THERMAL TRANSPORT IN LOW DIMENSIONAL GRADED STRUCTURES AND SILICON NANOWIRES YANG NUO NATIONAL UNIVERSITY OF SINGAPORE 2010 THERMAL TRANSPORT IN LOW DIMENSIONAL GRADED STRUCTURES AND SILICON NANOWIRES YANG NUO B.S. (UNIVERSITY OF SCIENCE AND TECHNOLOGY OF CHINA) 2000 M.ENG. (CHINESE ACADEMY OF SCIENCE) 2003 A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2010 ii c ⃝ Copyright by YANG NUO 2010 All Rights Reserved i Acknowledgements I was extremely fortunate for the opportunity to work with, learn from, and establish friendships with some of the finest people during my time in Singapore.First and foremost, I would like to thank my advisor, Professor LI Baowen for his support, energy, encouragement, and insightful advice over the past years. There would be no this research work without his far-sight and guidance. I would also like to thank my collaborators, Prof. ZHANG Gang, Prof. WANG Lei and Dr. LI Nianbei for their sagacity and hard working. Additionally, I am appreciative of the colleagues in our group, Prof. WANG Wenge, Dr. YAN Yonghong, Dr. LAN Jinghua, Dr. Tang Yunfei, Mr. LO Wei-Chung and Mr. ZHANG Lifa for their valuable suggestions and comments. I am also grateful to all my friends in Singapore.A partial list includes: LI Pinghui, DAI Liang, LUO Jie, LI Zhipeng, SUI Yi, CHAI Bo, BAI Huixing, ZHAO Wei, LI Yangfan, LIU Furong, CHEN Xiaobing, ZHENG Jianguo, and ZHOU Lihong. I really enjoy the frisbee games under the sunset at West Coast Park. Finally, I would like to thank my parents and my parents-in-law. I am forever indebted for their love, support, and encouragement. Also, I am greatly appreciative of my dear wife Ying’s support and never-ending patience with me. Abstract ii Abstract Very recently, phononic (thermal) devices have been brought forward theoretically, in which the phonon is used as information carrier. It drives us to search materials fit for thermal devices, such as thermal diodes and thermal transistors. On the first part of this thesis, it is proposed that low dimensional graded materials are good candidates for thermal rectifier. The heat flux in the one dimensional harmonic/anharmonic chain with a mass gradient and the carbon nanocone were studied by using classical non-equilibrium molecular dynamics simulation. It was found that the heat flowed with asymmetric in anharmonic lattices with a mass gradient. Moreover, in a certain temperature region, negative differential thermal resistance was observed. It was also demonstrated that the structural asymmetry in carbon nanocone benefited the rectification ratio remarkably. It was found that there was a larger heat flux in the direction of decreasing diameter and the rectification in carbon nanocone was size independent. Possible applications in constructing thermal rectifiers and thermal transistors by using the graded material were discussed. The silicon nanowire (SiNW) has been shown to be an efficient thermoelectric material. The thermal conductivity of SiNW is crucial in thermoelectric applications. On the second part of this thesis, using classical nonequilibrium molecular dynamics simulation, it was studied that the reduction of the thermal conductivity of SiNWs with two isotope-doping methods: doping nanowires with isotope impurities randomly and isotopic-superlattice nanowires. It was shown that these two methods led to a large scale decrease of thermal conductivity of SiNWs. The thermal conductivity of isotopicsuperlattice structured SiNWs depended clearly on the period length of super- Abstract iii lattice. The mass effect on thermal conductivity was obvious. The heavier isotope atoms (42 Si) could decrease the conductivity much more than the lighter ones (29 Si). The remarkable isotopic effect observed in this work provides an efficient approach to decrease thermal conductivity of SiNW, which could be of great benefit to improve the thermoelectric performance. These improvements have raised the exciting prospect that SiNWs can be applied as novel nano-scale thermoelectric materials. It was also studied that the size effect on the thermal conductivity of nanowire structures. It was demonstrated that the thermal conductivity of SiNWs diverged with the longitudinal length, even when the sample length was much longer than the phonon mean free path at the room temperature, which meant Fourier’s empirical law was broken. The effect of fixed boundary on heat transport in SiNW was researched. It was reported that there was obvious difference between the heat flux of atoms close to boundary and the flux of atoms at the center of cross section. Contents Acknowledgements i Abstract ii Contents iv List of Tables viii List of Figures ix Introduction 1.1 Thermal Transport in Low-Dimensional Systems . . . . . . . . . 1.1.1 Thermal Transport in 1D Chains . . . . . . . . . . . . . 1.1.2 Thermal Transport in Quasi-1D Nano-Structures . . . . iv CONTENTS 1.1.3 1.2 1.2.2 1.4 1D Thermal Management Device Models . . . . . . . . Potential of Using Silicon Nanowires in Thermoelectrics . . . . . 10 1.2.1 1.3 v Thermoelectric Effects . . . . . . . . . . . . . . . . . . 10 Low Dimensional Thermoelectric Materials . . . . . . . 12 Molecular Dynamics Simulation . . . . . . . . . . . . . . . . . . 15 1.3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.3.2 Thermodynamic Properties . . . . . . . . . . . . . . . . 19 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 Thermal Rectification Effects in Low Dimensional Graded Structures 2.1 27 Thermal Rectification in 1D Mass-Graded Lattice . . . . . . . . 28 2.1.1 Simulation Method . . . . . . . . . . . . . . . . . . . . . 29 2.1.2 Abnormal Thermal Conductivity in Graded Harmonic Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.1.3 Abnormal Thermal Conductivity Graded Anharmonic Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1.4 Thermal Rectification in Graded Anharmonic Lattice . . 35 CONTENTS vi 2.1.5 Negative Differential Thermal Resistance . . . . . . . . . 38 2.1.6 Rectification Mechanism in 1D Graded Chain . . . . . . 38 2.2 2.3 Thermal Rectification in Carbon Nanocone . . . . . . . . . . . . 41 2.2.1 Simulation Method . . . . . . . . . . . . . . . . . . . . . 43 2.2.2 Thermal Rectification in Carbon Nanocone . . . . . . . . 45 2.2.3 Heat Bath, Temperature and Size Effect and NDTR . . . 49 2.2.4 Rectification Mechanism in Carbon Nanocone . . . . . . 53 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 Thermal Conductivity of Silicon Nanowires 58 3.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 3.2 Thermal Conductivity of Isotope Doped Silicon Nanowires . . . 61 3.2.1 Modeling Method . . . . . . . . . . . . . . . . . . . . . . 61 3.2.2 Thermal Conductivity of Random Doping Silicon Nanowires 63 3.2.3 Thermal Conductivity of Superlattice Structure Silicon Nanowires . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.3 Size Effect on Thermal Conductivity of Silicon Nanowires . . . . 75 CONTENTS vii 3.3.1 Modeling Method . . . . . . . . . . . . . . . . . . . . . . 76 3.3.2 Dependence of Thermal Conductivity on Longitudinal Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 3.4 Boundary Effect on Thermal Conductivity . . . . . . . . . . . . 83 3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 Conclusions and Future Works Publication list 89 111 BIBLIOGRAPHY 97 [36] P. 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Nuo Yang, Gang Zhang, and Baowen Li, ”Ultra-low Thermal Conductivity of Isotope-Doped Silicon Nanowires”, Nano Lett. 8, 276 (2008). (This paper has been highlighted by Nature Asia-Pacific.) 3. Nuo Yang, Gang Zhang, and Baowen Li, ”Carbon Nanocone - A Practical Thermal Rectifier”, Appl. Phys. Lett. 93, 243111 (2009). (This paper has been selected for the Jan. 5, 2009 issue of Virtual Journal of Nanoscale Science & Technology.) 4. Nuo Yang, Gang Zhang, and Baowen Li, ”Thermal Rectification In Asymmetric Graphene Ribbons”, Appl. Phys. Lett. 95, 033107 (2009). 5. Nuo Yang, Gang Zhang, and Baowen Li, ”The Length Dependence of Thermal Conductivity of Silicon Nanowires”, Nano Today 5, 85 (R) (2010). Conferences Oral Presentations 1. ”Ultra Low Thermal Conductivity of Isotope Doped and Superlattice Structured Silicon Nanowire” and ”A Solid-State Thermal Rectifier from Carbon Nanocone”. The 2008 Asian Conference on Nano-science and Nano-technology (AsiaNANO2008), Singapore, Nov,2008. 2. ”Reduce Thermal Conductivity of Silicon Nanowires by Doping Isotopic Atoms” The 9th Frontier Science Symposium, Singapore, Oct 2008. BIBLIOGRAPHY 112 3. ”Ultra-low Thermal Conductivity of Isotope-Doped Silicon Nanowires”. The 5th Dynamics Days Asia Pacific (DDAP5), Nara, Japan, Sep, 2008. 4. ”Carbon Nano-cone: An Excellent Thermal Rectifier”. Transmission of Information and Energy in Nonlinear and Complex Systems (TIENCS), Singapore, June, 2008 5. ”A Thermal Rectifier from Cone-shaped Carbon Nanotube”. International Conference on Materials for Advanced Technologies & International Union of materials Research Societies International Conference in Asia 2009 (ICMAT & IUMRS-ICA 2009), Singapore, Jul, 2009. 6. ”The Size Effect on the Thermal Conductivity of Silicon Nanowires”. International Conference on Materials for Advanced Technologies & International Union of materials Research Societies International Conference in Asia 2009 (ICMAT & IUMRS-ICA 2009), Singapore, Jul, 2009. [...]... nanometers In the following search, the ther- 1.1 Thermal Transport in Low- Dimensional Systems 3 mal transport in bulk and nano -structures are introduced firstly Secondly, the application of nano-material in thermoelectrics is shown At the end of this chapter, molecular dynamics simulation methods and research objectives are presented 1.1 1.1.1 Thermal Transport in Low- Dimensional Systems Thermal Transport in. .. at low temperature and weak coupling in 1D lattice and β was 1/3 when there was coupling between longitudinal and transverse modes [16] The sufficient condition for keeping Fourier’s law in low dimensional system is still an open question Generally, studies have been focusing on disorders, chaos and the breaking of momentum conservation Zhao et al reported that the boundary condition was dominant in. .. harmonic chain, which means there is no energy diffusion and mode coupling 1.1 Thermal Transport in Low- Dimensional Systems 4 In the last few decades, there are many studies on the heat conduction in 1D lattices It was found that there was an abnormal conduction in a chain with nonlinear potential and the Fourier’s law was violated [5–14, 16–18] It was found that thermal conductivity of FPU-like chains diverged... of phonons in a solid than it is to control the flow of electrons In recent years more attention has been directed toward the phonon management on energy transport in dynamical systems and the emerging field is described as phononics Thermal diodes, thermal transistors and thermal logic gates, which are the basic 1.1 Thermal Transport in Low- Dimensional Systems 9 components of functional thermal devices,... for thermal energy transport which would reduce the efficiency of the rectifier Besides thermal diode, there are also some reports on other thermal devices In 2006, Li, Wang and Casati found the negative differential thermal resistance in nonlinear lattices, which provided the possibility of building the thermal transistor by a three-segment FK device [29] In the following year, Wang and Li realized a thermal. .. size independent in a system without the breaking of momentum conservation, the 1D coupled rotor model [22, 23] On the other hand, it was proposed that the momentum conservation was the sufficient condition for divergent thermal conductivity [9] 1.1.2 Thermal Transport in Quasi-1D Nano -Structures Recently, many researches focus on the thermal transport in quasi-1D materials, like silicon nanowires and. .. There was much debating on searching for a universal exponent value of β In 2002, Narayan and Ramaswamy asserted β should be 1/3 in 1D momentum-conserving systems by a renormalization group approach of the hydrodynamic equation of heat transport in a liquid [13] In the following year, using mode-coupling theory, Levi, Livi and Politi derived the universal exponent β = 2/5 [12, 14] Wang and Li found that... regime and anomalous conductivity in regular regime [7] However, Lepri et al found the anomalous 1.1 Thermal Transport in Low- Dimensional Systems 5 thermal conductivity in Fermi-Pasta-Ulam (FPU) -β chain, which had positive Lyapunov exponent and was chaotic [8] Later, Li et al showed that chaos was not even a necessary condition for normal thermal conductivity by finding finite thermal conductivity in 1D... new thermal properties in nano-materials are still not clearly understood In 2000, Berber et al predicted the super high thermal conductivity of an isolated (10, 10) single wall carbon nanotubes (SWCNT), 6600 W/mK at room temperature, by using classical molecular dynamics (MD) methods [34] This is a significant result, which motivated huge interest on researching thermal 1.1 Thermal Transport in Low- Dimensional. .. MD simulations [43] They solved the Boltzmann transport equation to explain the possibility of diffuse boundary scattering causing the thermal conductivity drop, which was observed in the MD simulation A few years later, Li et al showed experimentally the ultra -low thermal conductivity of single SiNW [4] The ultra -low thermal conductivity of SiNW could be mainly caused by two factors First, the confinement . THERMAL TRANSPORT IN LOW DIMENSIONAL GRADED STRUCTURES AND SILICON NANOWIRES YANG NUO NATIONAL UNIVERSITY OF SINGAPORE 2010 THERMAL TRANSPORT IN LOW DIMENSIONAL GRADED STRUCTURES AND SILICON. Introduction 1 1.1 Thermal Transport in Low- Dimensional Systems . . . . . . . . . 3 1.1.1 Thermal Transport in 1D Chains . . . . . . . . . . . . . 3 1.1.2 Thermal Transport in Quasi-1D Nano -Structures. the length scale of nanometers. In the following search, the ther- 1.1. Thermal Transport in Low- Dimensional Systems 3 mal transport in bulk and nano -structures are introduced firstly. Secondly,