Thermal transport in carbon nanostructures

THERMAL TRANSPORT IN CARBON NANOSTRUCTURES NI XIAOXI (B.Sc., Nanjing University) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF PHYSICS NATIONAL UNIVERSITY OF SINGAPORE 2011 I would like to dedicate this thesis to my grandfather, QIU Mingsen, for inspiring my enthusiasm for mathematics; to my father, NI Xialin, for instilling my curiosity for science; to my beloved family and friends and to all the people who helped me towards its successful completion. Acknowledgements I would like to express my thanks and appreciation to my supervisor, Prof. Li Baowen, for being so enthusiastic, supportive and helpful over my past years at National University of Singapore. It is because his stimulating suggestions and encouragement that help me in all the time of research and development of this thesis. I would also like to thank to my co-supervisor, Prof. Wang Jian-Sheng, for his advice, guidance and kindness throughout my research work and thesis writing. My colleagues from Centre for Computational Science and Engineering supported me in my research work. I want to thank them for all their help, interesting and valuable hints and comments. I am also indebted to my collaborators, Prof. Liang Gengchiau, Mr. Leek Meng Lee and Prof. Zhang Gang for their assistance and encouragement. Especially, I would like to give my special thanks to my family, for encouraging me and convincing me to believe in myself and for always believing in me even when I not. ii Table of Contents Acknowledgements ii Abstract vi List of Tables viii List of Figures ix Introduction 1.1 1.2 1.3 1.4 Background of Thermal Transport in Nanostructures . . . . . . . . 1.1.1 Effect of Size due to Nanostructures . . . . . . . . . . . . . . 1.1.2 Thermal Transport Properties of Carbon Nanostructures . . Methods of Computing Thermal Transport Properties . . . . . . . . 10 1.2.1 Molecular Dynamics . . . . . . . . . . . . . . . . . . . . . . 11 1.2.2 Nonequilibrium Green’s Function Method . . . . . . . . . . 21 1.2.3 Quantum Molecular Dynamics . . . . . . . . . . . . . . . . . 31 Interesting Effects Related to Thermal Transport . . . . . . . . . . 34 1.3.1 Thermoelectric Effect . . . . . . . . . . . . . . . . . . . . . . 35 1.3.2 Thermal Rectification Effect . . . . . . . . . . . . . . . . . . 40 Objectives of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . 41 iii Applying Quantum Molecular Dynamics (QMD) to Thermal Transport in Carbon Nanostructures 45 2.1 Methodology and Implementation . . . . . . . . . . . . . . . . . . . 46 2.1.1 Overcoming Instability . . . . . . . . . . . . . . . . . . . . . 47 2.1.2 Overcoming Singularities in lead self-energy . . . . . . . . . 49 Results on Graphene and Carbon Nanotubes . . . . . . . . . . . . . 53 2.2.1 Test Runs on Graphene and Comparison with NEGF . . . . 53 2.2.2 Results on Carbon Nanotubes . . . . . . . . . . . . . . . . . 56 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 2.2 2.3 Modifying Thermal Conductivity in Carbon Nanostructures 3.1 3.2 59 Modification by Introducing Disorder . . . . . . . . . . . . . . . . . 62 3.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1.2 Formalism and Model . . . . . . . . . . . . . . . . . . . . . 65 3.1.3 Test Runs and Comparison . . . . . . . . . . . . . . . . . . 71 3.1.4 Effect of Isotope Disorder on One-Dimensional Harmonic Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.1.5 Effect of Isotope Disorder on Carbon Nanotubes . . . . . . . 74 3.1.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 Modification by Introducing Folds: Grafold . . . . . . . . . . . . . . 79 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 3.2.2 Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 3.2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 3.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 iv Thermoelectric and Thermal Rectification Effect in Carbon Nanostructures 88 4.1 Graphane Nanoribbons . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 4.1.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . 92 4.1.3 Thermoelectric Properties of Graphane Nanoribbons . . . . 94 4.1.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.2 Carbon Nanotubes - Graphene Nanoribbons Junction . . . . . . . . 102 4.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 4.2.2 Thermal Transport in Unzipped Carbon Nanotubes . . . . . 104 4.2.3 Thermal Rectification Effect . . . . . . . . . . . . . . . . . . 111 4.2.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Conclusion 114 5.1 Thesis Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.2 Future Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 Appendix: Beyond CPA: Force Constant Disorder 119 List of Publications 127 References 128 v Abstract Graphene has recently become the focus of scientific community due to its unique electronic, phononic and optical properties, and it has great potential to become the mainstream semiconductor material in future devices. The Nobel Prize in Physics for 2010 is awarded to Andre Geim and Konstantin Novoselov “for groundbreaking experiments regarding the two-dimensional material graphene”. This breakthrough has revealed plenty of new physics and potential applications of graphene. Prior to the discovery of graphene, carbon nanotubes are also found to have unusual properties, which are valuable for nanotechnology, electronics, optics and other fields of materials science and technology. In particular, owing to their extraordinary thermal and electrical properties, carbon nanotubes may find applications as building blocks being incorporated into future circuits. Moreover, graphene nanoribbons (GNR), which are patterned as thin strips of graphene (sometimes thought of as unrolled single-walled carbon nanotubes), are also known to display diverse transport properties compared to the infinite sheet, as GNR has electronic properties that range from metallic to semiconducting. This is due to the possibility of manipulating different ribbon width as well as the possibility of controlling vi the atomic configuration at the edges in the GNRs. Furthermore, one obtains bilayer graphene nanoribbons by stacking monolayer graphene nanoribbons, which exhibits quite different properties in terms of energy gaps, electronic conductance, and edge states etc. In this case, the room for manipulation and as a result, obtain diverse properties in carbon derivatives has made carbon nanostructures based nanoelectronics a widely regarded alternative to silicon-based devices for the future. On the other hand, as the size of devices shrinks to the nano-regime, heat dissipation becomes one of the key topics for nanotechnology. At the same time, phononic (thermal) devices have been brought forward theoretically, in which the phonon is used as information carrier. Both topics drive us to further study the thermal transport properties in nanostructures. On the first part of this thesis, it is proposed that classical molecular dynamics along with quantum thermal baths (quantum molecular dynamics) can be implemented to study the thermal transport properties of carbon derivatives. This method is capable of numerically predicting the quantum effect in heat conduction. Followed by the second part, the question of minimizing thermal conductivity in carbon derivatives is addressed to meet the requirement of maximizing electricity conversion efficiency in thermoelectric application. On the third part, interesting topics in thermal transport applications, like thermoelectric and thermal rectification effects are further studied in carbon nanostructures, which opens broader room for applications of carbon derivaties in energy management. vii List of Tables 1.1 Transport regimes for phonons: L is a device characteristic length and O denotes the order-of-magnitude of a length scale; the listed mean free path Λ and coherence length l are typical values but these values are strongly dependent on material type and temperature. . . 1.2 Measured thermal interface resistance r between graphene and SiO2 /Si (300 K). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Measured and calculated thermal conductivity κ of graphene (300 K). 10 viii List of Figures 2.1 The surface density of states Ds (ω) vs. frequency for a (n, 2) zigzag graphene strip with (1,2) of atoms as a repeating unit cell. The delta peaks are located at 566, 734, 1208, 1259, 1287, and 1632 cm−1 . The rest of the peaks not diverge as η → 0. . . . . . . . . 2.2 49 Normalized vibration amplitudes vs. reduced coordinate x of each carbon atom. From (a) to (f) are six edge modes in (10, 2) graphene strip (blue solid line) and (20, 2) graphene strip (red dotted line). The frequency ω for each mode given in the figure is in cm−1 . . . . 2.3 The structure for an armchair graphene strip with (n, m) = (4, 2). The red box is the chosen periodic cell. . . . . . . . . . . . . . . . . 2.4 52 54 A comparison of temperature dependence of thermal conductance for an armchair graphene strip with (n, m) = (4, 2) between, solid line: NEGF, circle: QMD with velocity Verlet, square: QMD with fourth order Runge-Kutta. Insert (a) shows the λ dependence of the same system at 300 K for QMD with velocity Verlet (circle) and NEGF (solid line). . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ix Chapter 5. Conclusion Finally, we write the spectral decomposition of the averaged Green’s function: G (l, l′ ; ω) = N N ′ eik(l−l ) = G k; ω k εα (kj)εβ (kj) G kj ′ eik(l−l ) kj (35) Therefore, f (ω) M = aLl ηl aR l G kj f (ω) M = ηl λ2l = ch η h + ηl λl + G kj − (1−ch ηh ) f (ω) M λ2 = ch η h − ch η h 1− We need to solve ε˜ = G kj + aLl ηl f (ω) M ˜ Σ , d˜ = M ω2 1− λ + f (ω) M ηl† aR l ηl λl + ch η h + f (ω) λ M = d˜ − d˜ G kj − (1−ch ηh ) + 1− f (ω) M G kj − λ λ ch η h ch η h + (1−ch ηh ) λ G kj (36) cd η d separately, and we write: λ f (ω) + 1+ 1− M λ d˜ ω (1 − ε˜) − ωef f (37) ˜ We will use equations of We need independent equations to solve for ε˜ and d. ¯ and F ′ , where motion for the other Green’s functions, a similar treatment to H ′ Fαβ (l, l; ω) = Fαβ (l, l; ω) + M (l) ω δαβ δll′ (38) We get the spectral decomposed versions: ¯ kj H + 1+ i ω Md M hλ = d˜ − d˜ Md M hλ − d˜ + λ λ −1 − d˜ − f (ω) ω (1−˜ ε)−ωef f (39) Taking the long wavelength limit (acoustic mode), ωef f k = 0, j = acoustic = ¯ k = 0, j H + 1+ Md M hλ i ω = d˜ − d˜ − d˜ + λ Md M hλ − d˜ −1 ω (1−˜ ε) λ − f (ω) = (40) ω2 125 Chapter 5. Conclusion d˜ − d˜ + 1+ Md M hλ Md M hλ λ −1 − f (ω) − d˜ + λ − d˜ (41) (1−˜ ε) And for F ′ : + 1+ = d˜ − d˜ F ′ kj M h ω2 Md M hλ − d˜ Md M hλ −1 =1 f (ω) (42) ω (1−˜ ε)−ωef f Taking the long wavelength limit (acoustic mode), ωef f k = 0, j = acoustic = F ′ k = 0, j + 1+ Md M hλ M h ω2 − d˜ d˜ − d˜ + 1+ Md M hλ Md M hλ = d˜ − d˜ ω (1−˜ ε) −1 − d˜ 2 = Md M hλ −1 ch M h +cd M d M h ω2 f (ω) (1−˜ ε) = ch M h +cd M d Mh f (ω) (43) (44) Thus ǫ˜ and d˜ can be solved. Now we know how to determine the averaged Green’s function but the calculation of the transmission function really is an average of a product of two Green’s functions. Therefore we have to further extend the theory in order to include vertex corrections so that the averaged transmission function can be evaluated. This is left as future work. 126 List of Publications [1] X. Ni, G. Liang, J.-S. Wang, and B. Li, “Disorder enhances thermoelectric figure of merit in armchair graphane nanoribbons”, Appl. Phys. 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Thermal Transport Properties of Carbon Nanostructures Carbon Nanotubes In 2000, Berber et al calculated a super high thermal conductivity of 6600 W/mK in an isolated (10, 10) single wall carbon nanotubes (SWCNTs) at room temperature by using classical molecular dynamics (MD) methods [13] This surprising result has motivated continuous interest in understanding thermal transport properties of carbon. .. thermoelectric materials and summarizes the progress in developing thermal rectification models 1.1 Background of Thermal Transport in Nanostructures In general two types of carriers contribute to thermal conductivity - electrons and phonons In nanostructures, phonons usually dominate and the phonon transport properties of such structures are of a particular importance for thermal conductivity As the characteristic... well as electronic engineering The potential of carbon nanodevices is far beyond our imagination and they will significantly 1 Chapter 1 Introduction impact our daily lives Following the prediction of Moore’s law, the size of devices will keep shrinking further into the sub-10 nm range in the near future, which will attract more attention on the topic of thermal transport management in such nano-scale... GNRs [27] 2200 First principle graphene [29] 200 − 5000 First principle graphene [30] phonon Boltzmann equation Table 1.3: Measured and calculated thermal conductivity κ of graphene (300 K) dominates the thermal conductivity at low temperatures The published results of thermal transport in graphene nanostructures are summarized in Table 1.2 and 1.3 1.2 Methods of Computing Thermal Transport Properties... thermal conductance as a function of the number of hexagonal rings m contained in a single supercell (circles) with fitting curve proportional to e−0.178m (dashed line) 4.5 99 Temperature dependence of ZT values for disordered AGANRs: 5 hexagonal rings contained in each supercell with γ = 0.45 (triangles), and 10 hexagonal rings contained in each supercell with γ = 0.30 (squares) ... longitudinal and transverse modes [10] Yet the sufficient condition of Fourier’s law in low dimensional systems is still unknown On the other hand, many researches also explore the thermal transport properties in quasi-1D nanostructures, like nanowires [3] and nanotubes [4] In the following subsections, we briefly review the results for low dimensional carbon nanostructures 6 Chapter 1 Introduction 1.1.2 Thermal. .. nonequilibrium Green’s function (NEGF) The universal thermal conductance for perfect ballistic systems will be discussed in this subsection, and the issue of introducing interactions between phonons will also be addressed NEGF method may be computationally intensive when solving that nonlinear problem, yet in principle, it can give exact results Models In the following subsections, a general junction model will... published results of thermal transport properties of carbon nanostructures are summarized 1.1.1 Effect of Size due to Nanostructures The thermal properties of nanoscale devices are complicated, unlike bulk materials, because of different boundary effects It has been discovered that in many cases, phonon-boundary scattering effects dominate the thermal conduction processes 4 Chapter 1 Introduction due to... represent carbon atoms (b) Except for the edge carbon atoms, the others are bound to the hydrogen atoms in an alternating manner (c) Configuration of a single supercell of disorderd AGANR, in which n = 3, and γ = 0.3 The supercell contains 5 hexagonal rings and corresponds to the square points in Fig 4.2 In all three cases, periodic boundary conditions and the same abinitio methods are adopted 94 xii 4.2 ZT... (squares) hexagonal rings contained in each supercell The optimum ZT values were chosen for each configuration, and each point in the figure corresponds to the mean value of ZT for fixed γ with error bars equal to the deviations in ZT arising from variations in the positions of hydrogen vacancies Insert: Phonon transmission coefficient for perfect AGNR (solid line), perfect AGANR (narrow solid line), and disordered . efficiency in thermoelectric ap- plication. On the third part, interesting topics in thermal transport applications, like thermoelectric and thermal rectification effects are further studied in carbon nanostructures, . Figures ix 1 Introduction 1 1.1 Background of Thermal Transport in Nanostructures . . . . . . . . 3 1.1.1 Effect of Size due to Nanostructures . . . . . . . . . . . . . . 4 1.1.2 Thermal Transport. summarizes the progress in developing thermal rectification models. 1.1 Background of Thermal Transport in Nanos- tructures In general two types of carriers contribute to thermal conductivity -