of orientation over most of the temperature range, while for the 关110兴-oriented nanowires, there is a little more dependence on diameter, but this may be due to the large difference in diameters. There are several reasons why nanostructures exhibit lower thermal conductivity than the corresponding bulk materials: 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 diameter of nanowires increases, so does the thermal conductivity, mainly because the boundary scatting rate decreases.33 The phonon-phonon interaction increases with size reduction due to the confinement, which causes the increase of thermal resistance and the decrease of heat conduction.34 The significant decreasing of thermal conductivity is primarily associated with the high surface to volume ratios of the nanowires. Specifically, the relatively large fractions of surface atoms enhance surface scattering of phonons and decrease the phonon mean free path, resulting in lower conductivity that is proportional to the mean free path. As the diameter of the nanowire increases, so does the thermal conductivity, mainly because the boundary scattering rate decreases. In conclusion, we have calculated the thermal conductivity of GaN nanowires using a NEMD technique. We determined the thermal conductivity of GaN nanowires with different diameters at different temperatures and found that the