Tunable thermal conductivity of Si 1 − x Ge x nanowires Jie Chen, Gang Zhang, and Baowen Li Citation: Applied Physics Letters 95, 073117 (2009); doi: 10.1063/1.3212737 View online: http://dx.doi.org/10.1063/1.3212737 View Table of Contents: http://scitation.aip.org/content/aip/journal/apl/95/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Phonon mean free path spectrum and thermal conductivity for Si1−xGex nanowires Appl. Phys. Lett. 104, 233901 (2014); 10.1063/1.4882083 The influence of phonon scatterings on the thermal conductivity of SiGe nanowires Appl. Phys. Lett. 101, 043114 (2012); 10.1063/1.4737909 Phonon coherent resonance and its effect on thermal transport in core-shell nanowires J. Chem. Phys. 135, 104508 (2011); 10.1063/1.3637044 Diameter dependence of SiGe nanowire thermal conductivity Appl. Phys. Lett. 97, 101903 (2010); 10.1063/1.3486171 Thermal conductivity of Si/SiGe superlattice nanowires Appl. Phys. Lett. 83, 3186 (2003); 10.1063/1.1619221 This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 222.66.175.223 On: Sat, 25 Apr 2015 09:07:21 APPLIED PHYSICS LETTERS 95, 073117 共2009兲 Tunable thermal conductivity of Si1−xGex nanowires Jie Chen,1 Gang Zhang,2,a兲 and Baowen Li1,3 1 Department of Physics and Centre for Computational Science and Engineering, National University of Singapore, Singapore 117542, Singapore 2 Institute of Microelectronics, 11 Science Park Road, Singapore Science Park II, 117685 Singapore 3 NUS Graduate School for Integrative Sciences and Engineering, 117456 Singapore 共Received 27 May 2009; accepted 3 August 2009; published online 21 August 2009兲 By using molecular dynamics simulation, we demonstrate that the thermal conductivity of silicon-germanium nanowires 共Si1−xGex NWs兲 depends on the composition remarkably. The thermal conductivity reaches the minimum, which is about 18% of that of pure Si NW, when Ge content is 50%. More interesting, with only 5% Ge atoms 共Si0.95Ge0.05 NW兲, SiNW’s thermal conductivity is reduced to 50%. The reduction of thermal conductivity mainly comes from the localization of phonon modes due to random scattering. Our results demonstrate that Si1−xGex NW might have promising application in thermoelectrics. © 2009 American Institute of Physics. 关DOI: 10.1063/1.3212737兴 Silicon and germanium can form a continuous series of substitutional solid, Si1−xGex. Single crystalline Si1−xGex nanowires 共NWs兲 have been grown and the electronic band gap modulation with composition has been reported.1 Recently, the experimental synthesis of core-shell structures2 provides intriguing opportunities for the development of NW based devices. One of the promising applications for NWs is as thermoelectric cooler.3,4 In thermoelectric application, low thermal conductivity is preferred to increase the figure of merit.5–7 Compared with bulk Si, SiNWs exhibits 100-fold reduction in thermal conductivity because of the strong boundary inelastic scattering of phonons.3,4,8 However, it is still indispensable to reduce the thermal conductivity of NW further in order to achieve higher thermoelectric performance. One possible way to achieve this is compound. In this case, Si1−xGex NW seems to be a promising candidate because both Si and Ge belong to the same group in the periodic table, have the same crystal structure, and display total solubility. In spite of an increasing number of works devoted to the electronic and optical properties,9–11 very little has been done for thermal conductivity of Si1−xGex NW. In this letter, we investigate the thermal conductivity of Si1−xGex NWs with x 共Ge content兲 changing from 0 to 1. The thermal conductivity calculated in this paper is exclusively from the lattice vibration 共phonon thermal conductivity兲, because phonons dominate the heat transport in SiNWs.12 In our simulations, nonequilibrium molecular dynamics 共NEMD兲 method is adapted to calculate the temperature distribution and thermal conductivity. To derive the force term, Stillinger–Weber 共SW兲 potential13,14 is used for Si and Ge. SW potential is widely used in the study of the thermal properties of silicon and germanium.15–18 It is a Keating type19 potential and consists of a two-body term and a three-body term that can stabilize the diamond structure of silicon and germanium. The details of the potential can be found in Ref. 15, and the parameters of Si–Ge interactions are taken to be the arithmetic average of Si and Ge parameters for Si–Ge, and the geometric average for Si–Ge and Si–Ge.20 a兲 Electronic mail: [email protected]. We study the thermal conductivity of NWs along 共100兲 direction with cross section of 3 ⫻ 3 unit cells 共lattice constant is 0.543 nm兲, which corresponds to a cross section area of 2.65 nm2. The atomic structure is initially constructed from diamond structured bulk silicon. Then Si atoms are randomly substituted by Ge atoms in the NW and the geometry is relaxed to its closest minimum total energy. The two ends of NWs are put into heat bathes with temperature TL and TR for the left and right end, respectively. Both Langevin,21 and Nosé–Hoover22,23 heat bathes are used to ensure our results are independent of heat bath. All results given in this letter are obtained by averaging about 1 ⫻ 108 time steps, a time step is set as 0.8 fs. Free boundary condition is used to atoms on the outer surface of the NWs. The thermal conductivity is calculated from the Fourier law, = −JL / ⵜT, where JL is the local heat current along the longitudinal direction, and ⵜT is the temperature gradient. The MD calculated temperature TMD is corrected by taking into account the quantum effects of phonon occupation, using the relation: 3NkBTMD = 兰0DD共兲n共 , T兲បd, where T is the real temperature and TMD is the MD temperature, is the phonon frequency, D共兲 is the density of states, n共 , T兲 is the phonon occupation number given by the Bose–Einstein distribution, and D is the Debye frequency, Debye temperature TD = 645 K for Si.16,17 Correspondingly, according to the Fourier law, the final effective thermal conductivity is rescaled by k = kMD共兩ⵜTMD兩 / 兩ⵜT兩兲 = kMD关TMD / T兴. Using this approach, we perform a quantum correction to temperature and calculate the rescale rate ␣ = TMD / T for SiNWs. When TMD is at room temperature, 300 K, the rescale rate ␣ = 0.91 for silicon, which gives a quite small quantum correction effect on thermal conductivity. Moreover, in our following study, we fix the simulation temperature at room temperature, and mainly focus on the compositional dependence of thermal conductivity. Therefore, in the following part, we do not do the quantum correction to MD temperature and thermal conductivity. We study the thermal conductivity of Si1−xGex NWs with Ge atoms randomly distributed, 0 ⱕ x ⱕ 1. The NW we studied has a length of 20 unit cells, which corresponds to 10.86 nm. For each Ge content x, in order to reduce the fluctuation, 0003-6951/2009/95共7兲/073117/3/$25.00 95,is073117-1 © 2009 American InstituteDownloaded of Physics to IP: This article is copyrighted as indicated in the article. Reuse of AIP content subject to the terms at: http://scitation.aip.org/termsconditions. 222.66.175.223 On: Sat, 25 Apr 2015 09:07:21 073117-2 Appl. Phys. Lett. 95, 073117 共2009兲 Chen, Zhang, and Li FIG. 2. The phonon participation ratio vs Ge content x for Si1−xGex NWs with 0 ⱕ x ⱕ 1. FIG. 1. 共Color online兲 The thermal conductivity vs x at T = 300 K. The inset results are from Refs. 24 and 25. the localization with O共1兲 for delocalized states and O共1 / N兲 for localized states. P is given by, P−1 = N兺i共兺␣iⴱ␣i␣兲2, where i␣ is the vibrational eigenvector component. We show the results are averaged over 20 realizations. In Fig. 1 we the participation ratio versus Ge content x in Fig. 2. For both plot / 0 versus Ge content x at room temperature. Here is pure Si NW and pure Ge NW, a high participation ratio the thermal conductivity of Si1−xGex NWs, and 0 is the appears, indicating the delocalized characteristics of corresponding thermal conductivity of pure Si NW. The therphonon modes and corresponds to high thermal conductivity. mal conductivity of pure SiNW calculated with Nosé– However, in the Si1−xGex NWs 共0 ⬍ x ⬍ 1兲, with the impurity Hoover heat bath is 2.43 W/m K, and it is 3.18 W/m K with concentration increasing, the participation ratio decreases Langevin heat bath. The lowest is only 18% 共Langevin兲 significantly, indicates strong localization due to impurity and 15% 共Nosé–Hoover兲 of that of pure SiNW calculated scattering and corresponds to low thermal conductivity. The with the same heat bath. We also did the calculation with Ge content dependent participation ratio is consistent with NW of ten unit cells in the longitudinal direction. The lowest the changes in thermal conductivity. thermal conductivity is 17% of that pure Si NW, which demWe now turn to the reduction of thermal conductivity by onstrates that the composition dependence of thermal conusing superlattice 共SL兲 structure. Some experimental and ductivity is a general characteristic for Si1−xGex NWs. It is theoretical works30–32 have been carried out to study the efquite remarkable that with only 5% Ge atoms 共Si0.95Ge0.05 fects of interface and SL period on thermal conductivity of NW兲, its thermal conductivity can be reduced 50%. The best various kinds of SL structures. Here we study the thermal fitting gives rise to = A1e−x/B1 + A2e−共1−x兲/B2 + C, where A1, conductivity of SL structured Si/Ge NWs. In our simulations, B1, A2, B2, and C are fitting parameters. The decaying rates the SL NWs consist of alternating Si and Ge layers with B1 and B2 by Nosé–Hoover heat bath coincide with those by changeable period length in the longitudinal direction. It has Langevin methods 共as shown in Table I兲 indicating that the a fixed cross section of 3 ⫻ 3 unit cells and a fixed length of low thermal conductivity observed in Si1−xGex NWs is indeten unit cells in the longitudinal direction. In the Si/Ge SL pendent of the heat path used. structured NWs, the NWs and heat bathes may have three We also show the simulation24 and experimental25 results different contacts: both heat bathes are chosen to be Si; both for bulk Si1−xGex alloy in Fig. 1. Although the thermal conare Ge; and one is Si, the other one is Ge, as the same ductivity of NW is about two orders of magnitude smaller material adjacent to the heat bath, respectively. Figure 3共a兲 than that of bulk material, the dependence of on the Ge shows the thermal conductivity of the SL NWs versus the atom content is similar. period length for these three contacts. Langevin heat bath is There is significant difference between nano and bulk used here. The thermal conductivity calculated from different material in the thermal property. In nanoscale system such as NW-heat bath contacts has a good agreement with each carbon nanotube, anomalous thermal conductivity has been other. Therefore, the thermal conductivity of SL structured observed both numerically26,27 and experimentailly.28 In orSi/Ge NWs has weak dependence on the detailed heat bath der to study the physical mechanism of the reduction of thercontact. As shown in Fig. 3共a兲, decreases monotonically mal conductivity, we have studied the participation ratio P,29 with the period length decreasing from 4.43 nm 共32 layers兲, which is an important measure for the fraction of phonons until period length reaches a critical value of 1.11 nm 共eight participating in thermal transport, and effectively indicates layers兲. At this critical period length, the thermal conductivity is only one sixth of that of pure SiNWs. This reduction in TABLE I. The fitting parameters for the best fitting thermal conductivity thermal conductivity is due to the fact that when decreasing formula: = A1e−x/B1 + A2e−共1−x兲/B2 + C, for both Langevin and Nosé–Hoover heat baths. the period length of SL structured NWs with a fixed total length, the increasing number of interface will lead to an B1 A2 B2 C A1 enhanced interface scattering, which is responsible for the reduction in . As period length decreases further from the Langevin heat bath 2.50 0.056 1.47 0.065 0.65 critical value, there exists a rapid increase in . At room Nosé–Hoover heat bath 2.01 0.063 1.23 0.066 0.40 temperature, dominant phonon wavelength inDownloaded SiNW is to IP: This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms the at: http://scitation.aip.org/termsconditions. 222.66.175.223 On: Sat, 25 Apr 2015 09:07:21 073117-3 Appl. Phys. Lett. 95, 073117 共2009兲 Chen, Zhang, and Li have also investigated the thermal conductivity of SL structured Si/Ge NWs. The dependence of thermal conductivity on the period length is explained by the overlap of phonon power spectrum of different layers. The low thermal conductivity of Si1−xGex NW significantly enhances its figure of merit ZT and has raised the exciting prospect for application in on-chip thermoelectric cooler. J.C. would like to thank Nuo Yang, Donglai Yao, and Lifa Zhang for fruitful discussions. This work is supported in part by an ARF Grant No. R-144-000-203-112 from the Ministry of Education of the Republic of Singapore and Grant No. R-144-000-222-646 from National University of Singapore. J.-E. Yang, C.-B. Jin, C.-J. Kim, and M.-H. Jo, Nano Lett. 6, 2679 共2006兲. L. J. Lauhon, M. S. Gudiksen, D. Wang, and C. M. Lieber, Nature 共London兲 420, 57 共2002兲. 3 A. I. Hochbaum, R. Chen, R. D. Delgado, W. Liang, E. C. Garnett, M. Najarian, A. Majumdar, and P. 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Here the atoms in heat baths are: both heat baths are Si atoms 共black squares兲; both are Ge 共red circles兲; and atom in one heat bath is Si, in the other one is Ge, as the same material adjacent to the heat bath 共green triangles兲. 共b兲 The normalized power spectrum of phonons of different atoms 共Si or Ge兲 along the longitudinal direction with period length is 1.11 nm 共eight layers兲. 共c兲 The normalized power spectrum with period length is 0.28 nm 共two layers兲. 共d兲 Overlap ratio S vs period length. about 1–2 nm,33 which is quite close to the critical value of 1.11 nm in the present paper. When period length is smaller than the dominant phonon wavelength, ballistic transport will replace diffusive mode and dominates the transport characteristic, which gives rise to a rapid increase in thermal conductivity.15,34 In order to get a better understanding of the underlying mechanism of the period length dependence of thermal conductivity, we calculate the power spectrum of both Si and Ge layers of SL structured NWs with different period length. Figure 3共b兲 and 3共c兲 show the normalized power spectrum in two typical cases: SL structured NWs with period length of 1.11 nm 共eight layers兲; and SL structured NWs with period length of 0.28 nm 共two layers兲. It is clear that there exists a larger overlap of power spectrum in the case which has a larger thermal conductivity. It is well understood that in low dimensional systems, a large overlap of power spectrum means that heat flow can easily go through the system and, therefore, results in a high thermal conductivity.35,36 In order to quantify the above power spectrum analysis, the overlap 共S兲 of the power spectra are calculated as36 S= 兰⬁0 PSi共兲PGe共兲d . ⬁ 兰0 PSi共兲d兰⬁0 PGe共兲d 共1兲 In Fig. 3共d兲 we plot the overlap ratio S versus period length. A comparison between Figs. 3共a兲 and 3共d兲 demonstrates that a larger overlap S corresponds to a higher thermal conductivity. To summarize, we have investigated the composition dependence of thermal conductivity of Si1−xGex NWs with x changing from 0 to 1. A remarkable composition effect on thermal conductivity is observed. With only 5% Ge atoms 共Si0.95Ge0.05 NW兲, its thermal conductivity can be reduced to 50%. This composition dependence of thermal conductivity is explained by phonon participation ratio. In addition, we This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to IP: 222.66.175.223 On: Sat, 25 Apr 2015 09:07:21
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