Thermal Transport in Mesoscopic Dielectric Systems

Ping Yang Centre for the Physics of Materials Department of Physics McGill University Montréal, Québec Canada

A Thesis submitted to the Faculty of Graduate Studies and Research in partial fulfillment of the requirements for the degree of Doctor of Philosophy

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Contents

Abstract xii Résumé xiv Statemnet of Originality xvi Acknowledgments xvii 1 Introduction 1 1.1 Theoretical approaches to thermal transport 6 1.1.1 Boltzmann transport equation . 7 1.1.2 Molecular dynamics...... 10 1.1.3 Transmission function ...... 14 1.2 Thermal transport experiments at the mesoscopic scale 17 1.3 Two probe formula of thermal transport 22 1.4 Summary of this thesis ...... 26 2 Theoretical Formalism 28 2.1 Introduction...... 28 2.2 Phonon scattering for thermal transport 29 2.2.1 Harmonie crystal ...... 30 2.2.2 Anharmonic crystal ...... 32 2.3 Anharmonic one-dimension al chain . 35 2.4 Heat flux in a multi-terminal system 38 2.4.1 Classical equation of motion . 39 2.4.2 Phonon subbands ...... 41 2.4.3 Displacement field in terminal wires . 43 2.4.4 A useful expression for transmission amplitudes 45 2.4.5 Quantization of energy flux ...... 50 2.4.6 Universal quantum of thermal conductance .. 53 2.5 Summary ...... 56 3 Thermal Transport in a T-Shaped Quantum Wire 58 3.1 Introduction...... 58 3.2 Model and formulation ...... 60 3.3 Features of transmission coefficients: w -t 0 limit 65 3.4 Features of transmission coefficients: periodicity 74 3.5 Thermal conductance . 79 3.6 Summary ...... 85

v Contents vi

4 Thermal Transport in a Four-Terminal Deviee 86 4.1 Introduction...... 86 4.2 Four-terminal thermal conductance ...... 87 4.2.1 Thermal conductance measurement among multi-reservoirs 90 4.3 An example: four-terminal mesoscopic dielectric system .. . 92 4.3.1 Transmission coefficient ...... 98 4.3.2 Thermal conductance in two-terminal measurements 104 4.3.3 Thermal conductance in four-terminal measurements 106 4.4 Summary ...... 109 5 Generalized Discussion for Thermal Conductance 111 5.1 Introduction ...... 111 5.2 Thermal transport in ID with FES .. 113 5.3 Thermal transport in 2D with FES . . 115 5.3.1 Two-terminal device with FES . 115 5.3.2 Four-terminal device with FES 119 5.4 Summary 121 6 Conclusion 122 Appendices 125 A.1 A relation for transmission amplitudes 125 A.2 Energy current formula ...... 131 A.3 Universal quantum of thermal conductance . . 139 References 140 List of Figures

1.1 Schematic diagram of an initial configuration to be simulated by MD. The left and right reservoirs represented by atoms EB and 0 are at temperatures Tl and T2 , respectively...... 14 1.2 Conceptual picture of an electron heating experiment. Joule heat Qin is put into the electrons. Part of heat is transferred to the phonons by the electron-phonon interaction Qep. This heats up the phonons which are then cooled by phonon diffusion Qph and by returning sorne of the heat to the electrons, Qpe. Heat also flows to the contacts via electron diffusion Qel. Plot derives from Ref.[23]...... 18 1.3 A suspended and mono-crystalline device which can directly measure thermal conductance on nanostructures. Left figure, a device consists of i+GaAs plate used as a quasi-isolated phonon cavity (3 x 3/-lm x 300 nm thickness), and four 5.5 /-lm long i-GaAs bridges (cross section 200 nm x 300 nm). The bridges suspend the cavity above the substrate by 1 /-lm. An integral pair of n+GaAs resistive transducers (120 nm line-width, 150 nm thickness) meander above the underlying GaAS cavity. Right figure, schematic diagram of principal components of the suspended device: isolated cavity, suspended bridges, and the supports with external reservoir, which fix the sample to the substrate. Figure derives from Ref.[U]...... 20 1.4 The measured thermal conductance data. The measured thermal con ductance is normalized by the expected low-temperature value for 16 occupied modes, 16 Go. Measurement error is approximately the size of the data points, except where indicated. For temperatures above Tco ~ 0.8K, A cubic power-Iaw behavior consistent with a mean free path of rv 0.9 /-lm. For temperatures below Tco , a saturation in G at a value near the expected quantum of thermal conductance is observed. Consistent with expectations, to within experimental uncertainty, G never exceeds 16Go once it finally drops below that value. Plot derives from Ref.[U]...... 21

vii List of Figures viii

2.1 Temperature distribution along an anharmonic chain consisting of 60 carbon atoms is obtained by classical molecular dynamics simulation. A temperature bias !:1T is applied at the two ends of the chain. (1) !:1T = 65-55(K); (2)!:1T = 120-100(K); (3)!:1T = 170-150(K); (4) !:1T = 210 -190(K); (5) !:1T = 260 - 240(K); (6) !:1T = 320 - 300(K). The black circle represents the reservoir where !:1T is applied. The empty circles are the atoms...... 36 2.2 Temperature dependence of the thermal conductivity /'f, for the same chain in Fig.2.1 for temperature below 320K. The inset is the experimental results from 50K to 400K...... 36 2.3 Temperature distribution along the same chain in Fig.2.2 but at very low temperatures. (l)!:1T = 1.5 - 0.5(K); (2) !:1T = 2.5 -1.5(K); (3) !:1T = 3.5 - 2.5(K); (4) !:1T = 4.5 - 3.5(K)...... 37 2.4 Schematic diagram of a general multi-terminal device. Each terminal wire is connected to a reservoir far away with different temperature T. The terminal wires are perfect crystals without any scattering. Phonon scattering occurs inside the center scattering region as weIl as at the contacts between the scattering region and the terminal wires. A local coordinate system on each terminal wire is set up for mathematical analysis...... 40 2.5 Schematic diagram of the incident wave, the reflected wave, and the transmitted wave in any terminal of a multi-terminal device. n, n', nif, and nlll are the mode indices. k is the wave vector for different modes. 44 3.1 Schematic diagram for the T shaped wire ...... 62 3.2 Transmission coefficient Tm(w) versus the incident phonon frequency w for the side-stub length L = 20nm. Wm is the mth mode cutoff fre quency. The horizontal axis is scaled by !:1w =W m+1 - Wm = 1raV where the velo city is fixed at v = 5000m/ sec. The geometrical parameters are: a = lOnm, b = lOnm. (a) the O-th mode case, e.g., the massless mode. (b) the lst mode case. (c) the 2nd mode case. (d) the 3rd mode case...... 66 3.3 Transmission coefficient Tm(w) versus the incident phonon frequency w for the side-stub length L = 25nm. Other parameters are the same as in Fig.3.2 ...... 66 3.4 Transmission coefficient Tm(w) versus the incident phonon frequency w for the side-stub length L = 28nm. Other parameters are the same as in Fig.3.2 ...... 67 3.5 Transmission coefficient Tm(w) versus the incident phonon frequency w for the side-stu b length L = 30nm. Other parameters are the same as in Fig.3.2 ...... 67 List of Figures ix

3.6 Transmission coefficient Tm(w) versus the incident phonon frequency w for the side-stub length L = 35nm. Other parameters are the same as in Fig.3.2 ...... 68 3.7 Transmission coefficient Tm (w) versus the incident phonon frequency w for the side-stub length L = 40nm. Other parameters are the same as in Fig.3.2 ...... 68 3.8 Transmission coefficient Tm(w) versus the incident phonon frequency w for the side-stub length L = 43nm. Other parameters are the same as in Fig.3.2 ...... 69 3.9 Transmission coefficient Tm(w) versus the incident phonon frequency w for the side-stub length L = 45nm. Other parameters are the same as in Fig.3.2 ...... 69 3.10 Schematic of the subband threshold energy (cutoff frequency) for the leads and the stub region with different ratios: (a) ~ = 2; (b) ~ = 2.5; (c) ~ = 2.8. The left column is the subband energies of the left lead; the middle column is for the stub-region; and the right column is for the right lead. The T -shaped wire is also shown to remind the geometry. 72 3.11 Transmission coefficient Ta versus the incident phonon frequency w with different stub lengths L. The geometrical parameters are: a = lOnm, b = lOnm.(a) L = 20nm, (b) L = 25nm, (c) L = 28nm, (d) L = 50nm...... 75 3.12 Mode matching profile of the lead and the stub at the stub-Iead inter face. w is the incoming frequency and k is the wave vector. The length ratio ~ = 3...... 76 3.13 Transmission coefficient Ta(w) and its components, Tao, 'li 0,120 ,and 730 versus the reduced frequency w j tlw. tlw = Wm+1 - Wm = V;. Here, v = 5000mj sec, a = lOnm, b = lOnm, and L = 30nm. Solid line: total transmission coefficient Ta(w). Dashed Hne: Tao. For wj tlw < 1, only Tao has contribution to the total transmission Ta, therefore the dashed Hne coin cid es with the solid Hne. Dotted Hne: 'lio; long dashed Hne: 120; and dash-dot Hne: 730' ...... 77 3.14 (a) Tao versus wj tlw. Tao is finite for any w. (b) 'lio versus wjtlw. 'lio becomes nonzero when w j tlw = 1. (c) 120 versus wj tlw. 120 becomes nonzero wh en wjtlw = 2. (d) 730 versus wjtlw. 730 becomes nonzero when wj tlw = 3. Other parameters are the same as in Fig.3.13. . .. 77 3.15 Transmission coefficient Ta(w) versus stub length L with fixed incident frequency. (a) w = 2Œ (b) w = 27TV (c) W = 37TV (d) w = 47TV. 5a ' 5a ' 5a ' 5a a = b = lOnm, Lo = lOnm, and v = 5000mj sec...... 78 3.16 Transmission coefficient Ta(w) versus stub length L with fixed incident frequency. (a) w 67TV (b) w ill (c) w = 87TV (d) w = 97TV. = 5a ' = 5a ' Sa ' Sa a = b = lOnm, Lo = lOnm, and v = 5000mj sec...... 78 List of Figures x

3.17 Thermal conductance G versus temperature T. Here a = lOnm, b = lOnm. Solid line: L = 30nm; dotted line: L = 50nm; dashed line: L = 70nm; long dashed line: L = 100nm; dot-dashed line: L = 120nm. Inset: thermal conductance G versus temperature for T < 1K. 80 3.18 Thermal conductance G/T versus temperature T. Here a = lOnm, b = lOnm. The solid line: L = 30nm; dotted line: L = 50nm; dashed line: L = 70nm; long dashed line: L = 100nm; dot-dashed line: L = 120nm...... 81 3.19 (a) Thermal conductance Gand (b) G/T versus temperature T with stub length L = 30nm and width b = lOnm, for different wire widths, a = 5nm, a = lOnm, and a = 20nm...... 83 3.20 (a) Thermal conductance Gand (b) G/T versus temperature T with stub length L = 100nm and width b = lOnm, for different wire widths, a = 5nm, a = lOnm, and a = 20nm...... 83 3.21 Thermal conductance G versus the stub length L with different b = lOnm, 20nm, and 50nm. The wire width is a = lOnm. Temperature is (a) 1K, (b) 2K, and (c) 4K...... 84 4.1 Schematic diagram for the four-terminal system with an arbitrary scattering region...... 88 4.2 Schematic view of a 2D four-terminal device where region V is the scat tering region. The terminal wires have different widths as indicated, the system has no obvious geometric symmetry. The incoming phonon waves is in terminal wire 1 with width al. Horizontal direction is along the x-axis, the vertical direction is y...... 93 4.3 Transmission coefficient 7j1,OO (a) and 7j1 (b) versus incident phonon frequency w. The parameters: al = a2 = a3 = a4 = 3Onm, L = 100nm, b = 20nm, and v = 5000m/ sec...... " 100 4.4 Transmission coefficient 7j1,OO (a) and 7j1 (b) versus incident phonon frequency w. The parameters: al = a2 = a3 = a4 = 30nm, L = 100nm, b = 30nm, and v = 5000m/ sec...... 100 4.5 Transmission coefficient 7j1,OO (a) and 7j1 (b) versus incident phonon frequency w. The parameters: al = a2 = a3 = a4 = 30nm, L = 100nm, b = 40nm, and v = 5000m/ sec...... " 101 4.6 Transmission coefficient 7j1,OO (a) and 7j1 (b) versus incident phonon frequency w. The parameters: al = a2 = a3 = a4 = 3Onm, L = 100nm, b = 50nm, and v = 5000m/ sec...... " 101 4.7 Transmission coefficient 7j1,OO (a) and 7j1 (b) versus incident phonon frequency w. The parameters: al = 40nm, a2 = 20nm,a3 = 50nm, a4 = 30nm, L = 100nm, b = 20nm, and v = 5000m/ sec...... 102 List of Figures xi

4.8 Transmission coefficient TjI,OO (a) and Tjl (b) versus incident phonon frequency w. The parameters: al = 40nm, a2 = 20nm, a3 = 50nm, a4 = 30nm, L = 100nm, b = 30nm, and v = 5000m/ sec...... 102 4.9 Transmission coefficient TjI,OO (a) and Tjl (b) versus incident phonon frequency w. The parameters: al = 40nm, a2 = 20nm, a3 = 50nm, a4 = 30nm, L = 100nm, b = 40nm, and v = 5000m/ sec...... 103 4.10 Transmission coefficient TjI,OO (a) and Tjl (b) versus incident phonon frequency w. The parameters: al = 40nm, a2 = 20nm, a3 = 50nm, a4 = 30nm, L = 100nm, b = 50nm, and v = 5000m/ sec...... 103 4.11 Two-terminal thermal conductance Gij (a) and Gij/T (b) versus tem perature T. The parameters: al = 40nm, a2 = 20nm, a3 = 50nm, a4 = 30nm, L = 100nm, b = 30nm, and v = 5000m/ sec...... 105 4.12 Four-terminal thermal conductance G14,23 versus temperature T. The parameters: al = 40nm, a2 = 20nm, a3 = 50nm, a4 = 30nm, L = 100nm, b = 30nm, and v = 5000m/ sec...... 106 4.13 Temperature ratio 0!14,23 versus temperature T. The parameters: al = 40nm, a2 = 20nm, a3 = 50nm, a4 = 30nm, L = 100nm, b = 30nm, and v = 5000m/ sec...... 108 4.14 Four-terminal thermal conductance G14,23 with different width b versus temperature T. Other parameters: al = 40nm, a2 = 20nm, a3 = 50nm, a4 = 30nm, L = 100nm, and v = 5000m/ sec...... 108 4.15 Temperature Ratio 0!14,23 with different width b versus temperature T. Other parameters: al = 40nm, a2 = 20nm, a3 = 50nm, a4 = 30nm, L = 100nm, and v = 5000m/ sec...... 109 Abstract

Although the study of thermal transport in condensed matter has a very long history, it continues to be an active field of work due to its importance in many applications. The research subject reported in this thesis is on theoretical investigations of thermal energy transport in systems whose linear dimension is less than the wavelength of thermal phonons. Such situations occur in mesoscopic and nanoscopic scale dielectric structures which can now be fabricated in a number of laboratories. Due to the small system dimensions, phonons must be treated as waves. Thermal energy transport, therefore, must be treated as phonon wave propagation through the system. After reviewing the general physics of thermal energy transport in the classical regime, we derive, for dielectric materials, a formula for thermal energy flux in devices involving multi-terminals each connected to a thermal reservoir at local equilibrium. The energy flux is driven by a temperature bias and traverses the system by virtue of phonon wave scattering. A multi-terminal thermal conductance formula is derived in terms of phonon transmission coefficient. Using our theoretical formulation, we investigate thermal transport properties of both two-terminal and four-terminal di electric devices by solving the quantum scattering problem using a mode matching numerical technique. For thermal transport in a T-shaped dielectric nanostructure with two-terminals at low temperature, due to quantum interference the transmission coefficient of phonons becomes quite complicated. We found that the value of phonon transmission coeffi cients at zero energy may be unity or zero depending on a geometrical ratio of the nanostructure. The transmission has an oscillation behavior with quasi-periodicity and irregularity. The thermal conductance is found to increase monotonically with temperature - a result that we conclude to be generally true for any two-terminal device. We confirm the existence of the universal quantum of thermal conductance which exists at the low temperature limit, and such a quantum is robust against aIl the system parameters. The physical behavior of four-terminal thermal conductance for mesoscopic dielec-

xii Abstract xiii tric systems with arbitrary shapes of scattering region is also investigated in detail. If we make a two-terminal measurement in the four-terminal deviee, the two-terminal conductance is a monotonically increasing function of temperature, and is equal to the universal quantum of thermal conductance masked by a geometric factor. If we make a four-terminal measurement, the four-terminal conductance has a non-monotonie dependence. In the low temperature limit, we predict that the four-terminal conduc tance diverges inversely proportional to temperature. Finally, we discuss an interesting theoretical problem on the general behavior of thermal conductance for multi-terminal systems when thermal carriers satisfy frac tional exclusion statistics. Our analysis allows us to conclude that results for fractional exclusion statistics are quite difIerent from those of the Bose-Einstein statisties. Résumé

Bien que l'étude du transport thermique en matière condensée ait une très longue histoire, elle continue d'être un domaine actif due a son importance dans plusieurs applications. Le sujet de cette thèse est une étude théorique du transport d'énergie thermique dans les systèmes dont la dimension linéaire est inférieure à celle de la longueur d'onde des phonons thermiques. De telles situations se manifestent à l'échelle mésoscopique et nanoscopique de structures diélectriques qui peuvent maintenant être fabriquées dans un grand nombre de laboratoires. En raison des petites dimensions du système, les phonons doivent être traités en tant qu'ondes. Le transport d'énergie thermique, donc, doit être traité en tant que propagation d'onde de phonon à travers le système. Après une révision générale de la physique du transport d'énergie thermique dans un régime classique, nous dérivons, pour des matériaux diélectriques, une formule pour le flux d'énergie thermique dans les dispositifs qui ont plusieurs terminaux, cha cun étant connecté à un reservoir thermique à l'équilibre local. Le flux d'énergie est généré par un déséquilibre de température et traverse le système par diffusion d'ondes de phonon. Une formule de conductance thermique a plusieurs terminaux est dérivée en termes du coefficient de transmission de phonons. En utilisant notre formulation théorique, nous étudions les propriétés du transport thermique de dis positifs diélectriques à deux et quatre terminaux en résolvant le problème de diffusion quantique en utilisant une technique numerique d'adaptation de mode. Pour le transport thermique dans une nanostructure diélectrique en forme de T avec deux terminaux a basse température, due a l'interférence quantique, le coeffi cient de transmission de phonons devient très compliqué. Nous obtenons que la valeur du coefficient de transmission de phonons à énergie nulle peut être égale à zero ou un, dépendant du ratio géométrique de la nanostructure. La transmission a un com portement oscillatoire avec une quasi-périodicité et une irrégularité. Nous obtenons une augmentation monotone avec la température de la conductance thermique - un résultat que nous concluons généralement vrai pour tout dispositif à deux terminaux.

xiv Résumé xv

Nous confirmons l'existence d'un quantum de conductance thermique qui existe à basse temperature, et que ce quantum est robuste envers tout autre paramètre du système. Le comportement physique de la conductance des systèmes diélectriques mésoscop iques à quatre terminaux, à forme géométrique de région de diffusion arbitraire est aussi étudié en détail. Étant donné que nous faisons une mesure à deux terminaux dans un dispositif à quatre terminaux, la conductance à deux terminaux et monotone avec la température, et est égale au quantum universel de la conductance thermique masquée par un facteur géométrique. Lorsque nous faisons une mesure à quatre terminaux, la conductance a une dépendence non-monotone. Dans la limite des basses températures, notre prédiction est que la conductance a quatre terminaux diverge d'une manière inversement proportionelle à la température. Finalement, nous discutons d'un problème théorique intéressant sur le comporte ment général de la conductance thermique pour des systèmes a plusieurs terminaux quand les statistiques d'exclusion fractionelles sont satisfaites par les porteurs d'énergie thermiques. Notre analyse nous permet de conclure que les resultats pour les statis tiques d'exclusion fractionelles sont différents de celles des statistiques de Bose-Einstein. Statement of Originality

This thesis reports a study on thermal transport in mesoscopic scale dielectric systems where wave phenomena dominates the physics. The original contributions to this subject include the following.

• In Chapter 2, a molecular dynamics simulation for thermal conductivity is car ried out to illustrate how one approaches thermal transport physics by classical MD. We then derive a multi-terminal formula of thermal flux for the first time, by means of second quantization. This formula casts the thermal energy trans port into a phonon wave scattering problem, and is used extensively in the rest of the thesis.

• In Chapter 3, thermal transport in a T-shaped wire at low temperature is investigated for the first time. A mode-matching method for solving the problem of phonon wave scattering is developed. Transmission features of thermal flux

and the behavior of the two-terminal thermal conductance are discussed III detail. This work is summarized in a manuscript to be published[l].

• In Chapter 4, we report for the first time the physics associated with four terminal thermal conductance in detail. A four-terminal thermal conductance is derived for mesoscopic dielectric devices with arbitrary central scattering region. We apply this formula to a specific system by solving the wave scattering problem with mode matching. We then discuss the physical behavior of the four-terminal conductance and compare it with that of two-terminal situations. Partial work on this topie has been published[2].

• In Chapter 5, a general discussion for thermal conductance in 2D and the four terminal case is made concerning a purely theoretical problem when the thermal energy carriers satisfy the fractional exclusion statistics. Here, we work out the general temperature dependence of thermal conductance in the quantum regime. The work reported here has been part of the content of Ref. [2].

XVI Acknowledgments

Most of all 1 would like to thank my research supervisor Prof. Hong Guo for his unre served guidance and unstinted support. He brought me the new method of thinking on research, which is of much benefit to me. Additionally he read carefully and cor rected seriously each draft of this thesis. The final version of the thesis has his great contribution. 1 doubt 1 would have been able to complete this thesis without his support. For these, 1 keep them in my memory forever.

1 am very grateful to Dr. Qingfeng Sun with whom 1 had a pleasant collabora tion, for giving me the great help on my research projects.

1 am also grateful to Dr. Qingrong Zheng for many interesting discussion on physics and helpful suggestion on my research projects.

Many thanks also to S. K. Tommy Mark, Martin Grant, Paula Domingues and Richard Harris, for all the help.

1 would like to thank Christian Voyer for his French translation of my thesis ab stract.

1 wish to thank my friends of Quanyong, Huabin, Yan, Yi Liu, Xubo, Wengang, Chao-Cheng, Li, Yongfang, Tao, Yu, Shan Zhu, Yongxiang, Xuedong, Jae-Ho, Yi and Xiaobin, for their sincere friendship.

Finally 1 especially like to thank my wife, Ying Song, for her understanding and support.

xvii 1

1ntrod uction

The subject of this work is thermal transport in mesoscopic dielectric system. We note that the study of thermal transport in condensed matter has a long history[3]. The early and typical macroscopic approach to deal with non-equilibrium thermal phenomena was based on defining thermal transport coefficients through phenomeno logical constitutive equations. Taking the hypothesis that the system in consideration was close enough to a global equilibrium state, transport coefficients were then com puted assuming the proportionality between the thermal energy flux and the thermo dynamic driving force[4]. J.B.J Fourier was the first to propose a phenomenological relation for thermal transport between the energy flux and the temperature gradient that drives the flux. It is called Fourier's law :

J(x, t) = -r;SlT , (1.1) where thermal flux density J represents the thermal energy transporting through an unit surface per unit time, and T(x, t) is the local temperature at position x and time t. The coefficient fi, is called thermal conductivity which depends on material details of the system. Formula (1.1) is assumed to be valid near thermal equilibrium, i.e., under a very small temperature gradient. In fact, the definition of local energy flux density J(x, t) and the local temperature field T(x, t) rely on the local thermal equilibrium hypothesis, i.e., on the existence of a local temperature for a macroscopically small but microscopically large volume at each location x for each time t. Fourier's law

1 Chapter 1. Introduction 2 becomes slightly more complicated for anisotropic crystals because the anisotropy usually leads to different thermal flux in different crystalline orientation:

J/t = - L K/tvaTjâxv , (1.2) v where K/tv is an element of the thermal conductivity tensor. In a solid, heat can be carried by mobile electrons, magnetic excitations, phonons, and other elementary excitations. Quantity K can then be written as a sum of these different contributions:

(1.3)

A metal typically has high values of thermal conductivity, a fact that is closely related with their high electrical conductivity. In the simple Drude theory of electrical conductivity, free electrons are treated as classical gas-like particles. There exists sorne average distance, the mean free path or momentum relaxation length lm, within which the electrons move freely without suffering elastic scattering due to impurities. The electronic heat conductivity in Eq.(1.3) can then be expressed as[3]

(1.4) where Ce is the heat capacity per electron, ne is electron density, V is the mean electronic velocity, and l is the mean free path (nece is the electronic heat capacity per unit volume). From this equation, it follows that Ke can be large because met aIs contain large numbers of electrons which are relatively free to move through the metal and can transport energy from one region to another[5]. In crystalline insulators there essentially exists no mobile electrons, heat is there fore conducted by atoms vibrating around their fixed equilibrium positions. According to classical lattice dynamics theory, thermal transport in insulators can be consid ered from the point of view of collective lattice vibrations in terms of the propagation of "sound waves". In a modern theory, these vibrations give rise to the notion of Chapter 1. Introduction 3 phonons which mediate heat conduction. Phonons are sim ply the quantized form of the collective modes of lattice vibration. The heat conductivity due to phonon propagation can then be expressed as[3]

1 ""ph = 3Cvl , (1.5) where v is the mean phonon velocity, approximately equal to the velocity of sound in the crystal, C is the heat capacity contributed by the crystal, and l the mean free path of phonons. Typically, these quantities have values as: v rv 6916 m . sec-l,

1 8 C rv 0.703 J. K- . g-l and l rv 10- m at 298 K for crystalline Si, giving thermal

1 1 conductivity "" rv 148 W . m- . K- [6]. In fact, the physics of heat conduction in a crystal is rather similar to heat conduc tion in agas. Heat conduction of gas can be explained as follows. By equipartition theorem, gas molecules at higher temperature have larger kinetic energy than those with lower temperature. Collisions between these gas molecules tend to "equilibrate" the system thereby pointing to a direction of energy flow. In a crystal, if the lattiee vibration is small enough to remain harmonic, we can represent thermal energy by a number of non-interacting "particles" or phonons. In the harmonie approximation, phonons behave with many of the properties of an ideal gas. In thermal equilibrium, the mean phonon occupation number is given by temperature and is represented by the Bose-Einstein distribution

1 n( w) = ---':-/'iw-- (1.6) ekBT - 1 where w is the angular frequency of vibration, kB is the Boltzmann constant, and T temperature of the system. The number of phonons is not conserved. As the temperature is raised by the input of thermal energy, more phonons are excited. When a temperature gradient exists in a sample, the density of phonon gas becomes non-uniform. Eq.(1.6) indicates that the density is high at high temperature and low Chapter 1. Introduction 4

at low temperature. One therefore deduces that the phonon gas will "drift" along the direction of temperature gradient, Le., a diffusive motion of phonons is expected. Because a phonon is the energy quanta of a specifie normal mode of the lattice vibration, the directional drift means that there is a thermal flux: the flux flows along the average motion direction of phonons. Heat conduction in a crystal can therefore be thought of the result of diffusive motion of phonons. Like that of gas molecules, the me an free path of phonons is determined by collisions between the phonons which depend closely on the average temperature of the system. In different temperature ranges, heat phenomena can be explained by different theories:

1. In the high temperature regime, T » e D where e D is the Debye temperature, Bose-Einstein distribution becomes

(1.7)

In this case the number of phonons n with frequency w is proportional to temper ature T. When temperature rises, the probability of phonon collision increases because there are more phonons, and therefore the phonon me an free path 1 decreases. In fact the mean free path is inversely proportional to temperature T. Furthermore, at high temperatures the phonon heat capacity obeys the law of Dulong and Petit [7] which states that the molar heat capacity of an simple

solids is equal to 3R at high temperatures (R = 8.3143 J. K-l. M ole-l, the gas constant) and is therefore temperature-independent. By formula Eq.(1.5), we should expect a decreasing of thermal conductivity with increasing temperature in the high-temperature regime. This was confirmed by many experiments and

/'î, is generally given[8] by

1 /'î,rv - (1.8) Tx '

where exponent x is somewhere between 1 and 2. Chapter 1. Introduction 5

2. At any temperature T « e D, only those phonons with energy comparable to or less than kBT will be present in appreciable numbers. In this case, the phonon occupation number reduces to 1 _2IL n = ---"'6:-Q-- ~ e k BT (1.9) ekBT - 1 Sinee mean free path is proportional to the inverse number of particles present, 6 we expect 1 rv e kBQ,. Henee, when temperature drops, the mean free path increases rapidly. We should mention that aside from phonon collisions, there are other factors in a crystal which limit the phonon mean free path. These include defects, dislocations, grain boundaries, impurities, surfaees and so on. Due to these factors, phonon mean free path cannot increase infinitely by lowering the temperature. It is known that at about 20K or so, phonon mean free path be cornes approximately a constant due to effects of boundary scattering[9]. When this happens, heat capacity C provides a competing effect because C ex: T 3 at low temperature in terms of the Debye model. Then, according to (1.5), a maximum of thermal conductivity is first reached and followed by a rapid fall as '" ex: T3 when temperature is decreased. In this case, the relation of thermal conductivity to temperature cornes mainly from the heat capacity.

The heat capacity at low temperature can be well described by the simple Debye model[lO]. Debye model assumes that a solid is a continuum of material carrying 3N traveling sound waves (normal modes) where N is the number of atoms in the solid. By solving the wave equation, we can find the number of vibrational modes within a given frequency range, Le., density of states (DOS) of vibrational modes. At a given temperature T, vibration al modes with 1ïw :::P kBT are not excited therefore they hardly contribute to the heat capacity. The heat capacity is contributed mainly by those modes with 1ïw :::; kBT. At the low temperature limit, heat capacity is deter mined by vibrations of the lowest frequency, i.e., the elastic waves with the longest Chapter 1. Introduction 6 wavelengths. The Debye model is approximately correct at the low temperature limit and predicts[lO] C ex: T3 when T -t OK. Although the Debye model (and the less accurate Einstein model[lO]) can explain sorne equilibrium thermal properties of materials, other theories and approaches are still needed to deseribe features of phonon transport. There are three approaches whieh are popular for ealeulating thermal transport properties of materials.

• The Boltzmann transport equation (BTE).

• The molecular dynamics (MD) numerical method which is based on the fiuctuation dissipation relation of linear response.

• The direct caleulation of phonon wave transmission.

These approaches have been applied to a wide range of thermal transport problems and in the following we provide a very brief discussions of them. We then move on to discuss the situation of reeent experimental studies of thermal transport at mesoscopie seale for which the wave phenomenon is important - a situation best studied by the last technique. In particular, we will discuss the experimental work of Roukes et.al[ll] who reported the first measured data on the quantum of thermal energy, and we present a theoretical analysis of this situation following the work of Kirczenow et. al. [12]. We then summarize the work to be presented in this thesis.

1.1 Theoretical appraaches ta thermal transport

In this section, we present a short review of each of the three theoretieal approaches which find applications to thermal transport. The first two, Boltzmann equation and MD, are most popular but cannot deal with wave phenomena of phonon transport. The direct calculation of phonon wave transmission, Le., the transmission function Chapter 1. Introduction 7 approach, which has only been developed in recent years, is designed for wave phe nomena analysis and will be the method of choice of this work.

1.1.1 Boltzmann transport equation The Boltzmann transport equation[13] has been successfully applied to thermal trans port: its development can be considered as the most important milestone for the the ory of thermal transport in solids. The main ide a behind the Boltzmann transport equation cornes from the kinetic theory: lattice vibration responsible for heat trans port can be viewed as a gas of phonons. Then, phonon transport can be described by treating phonons as particles. In the semi-classical limit, the Boltzmann transport equation has the following form

F an8t + i . \7 k n + v . \7 r n = (an)8t ' (1.10) col where n( r, k, t) is the carries (phonons) distribution function, v = -k \7 ke is the carrier group velocity, e is the energy dispersion of the phonon modes, and F the external force. The right hand side of the equation is the rate of distribution change due to collisions of the phonons. Unfortunately, the Boltzmann equation is difficult to solve even for the simplest situations because it involves variables in both real and momen tum spaces, as well as time. To make the problem tractable, various approximations are widely applied. A particularly useful approximation is supposing that collisions of the carriers (phonons) between themselves or with other scattering centers, drive the distribution n toward the equilibrium distribution. Such a relaxation dynamics can then be de scribed by sorne relaxation time of the order of collision time. This way, the collision term is drastically simplified,

eq an) n- n (1.11) (at col ~ - T(W) Chapter 1. Introduction 8

eq eq /iw where n is the equilibrium distribution and is taken as Eq.(1.6): n = (e kBT -1tl,

T is the heat-carrier relaxation time which usually depends on the angular frequency

W (or energy) of the heat carriers. Under this relaxation approximation and in the absence of any external force and in the assumption that phonons only travel in one direction along with z-axis, the steady state Boltzmann equation of heat transport becomes:

onoT n- neq vz - - = - -".--,-- (1.12) ôT ÔZ T(W)

2 where Vz is the z component of phonon group velo city v and v; = !v , rI; is the temperature gradient in the z direction. When n is solved from (1.12), the thermal current density in the z direction[3] is obtained by the following equation,

Jz = L n(k)1ïw(k)vz(k) (1.13) k Usually in using Eq.(1.12), we assume that the distribution in the presence of tem perature gradient is not much deviated from the equilibrium distribution, so that we can replace ôn/ôT by ôneq/ôT in the left side of Eq.(1.12). Therefore, combining Eqs.(1.1), (1.12), and (1.13), the thermal conductivity yields

If we replace the summation by an integral over w, we obtain

(1.14) where f (w )dw is the number of phonon modes between w and W + dw per unit volume of the crystal. Consequently, by using the Boltzmann equation in the relaxation approximation, the thermal conductivity for phonon heat carries can be derived from Eq.(1.14)[3] by Chapter 1. Introduction 9

a linear dispersion relation, w(k) = vk, for each branch of the phonon spectrum in the Debye model

4 x kB (kB)3 3 (~DIT x e (1.15) '" = 27r2v h T Jo T(X) (ex _ 1)2dx , where e D the Debye temperature, and x - k';:T' Sorne further manipulations on Eq.(1.15) can be done. We note that the contribution to heat capacity from modes in frequency range w to w + dw is written as d~C!iwneqf(w)dw), where f(w)dw is the number of phonon modes in this range per unit volume of the crystal. In the Debye approximation, it can be shown[3] that the differential contribution to the heat capacity is

4 x 3kB (kB) 3 3 x e (1.16) C(x)dx = 27r2V3 h T (ex _ 1)dx

Therefore, using Eqs.(1.15) and (1.16), the thermal conductivity can be rewritten as

1 2 ineDIT '" = -v T(x)C(x)dx . (1.17) 3 0 A relaxation time can be expressed as the ratio of a mean free path to a velo city, so the conductivity may be written as

'" = ~v Jl(x)C(x)dx , (1.18) which is a logical extension of the simple kinetic equation in Eq.(1.5). Here T(X) is the combined relaxation time, i.e., TC. Generally, there are three main contributions to the scattering pro cess so that 1 1 1 1 -=-+-+- , (1.19) TC TU Tf TB where TU, TI, and TB are the relaxation times due to the Umclapp pro cess (U-process), the impurity scattering, and the boundary scattering, respectively. The Umclapp pro cess is produced by three-phonon collisions in which the sum of the phonon wave vec tors is not conserved but changes by a reciprocallattice vector. An important outcome Chapter 1. Introduction 10

of the Boltzmann transport equation[8] is the proof that crystalline anharmonicity[14] is necessary to obtain genuine diffusion of thermal energy through the Umclapp pro cess. On the other hand, impurity scattering cornes from several sources arising from the presence of atoms with different masses, point defects, dislocations and other crystalline imperfections. Finally, boundary scattering occurs when the mean free path l of the phonons becomes greater than or comparable to the smallest sam pIe dimension at low temperatures. In recent years, the research of phonon transport through mes os copie dielectric ma terials has attracted increasingly more attention because accurate experimental mea surements of thermal conductance in these systems st arts to be possible[ll, 15, 16]. Theoretically, this problem has been considered by two other approaches in addi tion to the Boltzmann transport equation. These are through numerical molecular dynamics (MD) simulations[17, 18, 19, 20, 21], and by calculating the transmission function[12, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31].

1.1.2 Molecular dynamics Molecular dynamics simulations of thermal transport start at the microscopic level where the time evolution of a set of interacting atoms is followed by integrating their equations of motion. In MD, we consider a system of N atoms and follow the law of classical mechanics, Le., Newton's law for each atom i in the system,

(1.20)

2 2 where ~ = d rddt is the acceleration of the i-th atom, Fi = -VriV(rl,r2,·· ·,rN) is the force acting on this atom due to its interaction with other atoms, and mi is the mass of the atom. In MD, the motion of atoms is governed by inter-atomic interac tions which can accurately include any anharmonic interaction between the atoms. Newton's equation is deterministic and preserves the time reversaI symmetry: given Chapter 1. Introduction 11 an initial set of positions and velocities, the subsequent time evolution is in princip le completely determined. However, in order to describe any thermal phenomenon, such as the flow of thermal energy from a high temperature place to a low temperature place, the concept of temperature must be introduced into the classical mechanics and time reversaI symmetry must be broken. There are several ways to do this. For instance one may introduce a noise force and a friction force into Eq.(1.20), mimick ing the fluctuating "fast variables" of the dynamics and the energy dissipation due to interaction of the atoms with a thermal bath. The fluctuation-dissipation theo rem then relates these forces and guarantees the relaxation to thermal equilibrium at large times[13]. Another popular way to introduce temperature into MD is through the "thermostat" [32, 33] which acts as a thermal bath in which the atoms can take or dump energy. The easiest way to introduce temperature, however, is through the numerical procedure of "velo city rescaling" using the equipartition theorem by which the statistical average of the kinetic energy (Ec) is related to the temperature

1 N 3 (Ec) = - Lmiv; = -NkBT . (1.21) 2i=1 2

Since the velo city Vi of the atoms in MD simulation is monitored, one can compute the kinetic energy of the entire system at each time step of the simulation. If the second equal sign of Eq.(1.21) is not satisfied for a given temperature T, one numerically sc ales aIl Vi 's such that it becomes satisfied. Such a rescaling is somewhat ad hoc, but it makes certain that thermal equilibrium condition will be satisfied at large times, even though it does not give the correct dynamics. The theoretical background for obtaining irreversible thermodynamics out of the reversible classical mechanics has been established decades ago, through the seminal works of Kubo[34], Green[35], Mori[36] and others[32, 33, 37, 38, 39, 40, 41, 42, 43]. We will not review these fundamental issues but refer interested readers to the original references. PracticaIly, in MD simulation the most important issue concerns the interaction Chapter 1. Introduction 12

model describing the interaction among the atoms. A simple but very useful model is the Lennard-Jones pair potential (LJ),

(1.22) where E and Œ are two parameters obtained by fitting results to experimental data. The LJ model provides a potential for qualitative investigations in MD, but occasion ally it can be used for sorne quantitative analysis if the the constants (E, Œ) are good enough for specifie materials. N evertheless, many better models of pair potentials have now been developed and widely used in MD for quantitative predictions. These include more complicated empirieal classical potentials and/or quantum mechanical ab initio potentials. In general, MD simulations have the advantage of not only obtaining a general physical picture, but also providing quantitative predictions[44]. For investigating thermal transport using MD, i.e., for calculating the thermal conductivity fi, , one applies the Green-Kubo (GK) linear response theory[18] - a result of the fluctuation-dissipation theorem which relates fi, to the correlation function of the thermal current[18]:

(1.23) where n is the system volume and J JJ. the thermal current density along direction p,. The angular bracket denotes an ensemble average, or in the case of an MD simulation, an average over time. Since a simulation is performed for discrete MD time steps ~t, the time integration in Eq.(1.18) is in fact a summation. lncluding the time average, in actual operation, the formula one uses is

(1.24) where TM is given by M~t and JJJ.(m + n) is the {tth component of the heat current at MD timestep m + n. Chapter 1. Introduction 13

To go further, the heat current is written as[14, 18] d J = dt ~ri(t)é , (1.25) where ri(t) is the time-dependent coordinate of i-th atom and éi(t) is the energy. If the potential is made up of not only pairwise terms but also three-body interaction terms, the potential energy is written as

(1.26)

Therefore the site energy of the i-th atom can be defined as

éi = ~mivl + ~ ~ V2(rij) + ~ ~k V3(ri' rh rk) (1.27) J J The definition leads to a thermal current defined by

(1.28)

where Fij is the force due to the pair potential and the three-body term Fijk is given by

(1.29)

To compute the thermal current fiowing through a crystal lattice[45] , the external disturbance is represented by a temperature gradient across the system. For instance, one may imagine that the sample is connected to two thermal reservoirs with different temperatures, as shown in Fig.1.1 by the initial atomic configuration of the atoms. In the simulation, the temperatures in the reservoirs are fixed through a thermostat. After a long time of relaxation, the sampIe will reach a steady state where the av erage thermal current becomes time-independent. Then, we can calculate thermal conductivity by Eqs.(1.20-1.29). This approach allowed sorne authors to study dif fusive energy transport[37], the thermal conductivity of amorphous solids[46], and other problems. Chapter 1. Introduction 14

Reservoir T, Reservoir T2

Figure 1.1: Schematic diagram of an initial configuration to be simulated by MD. The left and right reservoirs represented by atoms EB and CV are at temperatures Tl and T2 , respectively.

80 far, the two methods we discussed, the Boltzmann equation and the MD, treated thermal transport classically or semi-classically as mediated by collisions of particles, where the wave nature of the phonons was neglected. The wave nature necessarily leads to such phenomena as interference which may become important when the phonon wave length is comparable to the sam pie size. The problem studied in this thesis belongs to this category and the approach we will use is through calculating the transmission coefficients.

1.1.3 Transmission function The transmission function approach is very weIl fit to study cases of large phonon wavelength compared to the sample size so that phonons flow through the sam pie ballistically or semi-ballisticaIly, and wave phenomenon becomes a dominant factor. This situation occurs for very small samples at very low temperatures. This approach is based on the original idea of Landauer[47] which treats transport of carriers from the scattering point of view, where an incoming wave is scattered by the medium. Again, the scattering is provided by surfaces, interfaces, and isolated scatterers in the interior of the system. The scattering events are mathematically studied via the transmission function. It is important to note that when system size is compatible Chapter 1. Introduction 15 to phonon wave length, one is interested in thermal conductance rather than thermal conductivity. As was shown by Landauer[47], no matter how scatterers are distributed in a system, one can always calculate a transmission function, T(w), which describes the propagation of quasi-particles of energy 1iw between two reservoirs connected to the system. For thermal transport, the thermal conductance is expressed as[12]

rn G = fow ~~ 1iwT(w) 8r;;;:) , (1.30) where the phonon spectrum extends between frequencies 0 and a maximum wm . n(w) is the reservoir distribution, which is taken in the form of Bose-Einstein distribution for phonon transport. This formulation has yielded new insight into the problem of phonon scattering by surface roughness where there is the transition from ballistic to diffusive transport[22]; it provided an understanding to the low temperature "dip" in the quantized thermal conductance[25, 26]; it has been applied to investigate heat flow through a mesoscopic link[48] and mono-atomic chain[29]. Despite of these excellent works and unlike the case of electron transport, the use of transmission function approach in phonon transport is just at its beginning stage. For transport, it is important to note different length scales: the system size and carrier characteristic sizes. For different sizes, the system will have different physical features. For example, a system has different thermal conductivities in one-dimension and three-dimensions. Even in one-dimensional situation, transport properties will change with decreasing length of the system[49]. In general, there are three major length sc ales which we consider in thermal transport: the transverse size W and longitudinal sizes L of system, the mean free path of carrier lm which is mostly limited by scattering.

• lm < W, L, the transport is diffusive and is in the Drude regime where semi classical and classical theory describe the transport phenomena very weIl as Chapter 1. Introduction 16

mentioned above.

• W < L ~ lm, this is the ballistic transport regime. In the regime, the phonons do not suffer impurity scattering. Scatterings only occur at the interface be tween different systems and on the boundary surface of system. The transmis sion function approach is aimed at solving problems in this regime.

• W < lm < L, this is the crossover regime between the diffusive regime and the ballistic regime, named "quasi-ballistic" where a small amount of impurity scattering exists. The ballistic-diffusive heat-equation based on an approxima tion of the Boltzmann equation can describe the heat-conduction situation[50] in this regime.

When the longitudinal length in a system is far larger than other transverse sizes,

W « L ~ lm, the system is thought to be one-dimensional or quasi-one-dimensional. Due to confinement in the transverse direction, a series of phonon subbands ( or discrete modes ) are created in the one-dimension al system when phonons propagate through the channel. In this situation, the heat flux associated with a small temper ature difference !1T between the two ends of the one-dimensional system consists of the sum of the contribution of the individual phonon subbands[12, 23, 24]:

(1.31) where k is the wave vector along the longitudinal direction, wm(k) is the dispersion relation of the m-th discrete mode, n(wm(k)) is the phonon distribution, and v gm = dwm(k)jdk is the group velocity. The transmission coefficient T(w) is the phonon transmission probability through the system. In principle, it should contain all the information of scattering which could affect the phonon transmission. The transmission probabilities are generally calculated by the mode-mat ching method[51]. It states that when an incident wave is going through an interface of Chapter 1. Introduction 17

two materials, due to the mismatch of modes at the two sides of the interface, a partial wave is refiected and a partial wave transmitted. Then the transmission prob ability from the incident wave with wave vector k~n in mode a to transmission state k! in mode m is given by

(1.32) where Itmal is the transmission amplitude. Summing Eq.(1.32) over aIl incident modes a, the transmission coefficient of the mth mode is:

(1.33)

By Eq.(1.31) and Eq.(1.33), the thermal conductance G = QI ~T[l1, 12, 23] in one dimension al system is obtained as:

(1.34)

Transforming the integral to an integral over frequencies, Eq.(1.34) becomes

1 00 Dn(w) G=-L: 1 dwfiwTm(w) DT ' (1.35) 27r m Wcut where Wcut is the cutoff frequency of the m-th mode, i.e., the lowest frequency that can propagate. In this thesis, we will adopt Eq.(1.35) to investigate thermal transport in mesoscopic thermal systems.

1.2 Thermal transport experiments at the mesoscopic scale

In comparison with electron transport, thermal transport at mesoscopic and 1 or nanoscopic scale was much less studied until very recently. This is because meso scopic structures confining phon on waves are not as straightforward to realize as electron systems such as quantum wells and wires. Since there are no perfect thermal Chapter 1. Introduction 18

Electron g.a!:

Phonon g.a!:

Figure 1.2: Conceptual picture of an electron heating experiment. Joule heat Qin is put into the electrons. Part of heat is transferred to the phonons by the electron-phonon interaction Qep. This heats up the phonons which are then cooled by phonon diffusion Qph and by returning sorne of the heat to the electrons, Qpe. Heat also flows to the contacts via electron diffusion Qe/. Plot derives from Ref. [23]. insulators, it has been very difficult to make accurate and unambiguous phonon mea surements in confined geometry. Typically, the experimental strategy for phonons is to induce electron heating in wires that are thermally decoupled from the environ ment, either through physical suspension of the wire, or through very poor thermal contacts, e.g., Kapitas boundary[52, 53] resistance for heat transport from a metal to a substrate. In these measurements, what is actually measured is the electron energy loss rate. Fig.1.2 shows such an experimental setup. The earliest method to experimentally investigate phonon transport in mesoscopic system began in 1985 with suspended wires[54]. The work was carried out on an array of suspended, mono-crystalline n+GaAs wires[55, 56]. Because electron diffusion in these samples overwhelmed the indirect energy pathway which involves phonons, litt le evidence of phonon contribution to transport could be observed. In order to over come this difficulty, one available solution is to perform direct thermal conductance measurement on suspended structures, Le., structures which are physically separated from the substrate for most of their extent so as to have thermal isolation. With the Chapter 1. Introduction 19

advance of semiconductor fabrication and lithography technology, such systems can now be produced in severallaboratories. In addition, sorne other techniques have also been developed. For example, an intriguing report has been published in 1992 about acoustic waveguide modes and nanowire phonon subbands as observed by Seyler and Wybourne[57]. Although their sample was anchored to a substrate, it was argued that phonon confinement could still occur due to acoustic mismatch at the interface between the substrate and the metal nanowire. However, the search of the same phenomenon by other authors have, so far, proven elusive[58]. Perhaps the most successful experimental measurement of phonon transport at mesoscopic scale was the work of Tighe, Worlock and Roukes[59]. They carried out direct thermal conductance measurements on suspended mono-crystalline nanostruc tures. Their work demonstrated a good fabrication technique and has opened up the field of mesoscopic phonon physics. The essential idea of Tighe et. al. [59] is to clamp one end of a nonmetallic nanostructured beam while providing heat input Q to the other thermally-isolated end. The thermal conductance for small heat input is then

G = QI f::j.T [11, 12,23], where f::j.T is the temperature difference between the clamped and thermally isolated ends. But fabrication of nm-scale structures in this way was proven difficult to assure a thermally-isolated end without placing undue strain on the suspended beam[59]. To overcome this problem, Tighe et.al fabricated a device depicted in Fig.1.3, in which a phonon "cavity" is formed from insulating GaAs in the shape of a thin plate, and is suspended above the substrate by insulating GaAs beams. To measure thermal conductance of these insulating beams, the cavity is heated by a source transducer patterned above it. The cavity then cools through the long, narrow, and mono-crystalline insulating GaAs beam underneath. The temperature change in the cavity is measured by a second sensor transducer. This configuration can measure directly the parallel thermal conductance of the four nm-scale supporting beams. In this approach, it has become possible to pattern separately the mesoscopic insulators Chapter 1. Introduction 20

Figure 1.3: A suspended and mono-crystalline device which can directly measure thermal conduc tance on nanostructures. Left figure, a device consists of i+GaAs plate used as a quasi-isolated phonon cavity (3 x 3 J-tm x 300 nm thickness), and four 5.5 J-tm long i-GaAs bridges (cross section 200 nm x 300 nm). The bridges suspend the cavity above the substrate by 1 J-tm. An integral pair of n+GaAs resistive transducers (120 nm line-width, 150 nm thickness) meander above the underlying GaAS cavity. Right figure, schematic diagram of principal components of the suspended device: isolated cavity, suspended bridges, and the supports with external reservoir, which fix the sample to the substrate. Figure derives from Ref. [U].

(beams) which determine thermal transport, and the transducers which are used to induce and measure heat response. After adopting the above experimental approach, Roukes and his collaborators completed the first experiment[ll] in literature where a "quanta" of thermal con

2 13 ductance, Go _ 7f k1T/(3h) = (9.456 x 1O- W/ K2)T, in freely suspended one dimension al Si wires was discovered. This showed unambiguously that phonon propa gation in this material is indeed ballistic at low temperatures[60, 61]. The experiments of Roukes et. al. were possible because of three essential ingredients[ll]. First, ballis tic phonon transport through one-dimensional phonon waveguides was made possible by the experimental device setup[57]. Second, coupling between the lowest phonon modes and the thermal reservoirs was optimal over an appreciable temperature range (the phonon transmission coefficient must remain close to unity) [12]. Finally, the sam pIe configuration and measurement technique provided sufficient resolution to observe the contribution to G down to a single mode[59]. From Fig.1.4, as temperature de creases the reduced thermal conductance G/16Go in the four waveguides varies from larger than unity to less than unity. Once the reduced thermal conductance becomes Chapter 1. Introduction 21

0.1 -f-...... -I--~----I-...... +------...... , 60 100 000 11X1J 6.000 Temperatl.Ife (ml'\!

Figure 1.4: The measured thermal conductance data. The measured thermal conductance is nor malized by the expected low-temperature value for 16 occupied modes, 16 Go. Measurement er ror is approximately the size of the data points, except where indicated. For temperatures above Tco ~ 0.8K, A cubic power-law behavior consistent with a mean free path of '" 0.9 /-Lm. For temper atures below Tco , a saturation in G at a value near the expected quantum of thermal conductance is observed. Consistent with expectations, to within experimental uncertainty, G never exceeds 16Go once it finally drops below that value. Plot derives from Ref.[U]. less than unit y, it never exceeds unity again because the system becomes ballistic. Because there are four lowest phonon modes in each waveguide, Le., one dilatational mode, one torsional mode, and two fiexural modes [11 , 12], one therefore deduces that the quantity Go is the quantum of thermal conductance[12]: it represents the maximum value of energy transported per phonon mode[11]. The thermal conductance quantization is analogous to the electric conductance

2 quantization in terms of electron conductance quantum 2e / h of one-dimensional conductors[47, 49]. In fact, when a dielectric sample is heated, various phonons are generated. But in the pro cess of phonon transport at the nanoscale, heat cannot be carried by phonons of just any wavelength in a confined geometry[4]. Generally speaking, phonon with the longest wavelength has the lowest phonon energy. At lower Chapter 1. Introduction 22

temperature, the phonons with the lowest energy have more contribution to thermal transport[12]. For smaller samples, this longest wavelength gets shorter[62]. Thus the lowest energy increases as the sample becomes smaller. When the sample is so small that this lowest phonon energy exceeds the thermal energy, kBT, quantum behavior of the phon on motion can be expected[63]. When we connect heat sources with various phonons to an extremely narrow (nanometre-scale) bridge, i.e., a ballistic channel, only specific phonon modes can couple into the narrow bridge due to the narrow lateral confinement. Because only the specific modes are allowed, the thermal energy transport spectrum is quantized in these nanobridges. Therefore, dimensionality plays an important role when quantized phonon spectrum affect the fiow of heat. In an one dimension al system, due to confinement in transverse dimensions, phonons are free to move only in the longitudinal direction within the allowed modes. The experimental data of Roukes et.al.[ll] confirmed this physical picture. In the experimental data of Roukes et.al.[ll], it is rather surprising that phonon transport shows a similar quantized behavior as that of electron transport. The reason for this surprise is that an electric current transports charge which has mass, whereas a heat current is carried by transporting phonons without mass. In the next section, we discuss a theory provided by the work of Kirczenow et.al.[12] which explains why this is possible.

1.3 Two probe formula of thermal transport f>arallel to the experimental studies, there have been several theoretical investiga tions of confined phonons and their contributions to thermal conductance [12, 23, 24]. Kirczenow and his student[12] theoretically predicted that dielectric quantum wires should show quantized thermal conductance at low temperature in the ballistic phonon regime. The fundamental quantum of thermal conductance, also called "the Chapter 1. Introduction 23

universal quantum", was predicted to be Go mentioned ab ove , which was verified by the experiment[ll] of Roukes et. al .. The word "universal" means that Go do es not depend on peculiarities of particle statistics, whether or not the heat energy carriers are fermions, bosons, or even anyons[64]. Kirczenow's prediction therefore provided one of the most important insights to thermal transport in nanoscale structures. On the technical side, because the thermal conductance quantum Go plays an analo go us role as the conductance quantum for charge transport[65, 66], one can deal with thermal transport with a very similar theoretical method as that used in electron transport[67], namely using the transmission function approach discussed previously. The theoretical starting point to study thermal conductance quantum is based on the scattering theory just as solving the electron transport problem[12, 68, 69, 70, 71]. We assume that the thermal phonon wavelength at low temperature becomes much larger than the lattice constant. The net heat current carried between two thermal reservoirs of phonons can be written as, using Eq.(1.31)

(1.36) where wa(k) is the dispersion relation for phonon branch a, va(k) is the group veloc ity, nhot, ncold are the Bose-Einstein distributions, Eq.(1.6), for the hot/cold thermal reservoirs, and Ta(k) is the phonon transmission coefficient between these reservoirs. Same as that of the electron case, when we replace the wave vector k with the fre quency w in the above integral, the group velocity cancels the ID density of states, and Eq.(1.36) becomes (a derivation will be given in Chapter 2)

00 Q. = -2 1 L 1 dwhN.J(nhot(w) - ncold(w))Ta(w) (1.37) 7f a w",(k=O)

We consider low temperatures and the ballistic regime in which the phonon mean free path is assumed to be far greater than the dimensions of the sample. Then, the thermal conductance is controlled by phonon scattering at the surface between Chapter 1. Introduction 24 the sam pIe and the heat source. Although we cannot directly apply the local Fourier law[14, 72], Eq.(1.I), because for our case the thermal conductance G is no longer just an energy independent quantity, it is still possible to define a thermal conductance for a wire as the ratio of heat flux through the wire and the temperature difference between the reservoirs connected to the wire. Therefore, we define G - Qj ~T[l1, 12,

23]. In the limit of linear response, ~T « T, G depends solely upon the threshold energy wa(k = 0)[11, 12], transmission coefficient Ta(w), and sorne basic constants. It is easy to obtain, from Eq.(1.37),

(1.38) where we assumed a very small difference between temperatures of the two reservoirs, and X a - 1iwa(k = O)jkBT. For ideal coupling between the ballistic thermal conductor (the wire) and the reservoirs in Fig.1.l, there would be no phonon wave reflection at the contacts. Fur thermore, we assume that there is no phonon scattering inside the conductor. Then each phonon mode can transmit through the conductor with unity transmission co efficient Ta = 1 for aIl modes Œ. We then obtain, from Eq.(1.38) and for the lowest energy mode, i.e., the "massless" mode with wa(k = 0) = 0,

2 2 1f k T G G - B = 0 = 3h

This expression is not related to any material parameter and is therefore "universal" (further discussion of the universality will come below). The quantum of thermal conductance, Go = (9.456 x 1O- 13Wj K2)T, represents the value of energy trans ported per phonon mode in the ideal case. If scattering of phonons o ccurs , thermal conductance will be less than Go.

When kBT ~ 1iwa, i.e., no mode is excited except for the lowest energy mode which has four branches arising from one longitudinal (dilatational), one torsional, and two Chapter 1. Introduction 25

transverse (flexural) vibrations [73] , a thermal conductance of 4Go is expected. This has indeed been confirmed by experimental data[ll, 74] and details of every mode for phonon transport can be worked out. On the other hand, because other modes rather than the lowest energy modes have higher cutoff frequencies, they can only be excited and have contribution to phonon transport when temperature reaches to kBT 2: nwa(k = 0). Further discussion about temperature dependence of the thermal conductance will be made below. In Roukes et. al. experiment[ll], the observation of universal quantum of thermal conductance was certainly an important discovery, but there are still sorne issues which need to be discussed further. In Fig.1.4, above rv 1K, the thermal conductance rises closely following a cubic power law. This is because the high energy modes, i.e., those with nonzero cutoff frequencies, are now excited and contribute to thermal conductance. The universal conductance could be observed at temperatures below rv 0.08K. In between the temperature range of 0.1 - OAK, the thermal conduc tance unexpectedly decreases below the universal value. Sorne people suggested that conductance decrease was caused by scattering due to rough surfaces[25, 26, 27]. For higher energy modes in the ideal ballistic conductor case, thermal conductance shows a dependence on the intrinsic properties of the material and on the geomet rical parameters of the sample. Different from low energy modes, the higher energy modes have nonzero threshold energies (nwi(k) Ik=O# 0) and they strongly depend on material details, the geometrical size, and also particle statistics [64]. With the increase of temperature by which higher energy modes are excited, thermal conduc tance rises over the universal quantum and G becomes non-universal. We also note from Fig.1.4 that at higher temperatures the thermal conductance rises smoothly without any further "quantization". This is different from charge transport in which conductance quantization persists to high energy and a step-wise increase of charge conductance is often observed[65]. This difference is due to the nature of carriers, Chapter 1. Introduction 26 phonons are Bosons while electrons are Fermions. For phonons aIl energy above the threshold energy contributes to thermal conductance, see Eq.(1.37); while for elec trons the conductance is measured at the Fermi energy. PracticaIly, one may estimate that thermal conductance results from aIl phonons with energies up to nw '" kBT, it is this wide energy range which smears out any complex oscillations in the phonon transmission coefficient 'Ta (see Eq.(1.37)) so that G(T) becomes a smooth function of temperature T: no further quantum is expected after Go is exceeded.

1.4 Summary of this thesis

80 far we have introduced the quantum phenomena of thermal conductance. For ideal cases of one dimension ballistic dielectric systems, we could assume unity transmis sion coefficient. This leads to the derivation of the universal thermal conductance quantum[12] in a two-probe device. In general, the transmission coefficient is not equal to unity due to imperfect coupling between the conductor and the heat reser voirs, due to scattering caused by surface roughness, and due to elastic scattering of the phonons by defects. For these situations one needs to compute the transmission coefficient. Furthermore, the experimental setup of Roukes et. al. [11] was actually four-probe, see Fig. 1.3. What are the effects of phonon wave quantum interference to thermal conductance? What is the proper theory for four-probe thermal conductance in the quantum regime? Are there any universal behavior in a four-probe setup? What are the low temperature limits in the four-probe situations? These are the questions studied in this thesis. The following Chapters are organized as:

• In Chapter 2, we begin by reviewing heat conduction from the point of view of lattice dynamics and continuum medium theory. These analyses give an illustration as how thermal energy transport is treated classicaIly, using, for Chapter 1. Introduction 27

example, the molecular dynamics simulations. By using the continuum theory for phonon waves, we present a detailed derivation of the Landauer-Büttiker expression of thermal energy flux, Q, which will be used throughout the work. In particular, a general multi-probe formula will be derived for this quantity.

• In Chapter 3, thermal transport properties in a special ballistic dielectric sys tem, the T -shaped wire, will be studied. The phonon transmission coefficient exhibits complicated oscillation behavior due to quantum interference at the scattering region. In the low temperature limit, thermal conductance tends to the universal quantum even though there exists strong wave scattering in this wire.

• In Chapter 4, a four-terminal thermal conductance formula for mesoscopic di electric system with arbitrary central scattering region is presented. The for mula is then applied to a particular four-terminal device without spatial symme try, and we analyze the four-terminal conductance as a function of temperature and geometry factors. Distinctly different results are obtained at the low tem perature limit from that of the two-terminal devices.

• In Chapter 5, we discuss a purely theoretical issue of thermal conduction by car riers satisfying the fractional exclusion statistics. We discuss on whether or not there exists any universal behavior for the four-terminal thermal conductance in the low temperature limit, in general terms of the device system.

• Chapter 6 presents a short summary of this thesis. 2

Theoretical Formalism

2.1 Introduction

As discussed in the last chapter, the thermal phenomena in a crystal can be described by a set of parameters including heat capacity, thermal expansion coefficients, and thermal conductivity. These parameters can be calculated by considering vibrations of atoms around their equilibrium lattice sites[8, 75]. For small vibration amplitudes, these vibrations can be cast into normal modes which represent the collective ex citation of the crystal. Quantizing these modes, we obtain the notion of phonons. Phonon wave propagation leads to heat energy flow. In this chapter we will give the theoretical pro cess of quantizing the classical equation of motion, and deriving the formula of heat flux, Eq.(1.37) of Chapter 1, for a general multi-probe mesoscopic system in terms of a phonon transmission coefficient. The multi-probe system is schematically shown in Fig.2.5 which is motivated by the experimental device of Roukes et. al. [ll] (that device was shown in Fig.1.3). Our derivation follows Ref. [24] which derived a two-probe formula, but the multi-probe derivation is more complicated. This derivation sets the necessary tool for specifie calculations of heat transport to be presented in subsequent chapters. Before we derive the quantum formula, however, we will follow the presentation of Chapter 1 by presenting a simple example of heat transport in the classical regime.

28 Chapter 2. Theoretical Formalism 29

This example is an MD calculation of an ID atomic lattice of Moorse potential, where heat flow is driven by a temperature gradient. The purpose of this example to show the drastic difference of heat transport in classical and quantum regimes. This chapter is organized as follows. In section 2.2, we discuss some general knowl edge about important phonon scattering processes which contribute to thermal energy transport. In section 2.3, we give a simple classical MD simulation of thermal trans port for an ID atomic chain. In section 2.4 we derive the multi-probe formula for thermal flux in the quantum regime of thermal energy transport.

2.2 Phonon scattering for thermal transport

As discussed in Chapter 1, phonons play very important roles for thermal phenomena in solids. Phonons are quantized form of crystal lattice vibrations. To deal with the lattice vibration, we expand the potential energy of the atoms in a Taylor series in powers of the lattice distortion. Using u~ to denote the displacement of the l-th atom from its equilibrium position in the a = x, y, z direction, then

(2.1) where the Einstein summation convention is used by which repeated indices are summed. The coefficients ~, ~~" ~~;'all, and ~~;;:::alll are the respective deriva- tives:

(2.2)

o is the potential energy of the crystal at zero temperature when all the atoms sit at their equilibrium positions. Here we assume that the displacement of each atom from its equilibrium position is small compared with the inter-atomic spacing. In Chapter 2. Theoretical Formalism 30

equilibrium, potential of atoms should be minimum, hence the first-order term in (2.2) must vanish.

2.2.1 Harmonie crystal 3 If we neglect aU terms higher than the second-order in (2.1), the system is a "harmonic crystal". The Hamiltonian of a harmonic crystal is:

(2.3) where ù~ _ âu~/ et is the time-derivative which gives the vibration velocity. The ground state energy of such a harmonic crystal can be easily determined by finding the normal modes. The normal modes can be found by diagonalizing the Hamiltonian, i.e., by eliminating the cross terms of (2.3), as follows. Assuming the normal modes (or normal coordinates) are q(kj) and their momen tum conjugate are p(kj) , where k is the wave vector and j = 1,2,3 the polarization of the vibration. We write:

u~ = ç(la; kj)q(kj) , (2.4)

ù~ = ç(la; kj)p(kj) , (2.5) where ç is the transformation coefficient between the Cartesian coordinates and the normal coordinates. Again, Einstein summation convention is implicit. When equa tions (2.4) and (2.5) are substituted into (2.3), we obtain the restrictive relations for ç,

C(la; kj)ç(l'a'; kj) = Kbll'baa, ,

ç*(la; kj)ç(la; k'j') = Kbkk,bjj' , Chapter 2. Theoretical Formalism 31

where 6 is the Kronecker-6 function, and K is the reciprocal lattice vector. Using (2.4) and (2.5) to transform the Hamiltonian (2.3), we obtain a new expression for the Hamiltonian

2 w (kj) = tI>~~,Ç*(lalkj)ç(l'o/lk'j)6kl,-k ,

2 2 H = ~ t: (lp(kj)1 + w (kj)lq(kj) 12) , (2.6) where w(kj) are determined by the inter-atomic interactions. The u~ and ù~ are real quantities, but the normal coordinates q(kj) and momenta p(kj) are complex. In (2.6), the argument of the summation is a "phonon" [10] whose angular frequency is w, and each phonon is characterized by a particular mode defined by k and j. The total energy of the system is the sum of phonon energies. Because different k, j do not couple in Eq.(2.6), these phonons do not "interact". For an N-atom harmonic crystal, there are 3N independent oscillators, i.e., the sum in Eq.(2.6) has 3N terms. Each term has the form of energy of a harmonic oscillator. According to quantum mechanics, the energy of a harmonic oscillator can have a set of discrete values:

[n(kj) + ~l1ïw(kj) , (2.7) where n(kj) is an integer occupation number of a phonon mode at a given energy level. A state of the entire crystal composed of N atoms is specified by giving the oc cupation number for the 3N normal modes. The total energy is then the sum of energies of the individu al normal modes in (2.7):

E = t:[n(kj ) + ~]1ïw(kj) , where k has N values, and j is 1, 2, and 3 respectively. Chapter 2. Theoretical Formalism 32

Because there is no interaction between phonon modes, the phonon occupation number cannot be changed for every allowed energy. Thus the state of a system can be described by how many phonons are there in each of the allowed energies. The representation of phonon state is therefore:

where ki , etc., are the allowed values of k for each of the branches j. This state specifies the occupation number of each of the allowed phonon states. If the state is in equilibrium, the mean occupation number of each kind of phonon with frequency Iiw(kj) w(kj), n(kj) = (e kBT _1)-1, must be a constant, so that all pro cesses must keep the phonon states unchanged. The conclusions above are also appropriate for the non-equilibrium state[3, 10]. Therefore, when heat flow is going through a harmonic crystal with a temperature difference at the two ends of the crystal, the distribution of phonons in crystal is established and remains unaltered in the course of time. By the definition of total thermal flux carried by all modes Eq. (1.13),

J = L n(k)1iw(k)v(k) , (2.8) k we see that the thermal current will remain undegraded. Hence we conclude that a perfectly harmonie crystal will have an infinite thermal conductivity. The macros copie consequence of an infinite thermal conductivity is that there can be no temperature gradient inside the crystal even if a temperature difference exists at the two ends of it.

2.2.2 Anharmonic crystal

It is clearly unphysical to have an infinite thermal conductivity. This means that the anharmonic terms are important and must be considered in order to obtain meaningful results. Anharmonic terms are also important when temperature is high so that Chapter 2. Theoretical Formalism 33

the amplitude of atomic vibration becomes large. Typically, the anharmonic terms are small compared with the harmonie term so that they can be considered as a perturbation. When anharmonic terms are considered, various phonon modes will be coupled and Eq.(2.6) will no longer hold. This means change of states are now possible with a definite set of phonon occupation numbers in the course of time, by the creation, an nihilation, or scattering ofphonons. In the cubic, quartic, or higher terms ofEq.(2.1), the cubic term is the most important for thermal conductivity except possibly at very high temperatures (in that case quantum vibration theory is invalid). We will mainly discuss the effect of cubic term in (2.1) on the thermal flux and conductivity. The cubic anharmonicity leads to three-phonon collisions which involve interac tions among three modes: the energy in modes (kI , WI) and (k2' W2) before collision may merge into a mode (k3 , W3) after interaction. The reverse process is also available. The processes could be illustrated according to the change of the phonon number in the corresponding mode (ki' Wi),

(a). The pro cess that a single phonon decays into two phonons:

initial state InI, n2, n3) --+ final state: InI + 1, n2 + 1, n3 - 1)

(b). The pro cess that two phonons merge into one phonon:

initial state: InI, n2, n3) --+ final state: InI - 1, n2 - 1, n3 + 1)

Here ni is the occupation number of phonon states at an allowed energy. During the three phonon pro cesses , energy must be conserved. Hence we have

Correspondingly, momentum also has a conservation relation,

(2.9) Chapter 2. Theoretical Formalism 34 where K is a reciprocal lattice vector. In phonon collisions there are two types of processes, the normal pro cess (N-process) for which K = 0; and the umclapp process

(U-process) for which K =1= 0, in (2.9). N-process

When K = 0, the phonon momentum Z=i n~ is conserved before and after the scattering. Therefore an N-process does not change the direction of heat flow and keeps the phonon distribution unchanged[3, 8, 10] from the initial condition. If the initial condition is equilibrium, since phonon distribution cannot change, Le., n(kj) = n( -kj), from (2.8) the total phonon wave vector z= kn(kj) = O. Thus the kj net thermal flow is zero. If the initial condition is non-equilibrium, since N-process does not change phonon distribution[3, 8, 10] in the sample, these N-processes will not provide any net change to the thermal energy flow which might have existed inside the initial condition of the system. This can be easily seen as follows. Let's assume, without losing generality, that all phonons have the same velo city in parallel to k and possess dispersion relation w = vk, so that (2.8) becomes J = nv 2 z= n(k)k. In k an N-process, each of the modes k l and k 2 contributes one phonon: the total wave vector z=n(k)k is reduced by k l + k 2 and equal to z=n(k)k - (kl + k 2 ). After the k k N-process collision, z=n(k)k will be increased by k3 and is equal to z=n(k)k + k3' k k 2 When (2.9) is satisfied, the net change of the thermal flow is nv (k3 - k l - k 2 ) = O. We therefore conclude that the N-processes do not provide thermal resistance to the thermal energy flow hence they do not contribute to the thermal conductivity. U-process

When the two original wave vectors k l and k 2 of Eq.(2.9) are large enough, the sum k l + k 2 falls beyond the edge of the first Brillouin zone. Because of the spatial periodicity of a crystal, any vector outside the first Brillouin zone can be represented by a corresponding vector inside the first zone with the help of a reciprocal lattice vector K =1= 0, i.e., the vector k3 of Eq.(2.9)[3, 8, 10]. After this lattice operation, Chapter 2. Theoretical Formalism 35

however, k3 will have a different direction from the vector k l + k2 . This means that a U-process can produce a thermal energy flow in the direction of vector k3 - which has a different direction. Such drastic change in k results in a tendency which can return any phonon distribution to that of the equilibrium. Hence U-processes can change phonon distribution inside the sample, which means they can pro duce a temperature gradients inside the sample if an external temperature bias is applied at the two ends of the sample. Because thermal conductivity is equal to thermal flux divided by temperature gradient (see Eq.(l.l)), it is the U-processes which give rise to thermal resistance of a sample. In a harmonic crystal, for there are no interactions between the phonons, the mean free path of a phonon is infinite. Hence thermal conductivity K, -t 00 according to Eq.(1.5). For anharmonic crystals, U-processes provide effective phonon scattering which gives rise to a finite phonon mean free path thereby leading to finite thermal conductivity. This finite mean free path is thus the mean distance between two successive U-processes.

2.3 Anharmonic one-dimensional chain

Having understood the basic phonon scattering pro cesses relevant to thermal trans port, in this section we discussed a classical simulation of an one-dimensional (ID) atomic chain and calculate thermal transport in this chain. Although the main focus of this thesis is the quantum transport of thermal energy, this short "classical" section is provided to give an idea on how calculation can be done in the classical regime. We consider an ID carbon atomic chain consisting of 60 atoms where the in teraction between the atoms is via an anharmonic Morse-potential in the following form[76]: VMorse = De ([1 - exp - f3(r - ro)]2 - 1), where De represents the depth of the potential weIl, ro is the equilibrium distance between carbon atoms, and f3 is an Chapter 2. Theoretical Formalism 36

200 • -- -- Reservoir • ---- Reservoir 0---- ChainAtcrn o ---- Chain Atom (3) 320 a, (6) 160 ~ q g " f:' • 1- 280 ~ ~ j j (5) 120 ·b (2) a, ëm ëm a. a. E E 240 m ~ • 1- 00 b (4) 200 am- (1) .. .. 40 160 0 5 10 15 20 25 30 35 40 45 50 55 60 0 5 10 15 20 25 30 35 40 45 50 55 60

AtomNo. AtomNo.

Figure 2.1: Temperature distribution along an anharmonic chain consisting of 60 carbon atoms is obtained by classical molecular dynamics simulation. A temperature bias f:::.T is applied at the two ends of the chain. (1) f:::.T = 65 - 55(K); (2) f:::.T = 120 - 100(K); (3) f:::.T = 170 - 150(K); (4) f:::.T = 210 - 190(K); (5) f:::.T = 260 - 240(K); (6) f:::.T = 320 - 300(K). The black circle represents the reservoir where f:::.T is applied. The empty circles are the atoms.

MD Simulation 150 Experiment 5 ~ E 100 .Q 4 ~ 50 >. :t= > U ~ 3 0 '0 50 100 150 200 250 300 350 400 c: 0 0 0 ü (ij 2 E Qi oC 0 1- 0

0 50 100 150 200 250 300

Temperature (K)

Figure 2.2: Temperature dependence of the thermal conductivity fi for the same chain in Fig.2.1 for temperature below 320K. The inset is the experimental results from 50K to 400K. Chapter 2. Theoretical Formalism 37

5 5

~ (4) 4 4

Q' (3) Q' i=' 3 • • 3 i=' ~ \ ~ :::l :::l ; ~ (2) ; a. 2 • 2 a. E E (1) CD t- t- (1)

• •\ Reservoir \ •0 Chain Atom • 0 0 0 5 10 15 20 25 30 35 40 45 50 55 60

Atom No.

Figure 2.3: Temperature distribution along the same chain in Fig.2.2 but at very low temperatures. (1) t::..T = 1.5 - O.5(K); (2) t::..T = 2.5 - 1.5(K); (3) t::..T = 3.5 - 2.5(K); (4) t::..T = 4.5 - 3.5(K).

"interaction range" parameter. A Nosé-Hoover thermostat[32, 39] is applied in the thermal reservoirs. For simplicity, the reservoirs are just atoms fixed in temperature, indicated by the black circles in Fig.2.1. The two reservoirs have different tempera tures, i.e., a temperature bias ~T is applied as the boundary condition. Other atoms inside the chain are free to move in the MD simulation, according to classical mechan ics discussed in Chapter 1. After a long MD simulation, the system reaches a steady state, and Fig.2.1 plots the steady state temperature distribution along the chain. A clear temperature gradient is observed. From the figure, a temperature discontinuity appears at the interface between the end atoms of the chain and the thermal reser voirs: this is caused by contact resistance at the interface. But in general, we obtain a temperature gradient along the chain. We measure thermal flux by Eq.(1.28) and compute thermal conductivity by Fourier's law Eq.(1.1), the result is plotted in Fig.2.2 as a function of temperature. It is interesting to plot experimental results of bulk carbon thermal conductivity[77] , seen in the inset. No real comparison can be made because what we computed is just Chapter 2. Theoretical Formalism 38 an artificial 1D chain of atoms. Nevertheless, the thermal conductivity of the chain is more than one order of magnitude lower than that of the bulk value, because of the small cross section of the chain. A similar result for silicon has been obtained in both theory and experiment[44, 78]. The classical simulation, such as the example presented here, has its limitations. This is because the U-processes occur at high temperature. When temperature is low ered to below about 20K, phonon-phonon interaction strength drops rapidly[8, 61]. At sufficiently low temperatures, :::; 1K, phonon-phonon interactions hardly exist and the mean free path can in princip le exceed sample dimensions so as to be determined by scattering of defects, dislocations, and surfaces of the sample. Furthermore, when the sam pIe size is reduced to that comparable with the phonon wavelength, ther mal transport enters the ballistic regime where thermal energy transport shows the quantization phenomenon as discussed in Chapter 1. Classically, the temperature gradient disappears at very low temperature, as seen in Fig.2.3, which signifies that the U-processes vanish. Typically, at low temperatures :::; 1K, the phonon wavelength can be '" 1J.km , hence it is not too difficult to have a system size L that is less than this scale. In the next section, we derive the necessary formula for analyzing thermal transport in this quantum regime in multi-probe samples such as that of Fig.2.4.

2.4 Heat flux in a multi-terminal system

When phonon wavelength is comparable to the system size, quantum interference becomes appreciable. In this section we will derive a multi-terminal formula for heat flux in this regime. Our derivation follows that of previous work by Blencowe[24] who has derived a two-terminal expression. For the multi-terminal case, the derivation becomes much more complicated but the physical spirit is similar. The multi-terminal formula will be used in subsequent chapters of this thesis. Chapter 2. Theoretical Formalism 39

Our model of a multi-terminal device has an arbitrary scattering region which is connected to the outside world by an arbitrary number of terminal wires, see Fig.2.4. Each terminal wire extends to far away where they are attached to thermal reservoirs whose distribution is assumed to be Bose-Einstein distribution, Eq.(1.6). The terminal wires are further assumed to be perfect so that no phonon scattering happens inside the wires. AlI scatterings occur in the central scattering region and at the interface between the terminal wires and the scattering region. In the foHowing discussions, phonon scattering is limited to elastic events. For convenience, a local coordinate system (x,y,z) is set up for each perfect terminal wire in Fig.2.4. To calculate the partition of thermal energy flux amongst the terminal wires, we proceed as foIlows[24]. In the next two subsections, starting from a classical wave equation of motion, we write down the energy flux expression for a terminal wire. We then formulate the vibration modes which we quantize to obtain a quantum formula of energy flux in terms of a transmission coefficient. Such a procedure naturaIly derives the formula Eq.(1.37) of Chapter 1, but here our formula is appropriate for multi terminal system as weIl. FinaIly, in subsection 2.4, we derive the universal quantum of thermal conductance.

2.4.1 Classical equation of motion Because we consider low temperatures, the phonon wavelengths in our system gen eraIly exceed several hundred angstroms. Therefore, aH microscopic length sc ales such as the lattice constants and distance of atomic interaction, become irrelevant for our derivation. A continuum medium approximation can therefore be adopted.

Under this approximation, the displacement field u = (ux , uy, u z ) can be safely used to describe the vibrational state. u satisfies the classical wave equation[79]:

(2.10) Chapter 2. Theoretical Formalism 40

Scattering Region

Figure 2.4: Schematic diagram of a general multi-terminal device. Each terminal wire is connected to a reservoir far away with different temperature T. The terminal wires are perfect crystals without any scattering. Phonon scattering occurs inside the center scattering region as weIl as at the contacts between the scattering region and the terminal wires. A local coordinate system on each terminal wire is set up for mathematical analysis. where subscripts i,j, l, m indicate x, y, z coordinates (i,j, l, m = x, y, z); pis the mass density, and C ij1m is the elastic modulus tensor. Again, summation over repeated indices is implicitly indicated. According to the theory of elasticity, an elastic wave must satisfy a set of boundary conditions[62, 80]:

OUm T ij = Cijlm~ , i,j,l,m = x,y,z, (2.11) UXI where Ti (n) is the stress in the i-direction on a surface of orientation n. The ni ne quantities T ij which define the stress state of a distorted medium, constitute a second rank tensor - the stress tensor. Note that Ti(n) and nj are the components ofvectors T(n) and n. In the present system, because the walls of terminal wires (see Fig.2.4) are just free surfaces without restriction for wave propagation, the walls have no stress: Ti (n) = 0 Chapter 2. Theoretical Formalism 41 on the walls. This means the displacement field on the walls may not vanish while its derivative is zero. Therefore the boundary condition for (2.11) is

OUm 1 = 0 . (2.12) OXI S

Eq.(2.1O) is a wave equation, Le., when thermal energy flux propagates in a perfect and infinitely long terminal wire with a constant cross section, the media displacement

Ui is a wave. The modes of Eqs. (2.10) and (2.12) has the form:

(2.13) where a labels the terminal wire, r - (x, y, z) is the local coordinate in terminal wire a, and k is the longitudinal wave vector along the x-axis (see Fig.2.4). In Eq.(2.13), index n labels mode of the wave, and 'l1~k (y, z) represents the transverse wave function of the n-th mode; w~k is the angular frequency of the n-th mode in terminal wire a. The solution Eq.(2.13) satisfies an orthonormality condition:

(2.14)

2.4.2 Phonon subbands 80 far we have written a general solution, Eq.(2.13), of the classical wave equation (2.10) in the terminal wire. We now determine the special solution for a rectangular shaped wire of uniform cross-section and the dispersion relation between frequency w and wave vector k. The cross-section of the wire has width w (y-direction) and height d (z-direction), and the wire has an isotropie and homogeneous elastic modulus C. Eq.(2.1O) becomes:

!)2 - ~ _ 2'02- - 0 fJt2 V v U - , (2.15) where v = (C/ p) 1/2 is the sound velocity in the wire. Chapter 2. Theoretical Formalism 42

Starting from a trial solution fi = ei(wt-kx-kyy-kzz), substituting into Eq.(2.15), we obtain

2 2 k2 k2)- W_ (k z u= 2 u , + y+ v and the transverse wave-function has general solutions:

w(y) A cos kyY + B sin kyY W(z) D cos kzz + Esinkzz where A, B , C, and D are constants. ky , kz are wave numbers with

(2.16)

Applying boundary conditions: 8w(y)j8y = 8w(z)j8z = 0, we obtain B = E = 0, and then w(y) = AcoskyY, w(z) = D cos kzz, and fi = Fei(wt-kx)w(y)w(z), where F is another constant, and

(p, q = 1,2,3· .. )

According to Eq.(2.16), we have

W2=V27f2 [(~)2 + (~)2l +v2k2 (2.17)

We define the right hand side of (2.17) to be

2 (k) - 2 2k2 Wpq = Wcut + V , (2.18) where

Eq.(2.18) is the dispersion relation of the phonon waves inside the wire. For an incoming phonon wave with energy nw, the energy has two parts inside the wire: the Chapter 2. Theoretical Formalism 43

threshold energy nWcut and the propagation energy nvk. The pair of integers (p, q) defines a "phonon subband" (or a mode as we have been saying before) so that its energy is also called subband energy. If values of p, q are too large, phonon momentum k becomes imaginary by Eq.(2.18), therefore these subbands cannot propagate along

the wire. For this reason, Wcut in Eq.(2.19) is called cutoff frequency: any incoming

wave with frequency W less than Wcut cannot propagate. On the other hand, for a given incoming energy, there may be several phonon subbands which can propagate along the wire. Finally, combining the last three equations, we have

(2.20)

2.4.3 Displacement field in terminal wires

We now analyze in more detail the displacement field, Eq.(2.13), for terminal wires. The wave solution Eq.(2.13) forms a basis set for our subsequent scattering analysis of phonon waves inside the scattering region for the multi-terminal device shown in Fig.2.4: a wave in the form of Eq.(2.13) is incident from a terminal wire, it scatters by the scattering region, and then partially transmitted to the other terminal wires. This allows us to derive the proper expression for the thermal energy current inside a terminal wire. Such an expression gives a quantitative description of thermal energy partition amongst the terminal wires. Inside terminal wire a, three kinds of waves exist as shown in Fig.2.5. They are the incident wave with wave vector k in mode1 n coming from the thermal reservoir a; the refl.ected wave with labels n', k' going back to reservoir a; and the transmitted wave with n", k" which originated from a different terminal wire ( e.g., the right terminal wire of Fig.2.5). Therefore, a wave function in terminal wire a with labels

1 We now use "mode" to indicate phonon subband, and use a single index n to indicate aH the subband indices. Chapter 2. Theoretical Formalism 44

n,k Incident Wave Scattering n', k' n''', k'" Reflecting Wave .. Region .. n", k" Transmitting Wave ...... ---

Figure 2.5: Schematic diagram of the incident wave, the reflected wave, and the transmitted wave in any terminal of a multi-terminal device. n, n', nif, and n'" are the mode indices. k is the wave vector for different modes. n, k, u~k(r, t), can be written in terms of the basis sets in Eq.(2.13), as:

u~,k(r, t) = ü~,k(r, t) + L Ü~',_k,(r, t)t~fn(w) + L ü~",_k,,(r, t)t~f,:',,(w) (2.21) n' nIf Here, the terms on the right hand side are the incident wave with unity amplitude, the refiected wave with refiection amplitude t~fn(w), and the transmitted wave with transmission amplitude t~7,:",(w), respectively. Due to scattering at the scattering region of the deviee (see Fig.2.5), mode mixing is possible for the refiected and trans mitted waves, therefore a summation over mode index is necessary in Eq.(2.21). We have assigned wave vectors k a positive sign for waves propagating toward the scat tering region and a negative sign otherwise. We further assume that wave scattering is completely elastic, and have in terms of (2.20)

(2.22)

Sinee there may exist many waves with different labels n coming from different terminal wires 0', the general wave function is a summation of aH of them:

q n L a~kü~,k(r, t) + L La~kü~',-k,(r, t)t~fn(w) n n n' + L L L a~;;;k/llÜ~",_k,,(r, t)t~f,:/II(w) , (2.23) q/ll n/ll nIf Chapter 2. Theoretical Formalism 45

where a~k is the weight coefficient of contributions made by each wave labeled by n, k. This wave function represents the total displacement field at time t and location r in an arbitrary terminal wire 0". Noting that we did not consider the summation of wave vector k in (2.23): that summation will be done later when we de rive thermal flux. FinaIly, for convenience of further derivations, we rewrite Eq.(2.23) into the fol lowing form by changing sorne dummy indices:

(2.24) n a n n' 2.4.4 A useful expression for transmission amplitudes Before we derive the expression for energy current in a multi-terminal device, the scattering relation between terminal wires needs to be examined. In particular, it is important to deduce the transmission coefficient which is related to the transmission amplitudes of Eq.(2.21). We make a further simplification by assuming that the thermal reservoirs are "reflectionless": namely any wave going into a reservoir along our ballistic terminal wire disappears inside the reservoir without any reflection back at the wire-reservoir interface. Indeed, we have no interest to what happens in the reservoir except that it maintains thermal equilibrium. Since the terminal wire is attached to the reservoir, the phonon distribution function in the wire is also Bose Einstein type from the reservoir aIl the way to sorne relaxation length away from the scattering region (see Fig.2.4). This model mimics exactly that of electron transport in mesoscopic conductors[49]. So far, the displacement field in Eq.(2.24) is a complex field, in the following we elect to work with a real displacement field V:

1 V = 2(u+ u*) , (2.25) where u* is the complex conjugate of u. Putting (2.24) into (2.25), we obtain va' in Chapter 2. Theoretical Formalism 46

terminal (J':

U a' "21 (a'(U r, t ) + U a'*( r, t ))

1 (~ a' - a' ( ) ~ ~ ~ a - a' ( ) a'a ( ) "2 L..JankUn,k r,t + L..JL..JL..JankUn',-k' r,t tn'n W n a n n' (2.26) ua' actually has the same physical meaning as u, and the only difference is that ua' is real and U is complex. For further derivations, we re-write ua' into two parts V(1) and U(2):

u(1) + U(2)

1 a' _a' ( ) a'* -a'* ( )) "2 (~~ ankUn,k r, t + ~ ank Un,k r, t (2.27)

~ (L L La~kii~:,_k,(r, t)t~:~(w) a n n'

(2.28)

With the above preparations we can now write down the classical energy flux according to the theory of elasticity[62, 80J. In particular, for any terminal wire, we have

the stress vector :

the energy flow vector :

where P is the energy flux density vector, Le., the Poynting vector. The vector expresses that when an elastic wave propagates, energy is transported cross the unit area per unit time; its direction is the direction of energy transport. Therefore the Chapter 2. Theoretical Formalism 47 classical energy flux Q for the displacement field U is obtained by integrating the Poynting vector:

Q(x, t)

(2.29) where the integral is over the cross sectional surface A at a given location x (as shown in Fig.2.4, we have assumed that energy current flows along the local x direction). We now substitute (2.26) into (2.29), divide the energy current Q(x, t) into two parts Q(l)(X,t), Q(2)(x,t) corresponding to Eqs.(2.27,2.28), and taking statistical av erage, we have

(QU') = (Q(1)(x,t)) + (Q(2)(X,t)) (2.30)

The term Q(l)(X, t) in Eq.(2.30) is

(2.31 ) where we have applied Eq.(2.27). We now exchange indices of the second term in the right si de of Eq.(2.31): (n, k) ~ (n', k'); use energy conservation (2.22): w~k =

W~;k' = w; and apply an orthogonality relation derived in Ref.[24] which also follows from Eq.(2.1O):

. c d d (- U ~ - u* - u* ~ - u) u u ~ ~7r xjlm jA Y Z Un,k,jU[Un',k',m - Un',k',jU[Un,k,m = PWn,kVn,kunn' , (2.32) where V~ ,k = Bw~ ,k/ Bk is the group velo city of the wave. This procedure gives Chapter 2. Theoretical Formalism 48

"" 1 (J" 12 (J" = L...J ank -4pw Vnk . n 7r Using a similar procedure, the second term Q(2) (x, t) in Eq.(2.30) can also be obtained

More details of these derivations of (Q(1) (x, t)) and (Q(2) (x, t)) can be found in Appendix A.1. Putting aU things together, the statistical average of the energy flux QO" can be written as

(2.33)

This is one of the main expressions which we will use below. Before moving further, we derive an useful result concerning the transmission amplitudes. Because total energy must be conserved (no energy source inside the scattering region and the terminal wires, see Fig.2.5), we must have L (QO") = o. (J" Renee, from Eq.(2.33) we obtain 2 2 ""L...J "" L...J 1ank (J" 1 -4pw2 vnk (J" = ""L...J ""L...J ""L...J "" L...J "" L...J "" L...J -4pw a 0'n k a 0'1nlkl * tn(J" 'n0' t0"0'1n'nl *vn'k' (J" . (2.34) (J" n 7r (J" 0' n n' 0'1 ni 7r To simplify notation, we write Eq.(2.34) into a matrix form by defining vectors and matrices a, v, and t: Chapter 2. Theoretical Formalism 49

o o o o o o

u tU tU t 00 01 02 u tU tU t 10 11 12 t=(t~;~) = tM tM tM

The relation (2.34) is re-written as:

where a t is conjugate of the a. Sinee coefficients a= (a~k) is arbitrary, the above equation gives

ttvt = v . (2.35)

Then (2.35) can be written in the component form:

(2.36) n",u" It is not difficult to verify that for a two-terminal device, (2.36) leads to what is derived by Blencowe (see Eq.(8)(9) and (10) in Blencowe's paper[24]). From (2.36), one can further obtain:

0''' vn"k" 0'''0' ~ ~ -u-tn"n = Unn'Uuu' (2.37) (f" ,n" vnk Chapter 2. Theoretical Formalism 50

"II Hence the matrix vn~kll t~::~ in (2.37) is a Hermitian matrix, with Vnk

(2.38) and then (2.39)

In the process of deriving Eqs. (2.38) and (2.39), we changed sorne dummy indices of variables. Eq.(2.39) will be used in the quantization pro cess of the next subsection.

2.4.5 Quantization of energy flux

In the last subsection, we derived the classical thermal flux Eq.(2.33), which is rep resented by a sum of propagating waves. Here we will quantize the classical flux using the important relations Eqs.(2.14, 2.22, 2.32, 2.39). In the following deriva tions we only present the important steps and leave a more detailed derivation in Appendix A.2. To quantize the thermal flux in a terminal wire, we write the displacement field into its operator form following well known procedures of quantizing a classical oscillator[lO, 24, 81]:

Û(r, t) = L (>0 dkV2 li a [â~ku~,k(r, t) + â~lu~~k(r, t)] , (2.40) a,n Jo PWnk which is analogous to Eq.(2.25). â~k and â~l are phonon annihilation and creation operators, they satisfy the following commutation relation:

(2.41)

For phonons, creation/annihilation operators have the following expectation value

(2.42) Chapter 2. Theoretical Formalism 51 where the angular bracket, ( ...), denotes statistical average; no- is just the Bose

Einstein distribution function (1.6) for terminal wire (J. We now write the energy flux operator in terms of (2.29):

(2.43) and we quantize this expression.

To be specifie, consider a terminal wire (J'. The field operator Û(r, t) in (2.40) can be expanded in the basis u~:k(r, t) using (2.24). Referring to (2.26), we have

(2.44)

In order to obtain energy flux in terminal (JI, we take an expectation value to

A 0-' Q (x, t)

(2.45)

Substituting the field operator (2.44) into the energy current operator (2.45), we obtain energy flux in terminal wire Cl' expressed by the basis u~~(r, t). Because the derivation is rather tedious, we put the details into Appendix A.2. Here we only give an outline. We start by dealing with the first term on the right side of Eq.(2.45),

\ -Cxjlm l dydzat Ûj' al Û~)

00 00 ( -Cxjlm 1dydz [I: I: 1 dk 1 dkl---;===:=~,=0-'::== A n nl 0 0 2p WnkWnlkl Chapter 2. Theoretical Formalism 52

(2.46)

In terms of (2.41) and (2.42), after considering that the group velo city v~~ = 8w~~/8k is canceled by the ID density of states g(w~~) = 8k/8w~~, (2.46) is transformed to:

( -Cxilm LdYdz8tÛj'8lÛ~)

00 t;. 0" d ft 0" Vn'k' 0"0' 0"0'* [ (0") (0' )] "L.J 1,,' W 27r Wn'k,-:;;;-tn'ntn'n nu' Wn'k' - nu Wnk u,n,n' Wn'O nk 00 +'1,'C xjlm L 1 dk 1d Y d Z-2fi UnkJ'-0" alU -0"* n 0 A P' , nkm , ,

00 'C dk d d fi -0" -0"* 0"0' 0"0'* +'1, xJ'lm L 1 1 Y Z-2 Un' -k'J'VIUn'~ -k' mtn'ntn'n (2.47) , , 0 A P" l' l' 1 u,n,n ,n1 In (2.47), a lower limit with some special value appears in the first integral due to a change between wave vector k and vibrational frequency w. Next, we consider the second term in (2.45) which can be manipulated in a similar way:

00 -'1,'C xJ'lm L 10 dk 1 dY d Z-2fi Un-0"* k J'V/Un~ -0" k m n 0 A P' , , ,

00 'C dk d d fi -0"* -0" 0"0' 0"0'* -'1, xJ'lm L 10 1 Y Z-2 Un' -k' J' alU n, -k'mtn'ntn'n (2.48) , , 0 A P l' l' " 1 u,n,n ,n1 Combining the relations (2.47) with (2.48), using Eq. (2.39), we obtain the energy AU') flux (Q in terminal wire a' as: Chapter 2. Theoretical Formalism 53

Letting: (2.50)

where T;': (w) is defined as the transmission coefficient. It represents the probability that a phonon at mode n in terminal wire a transmits to mode n'in terminal wire a'. w~:o, the lower limit of the integration, is the cutoff frequency discussed in the

last subsection. Because w = W~k = W~:k' = W~::kll, we combine (2.49) with (2.50) to obtain the following energy flux formula for a multi-terminal device:

(2.51)

Expression (2.51) is a main result of this chapter and this thesis, which we will use for specific calculations in the rest of this work. In (2.51), nu'(w), nu(w) are Bose-Einstein distribution of phonons in different reservoirs, respectively: nu(w) = /iw (e~ _1)-1. Here Tu is the temperature of reservoir a and kB is the Boltzmann constant. Eq.(2.51) has a very clear physical meaning: the term with nu' is the transmission from reservoir a' into the scattering region via terminal wire a'; the term with na is the transmission from aH other reservoirs a to terminal wire a' - which flows in the negative direction. Therefore, the thermal energy flux in a terminal wire equals the net transmission inside that wire.

2.4.6 Universal quantum of thermal conductance Using the expression of thermal flux for multi-terminal devices, Eq.(2.51), we now investigate an interesting quantum phenomenon of thermal conductance [11 , 12, 23, 24], namely its universal quantum in two-terminal systems.

A two-terminal formula is obtained by setting a, a' = 1,2 in Eq.(2.51). We assume

that Tl and T2 are the corresponding thermal reservoir temperatures. The tempera

ture difference of two thermal reservoirs is D..T = T2 - Tl' The thermal conductance Chapter 2. Theoretical Formalism 54

Gis obtained from Eq.(2.51)[72], as:

(2.52)

Consider linear response, Le., temperature difference between reservoirs !J.T « T, Eq.(2.52) becomes

(2.53)

Note that Eq.(2.53) will be used in the following chapters. In this result, there is a lower integration limit that is given by the cutoff frequency. In general, the cutoff frequency has to do with the geometry of the structure (width of the terminal wire, for instance) and is not equal to zero, see Eq.(2.19). A special mode, however, has a zero cutoff wnk(k = 0) = O. This mode is called "massless". It cornes from the lowest vibrational state n = 0 (or p = q = 0 in Eq.(2.19)). The massless mode is assumed to be excited first by any external perturbation such as the application of a temperature gradient at the sample boundary. This is a reasonable assumption and hence the massless mode is the most important mode involved in thermal transport. The massless mode gives rise to the universal thermal quantum. For a perfect wire of thermal conductor, the transmission coefficient of every pos sible propagating mode is unity because there is no scattering of phonon waves:

L T;,; (w) = 1. To observe the contribution of the massless mode, we separate the n integral in Eq.(2.52) into two parts[12], Chapter 2. Theoretical Formalism 55

(2.54) where a is the index of massless modes, (3 is the index of higher energy modes. Non Nf3 are the total number of these modes, respectively. In Eq.(2.54), the first integral is due to the massless mode and the second is due to higher energy modes. Normally, higher energy modes need higher temperature to activate. Hence if temperature is controlled to be very low in an experiment, say, 1ïwf3o ~ kBT, only the massless modes can have any appreciable contribution to thermal transport. For a typical dielectric medium, this amounts to roughly T < lK which can be realized in many laboratories. From now on we assume this is the case and therefore neglect the second integral of (2.54). Eq.(2.54) becomes

l G = ~ ~ {OO dw1ïw-- ( I _ I ) 27r ~ Jo !:l.T e1iW/kBT2 - I e1iW/kBTl - 1 0: k~7r2 (Tl + T2) N 3h 2 0: k2 2 ~TN (2.55) 3h 0: where T is the average temperature between reservoirs; T = Tl !T2. The detailed processes of obtaining Eq.(2.55) are seen in Appendix A.3. In this expression, the reduced thermal conductance G /T is equal to the product of No: and an universal constant 7r2k~/3h = 9.456 x 1O-13W/ K 2. Because of No: is a positive integer, the thermal conductance produced by massless phonon modes in a perfect ID wire is therefore quantized in terms of a fundamental quantum 7r2k~T/3h. This conclusion has been confirmed by experiments[ll], shown in Fig.1.4 of Chapter 1. In particular, experiment data shows the universal thermal quantum at temperatures below about O.08K. For higher temperatures above IK, the well known cubic power-Iaw behavior was observed as modes with nonzero cutoff frequencies become excited and contribute to thermal transport. About IK, one therefore must consider the second term of Chapter 2. Theoretical Formalism 56

Eq.(2.54).

2.5 Summary

In this chapter we have briefly reviewed the theory of lattice vibration whieh can be used to describe thermal transport in the diffusive regime where harmonie and anhar monie effects to thermal conductivity was the focus of discussion. For an anharmonic crystal, the N-process and U-process have different contributions to thermal conduc tivity: an N-process does not produce thermal resistance and therefore has no direct influence on thermal flux. An U-process, on the other hand, is the source of thermal resistance which is essential in changing the thermal distribution and helping the relaxation dynamics to reach equilibrium state from non-equilibrium. As an exam pIe, we carried out a simple ID molecular dynamics simulation to thermal transport, showing how classical thermal transport can be studied by such a numerical proce dure. Importantly, in the classical regime, thermal transport is measured by a local quantity called thermal conductivity. The main focus of our discussion was on phonon transport in the mesoscopic regime where the wave phenomena must be considered: due to small sample size compared with thermal phonon wavelength, thermal current must be described in the form of wave propagation instead of diffusion. We adopted the Landauer the ory of electric conductance for our analysis, namely using a transmission function to account for thermal transport. Within this theory, we view thermal transport as a scattering process of phonon waves. Here, the experimentally relevant param eter is a global quantity, namely the thermal conductance: the classical notion of thermal conductivity becomes less useful. Following the derivation of two-terminal thermal conductance[12, 23, 24], we derived a multi-terminal energy current formula, Eq.(2.51), in terms of a transmission coefficient which must be evaluated by solving a Chapter 2. Theoretical Formalism 57

wave scattering problem specific for each samples. For a perfect ID wire without any scattering centers, our formula naturally deduces the universal quantum of thermal conductance[12]. Before moving on to the applications of our formula Eq.(2.51) and Eq.(2.52), we summarize sorne "jargon" which has been and will further be used frequently in this work.

• Vibrational mode. A vibrational mode has a frequency w, its quantized form is the phonon. In a quasi-ID wire at low temperature, there is a series of discrete vibration al modes due to the transverse size confinement by the wire walls: the corresponding energy is called phonon energy subbands with energy value nw. Every vibrational mode has a minimum frequency value, Le., the cutoff frequency. The cutoff frequency is determined by two characteristic sizes: the sam pIe size and phonon wavelength.

• Excitation. When a sam pIe is perturbed by a temperature difference applied to its boundary, sorne vibrational modes can be exciied which produces ther mal energy flow. Whether or not a vibrational mode is excited depends on the background temperature. We can estimate phonon occupation with the ~ Bose-Einstein distribution: n(w) = (e kBT - 1)-1 in terms of the background

temperature and the cutoff frequency. When nw » kBT, n(w) ---t o.

• Thermal reservoir. Thermal reservoir is an energy source which provides the energy for exciting vibrational modes. A reservoir is assumed to be infinitely large so that its temperature is fixed. The contact between a terminal wire and a reservoir is assumed to be "reflectionless", namely any phonon going into a reservoir sim ply disappears inside the reservoir without any reflection back to the wire and without any other effect. 3

Thermal Transport ln a T-Shaped Quantum Wire

3.1 Introduction

In the previous two chapters,, we introduced the concept and reviewed sorne recent investigations of thermal transport in the mesoscopic regime at low temperature. We derived an expression for thermal flux, Eq.(2.51), in terms of the phonon transmission coefficient; this expression will be used in the rest of the thesis. Of the many studies on low temperature thermal conductance of dielectric materials[ll, 15, 59, 74], a most interesting and important result is the notion of "universal" quantum of thermal 2 conductance in one-dimension al mesoscopic system, Go = 7r :J:T. If there are No modes with zero cutoff frequency[ll, 22, 25, 26], then the total thermal conductance becomes G = NoGo. The quantum Go was first predicted theoretically[12] and later verified experimentally[II]. In the previous theoretical analysis of mesoscopic thermal phenomena [12], one typically considered one-dimension al (ID) samples connected to two thermal reser voirs with Bose-Einstein distribution[23, 24, 72]. At low temperature, due to the large wavelength of phonons, a series of discrete phonon energy subbands or vibrational modes with sorne cutoff frequencies are formed in the one-dimensional sample due to transverse confinement, as we discussed in the last chapter. These vibrational modes are generally regarded as non-interacting, and they propagate freely inside the ID

58 Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 59 sample if there are no scattering centers. Because there is no interaction between the discrete vibrational modes, phonons transport ballistically in the ID sam pIe without energy loss and each mode contributes Go to the total thermal conductance G: a perfect sample has G = NoG o which is a finite number. Classically, as discussed in Chapter 2, a perfect sam pIe would have infinite thermal conductivity which is very different from the quantum situation here. The difference is that in the quantum regime, thermal energy is carried by a set of discrete subbands in the ID sample with a cutoff frequency. Similar to that of electric conductance in the mesoscopic regime[68, 82], for a perfect sample we can view the finite resistance for phonon prop agation as coming from the non-ideal connection between the thermal reservoir and the lD sam pIe. The "universal" thermal energy quantum Gois produced by the massless ph on on mode[12], its observation is only possible at low temperature « lK) as discussed in the last chapter. The essential physics goes as follows. Neglecting scattering between the ID sample and the reservoir, the phonon distribution in the sample is /iw same as that of the reservoir, i.e., a Bose-Einstein distribution n(w) = (ekBT - 1)-1. When temperature T -+ 0, only the massless mode Wcut ° contributes to the distribution [l2] and other modes cannot be excited. Such a massless mode gives Go by the theory of Chapter 2 (see Eqs.(2.54, 2.55)). For higher temperature, there is still a cutoff frequency nWcut somewhat greater than kBT: phonons with frequency higher than Wcut do not contribute to n(w). Nevertheless, when temperature is high enough, modes with non-zero cutoff frequency will open up and thus thermal conductance will increase to beyond the universal value[12, 23, 24]. Whether or not a mode with non zero cutoff frequency is excited not only depends on the environmental temperature, but also on the sam pIe geometry. In mesoscopic physics, the influence of geometrical structure was most clearly demonstrated in electron charge transport[67, 83, 84]. For example, electrons trans- Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 60 mitting through an ID wire with a side stub, Le., a T-shaped junction, has been inves tigated by several groups both experimentally and theoretically[51, 85, 86]. Charge transport through such a device reveals interesting features of electrical conductance such as the observed conductance plateaus, resonances, zero transmission at certain values of the side-stub length, etc. More recently, an investigation was carried out to investigate both charge and heat transport through a diffusive metallic wire cou pIed with a super-conducting wire, where the electric conductance and thermal conduc tance both varied with temperature[87]. In this chapter, we will investigate thermal transport in a T-shaped dielectric nanostructure schematically shown in Fig.3.1, in the low temperature mesoscopic regime, using the theoretical formalism of Chapter 2. Although electron transport and the electrical influence on thermal conductance in T-shaped metallic structure were studied in Ref. [51, 85, 86, 87], to the best of our knowledge, we are not aware of a theoretical investigation on phonon transport in a dielectric T -shaped wire. As we will show below, although the geometric structure of Fig.3.1 appears to be sim ple, rather complicated phonon transmission does appear due to interference effects. For instance, a slight change of the structure size will be possible to affect phonon propagation and thereby influence thermal transport. Our calculation is based on evaluating the thermal transport formula Eqs. (2.51) and (2.53). In order to deal with geometric scattering of the phonons in the T-shaped junction, we develop a mode-matching method for solving the wave equation (2.10).

3.2 Madel and formulation

In order to evaluate Eq.(2.53) for thermal conductance, we need to calculate the phonon transmission coefficient T(w) by solving a wave scattering problem. The T -shaped dielectric junction is considered to be a two-dimensional system in Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 61 the x-y plane, shown in Fig.3.l. The junction has two long and perfect leads with uniform width a, which join a rectangular stub with width band length L. We divide the structure into three regions: the left lead is region l, the right lead region III, and the scattering region is region II which includes the side-stub. The leads are connected to their thermal reservoirs with a temperature bias TL > TR , here

TL/ R is the temperature of the left (L) and right (R) reservoirs. The reservoirs are at thermal equilibrium with phonon distributions in the Bose-Einstein form, i.e., n(w) = nw (e "'ET -1) -1. We assume "refiectionless" contacts between the leads and the reservoirs: any waves going into a reservoir disappears inside the reservoir without being refiected back to the leads. This is a reasonable assumption because reservoirs are far away and we are not interested in them. Therefore, the phonon distribution inside the leads is the same as that of the related reservoirs. At low temperatures, phonon-phonon interactions can be safely neglected, i.e., we neglect interactions between different vibration al modes. This way, the leads act as phonon waveguides. Assuming that an incident wave cornes from the left lead (region 1), it then enters the scattering region II where it suffers elastic scattering due to the geometry, finally the wave partially transmits to the right lead (region III) and partially refiects back to the left lead. At low temperature, phonon wavelength is generally over a few hundred angstroms which can be greater than the width of the side-stub b and the width of the wire a. Such a wavelength is certainly much greater than any microscopie length such as the atomic bond length. Therefore, the scalar model of continuum medium theory can be used to describe the wave propagation. According to this theory, the displacement field u(x, y) satisfies a wave equation[62, 80] (see also Eq.(2.1O) of Chapter 2):

(3.1) where w is the frequency of the wave, v is the sound velo city. The medium surface Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 62

t t a l y II III a

Figure 3.1: Schematic diagram for the T shaped wire provides free boundary conditions so that

ou =0 on ' (3.2) where n is the unit vector perpendicular to the surface. To solve Eq.(3.1), we apply a mode-matching technique. The solutions in regions l, II, and III are written as follows:

00 ikmX iknX u(I) = \lIm(y)e + L r nm \lIn(y)e- , (3.3) n=O 00 ikaX ikax U(II) = L [Pam \li Ct (y )e + qCtm \li Ct(y )e- ] , (3.4) Ct=o 00 iknX u(III) = L tnm \lin (y)e , (3.5) n=O where m, 0:, and n are the mode index for waves in regions l, II, and III, respectively. rnm and tnm in Eqs.(3.3) and (3.5) are the reflection and transmission amplitudes which we wish to calculate. PCtm and qCtm in Eq.(3.4) are constants to be determined.

\li min (y) and \li Ct (y) are the wave functions in the y-direction in regions l, III and II, respectively. Because of the free boundary condition in our problem g~ = 0, \lImln(Y) Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 63

and 'lia (y) must have the following form:

m/mry wm/n(y) = - cos( ) , (3.6) I!a a This is different from the case of electron wave propagation for which the boundary condition is u = 0[24, 51]. In above equations, km, km and ka are the longitudinal wave vectors in different regions satisfying the dispersion relation Eq.(2.20):

(3.7) where Wm = m;v, Wn = n:v, and Wa = a~v are the cutoff frequency of modes m, n, and a. Again, integers m, n, a label the subbands of waves inside the wire. The

1 velo city is assumed to be v = 5000 m . sec- . To calculate the coefficients rnm , tnm , Pam, qam, we match the waves in different regions at the scattering region boundary. For phonon wave, the wave function and its derivative are continuous across boundaries, therefore we have

O~y~a ,

I ll u (x, y)lx=o = u (x, y)lx=o ' (3.8)

O~y~a, I ll I ll 8u (x, y) 1 = 8u (x, y) 1 8u (x, y) 1 8u (X'Y)1 . (3.9) 8x x=o 8x x=o 8x x=b 8x x=b a

Substituting wave functions (3.3)-(3.5) into Eqs.(3.8)-(3.1O), we obtain

00 00 I>lJa(y) [Pam + qaml = Wm(y) + L rnmWn(y) , O~y~a (3.11) a=O n=O 00 00 L ikawa(Y) [Pam - qaml = ikmWm(y) - L iknrnmWn(y) , O~y~a (3.12) a=O n=O 00 L ikaWa(Y) [Pam - qaml = 0 , a

00 onm + rnm = LAna [Pam + qaml , (3.17) a=O 00 kmAma - L knrnmAna = ka [Pam - qaml , (3.18) n=O 00 tnm = LAna [Pameikab + qame-ikab] , (3.19) a=O 00 "L....t t-nm k n A na = k a [ikabPame - qame -ikab] , (3.20) n=O a where Ana = Io w~(y)wa(y)dy, and tnm = tnmeiknb. From Eqs.(3.18) and (3.20), Pam and qam can be solved. We then substitute these quantities into Eqs.(3.17) and (3.19) to get the following equations:

00 [00 2k-1a k j 00 [00 e-ikab + ékabj_ L L AlaAna Ok b IOk b rlm + LOin + L AlaAnak~lkl Ok b Ok b tlm 1=0 a=O e-~ a - e~ a 1=0 a=O e-~ a - e~ a

(3.21) Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 65

(3.22)

(3.21) and (3.22) are a set of linear equations for quantities [lm and rlm which can be solved by standard matrix inversion. Afterwards the transmission and refiection coefficients can be expressed as

(3.23) n n L:0(W - wn)lrnmI2kn/km (3.24) n where function O(w - wn) is zero for W < Wn and is unity for W 2: Wn. Its existence sim ply refiects the nature of a waveguide: when the frequency of the incoming wave

W < Wn , in terms of the dispersion relation (3.7), the wave vector of the n-th mode is imaginary hence that mode cannot propagate. The transmission coefficient Tm(w) represents the probability that phonons transmit into the outgoing terminal when the incident energy is that of the m-th subband with total energy nw in the incoming terminal. Similarly, the refiection coefficient Rm(w) represents the probability that phonons are refiected into the incoming terminal. Quantities Tnm and Rnm represent the transmission coefficient and refiection coefficient between two single subbands, e.g., Tnm gives transmission probability from subband m in the incoming terminal wire to subband n in the outgoing terminal wire. Clearly, due to time reversaI invariance, if we exchange indices m +---t n, the transmission coefficient does not change.

3.3 Features of transmission coefficients: w ---t 0 limit

Fig.3.2-3.9 plots the transmission coefficient Tm(w) versus frequency (w - wm), for different lengths L of the side-stub in the T-shaped junction where we fixed the wire width a and side-stub width b. Here Wm is the cutoff of the m-th vibration al Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 66

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

Tm 0 0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 0 0.5 1.5 0 0.5 1.5

(O>-

Figure 3.2: Transmission coefficient Tm(w) versus the incident phonon frequency w for the side stub length L = 20nm. W m is the mth mode cutoff frequency. The horizontal axis is scaled by !:J.w == W m +l - Wm = 7raV where the velocity is fixed at v = 5000m/ sec. The geometrical parameters are: a = lOnm, b = lOnm. (a) the O-th mode case, e.g., the massless mode. (b) the Ist mode case. (c) the 2nd mode case. (d) the 3rd mode case.

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

Tm 0 0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0.5 1.5 0 0.5 1.5

(ro-rom)/~ro

Figure 3.3: Transmission coefficient Tm (w) versus the incident phonon frequency w for the side-stub length L = 25nm. Other parameters are the same as in Fig.3.2 Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 67

0.8 0.8

0.6 0.6

0.4 0.4

(a) (b) 0.2 0.2

Tm 0 0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 0 0.5 1.5 0 0.5 1.5

(Cl)-rom)1 ~ro

Figure 3.4: Transmission coefficient Tm(w) versus the incident phonon frequency w for the side-stub length L = 28nm. Other parameters are the same as in Fig.3.2

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

Tm 0 0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 0 0.5 1.5 0 0.5 1.5

(O>-O>m)1~ro

Figure 3.5: Transmission coefficient Tm(w) versus the incident phonon frequellcy w for the side-stub length L = 30nm. Other parameters are the same as in Fig.3.2 Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 68

0.8 0.8

0.6 0.6

0.4 0.4

(a) (b) 0.2 0.2

Tm 0 0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 0 0 0.5 1.5 0 0.5 1.5

(ro-rom)/~ro

Figure 3.6: Transmission coefficient Tm(w) versus the incident phonon frequency w for the side-stub length L = 35nm. Other parameters are the same as in Fig.3.2

0.8 0.8

0.6 0.6

0.4 0.4

(a) (b) 0.2 0.2

Tm 0 0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

( ro-rom)/~ro

Figure 3.7: Transmission coefficient Tm (w) versus the incident phonon frequency w for the side-stub length L = 40nm. Other parameters are the same as in Fig.3.2 Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 69

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

Tm 0 0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

0 1.5 0 (orro.)/Aco

Figure 3.8: Transmission coefficient Tm(w) versus the incident phonon frequency w for the side-stub length L = 43nm. Other parameters are the same as in Fig.3.2

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

Tm 0 0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

(co-com)/Aco

Figure 3.9: Transmission coefficient Tm(w) versus the incident phonon frequency w for the side-stub length L = 45nm. Other parameters are the same as in Fig.3.2 Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 70 mode in the incoming terminal-wire. AlI the transmission coefficients exhibit strong oscillations versus phonon energy 1ïw. Tm can reach unity for certain frequencies (energies), showing a clear resonance transmission behavior. For the O-th mode, Ta can dip to zero indicating anti-resonances where total reflection occurs. Ta also shows a good quasi-periodic variation versus energy 1ïw. Similar quasi-periodic behavior was reported for electron transport in T-shaped junctions[51, 85]. The behavior of other modes is much more irregular, in particular Tm f 0 for all m > O. The oscillation amplitude becomes smaIler as w increases, and Tm approaches unity for large w because high energy phonons can easily traverse from left lead to the right lead via the side-stub region (region II, see Fig.3.1). Interestingly, from Figs.3.2-3.9 we find that Tm(w) starts from either 0 or 1 at

(w - wm ) -7 O. In fact, we found that when the ratio of stub length L to the wire width a is an integer, Le., ~ = 1,2,3· . " Tm for aIl m starts with unity. Furthermore, when this ratio is a half integer, Le., ~ = ~'~"'" Tm for even m still starts with unity, but for odd m it starts with zero. When the ratio takes any irrational values, only Ta (for O-th mode) st arts from unity and aIl other Tm starts from zero. FinaIly, when the ratio is any other number, most of Tm st arts from zero and only a few starts from unity. Although these results are different from what reported before[12] (on a different system) where aIl Tm st arts from unity at w = 0, we can understand our results perfectly, as follows. As discussed in Chapter 2, for a given incoming phonon energy 1ïw, there can be sever al subbands inside the incoming terminal wire. If the incoming phonon wave is in a particular subband m, then it is most likely for this wave to exit to the out-going terminal in the same subband m because of the symmetry of the wave-functions. This is especiaIly true if the incoming wave energy is just above that of the subband m, Le., when (w - wm ) -70 so that according to (3.7) the wave's momentum k -7 O. Clearly, the scattering region can and will cause mixing of the subbands, namely Tnm f 0 for Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 71

m =1=- n, but the largest transmission coefficient is nevertheless Tmm.

ln the following discussion, we will therefore just focus on Tmm: the probability of an incoming wave in the incoming terminal in subband m to scattering into an out-going wave in the out-going terminal in the same subband. In this scattering process, the incoming wave couples to the waves inside the scattering region, and this coupling strength is proportional to the overlaps of the wave-functions in the two regions (regions 1 and II). More specifically, we will now assume that the terminal wires have the same width a (see Fig.3.1O). In Eqs.(3.3), (3.4), (3.5), and (3.6), the transverse wave function in the left lead and in the scattering region can be written as

l m7f'y u f'.J cos(--) , m a We can compute the overlap of these two wave-functions and we define[23] the cou pling strength as aam, where m, Œ are the quantum indices (subbands indices) in regions 1 and II, respectively. We have ra m7f' Œ7f' aam f'.J Jo COS (--;;:-y)cos( Ty)dy sin( a7ra + m7r) sin( Ci7ra _ !illr.) f'.J [ 2L 2 + 2L 2] a7ra + m7r a7ra _ m7r (3.25) 2L 2 2L 2 The same coupling strength can be written between regions II and III (see Fig.3.1O) because the out-going terminal has the same width a. When coupling strength aam takes a large value, we expect a large transmission from the incoming lead (region 1) through the scattering region, and out-going to the out-going lead. From Eq.(3.25), a maximum aam rv 1 is obtained when

L Œ Œ7f'a _ m7f') -70 i.e., - ==- (3.26) ( 2L 2 a m Therefore, the length ratio Lja determines the subband index ratio Œjm for large coupling strengths. The transmission amplitude tam, on the other hand, is propor- Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 72

2 tional to aam, henee the transmission coefficient Tam = Itaml2e "-J laaml • We hence conclude that Tam is large (order unity ) when ~ = ~.

--b-- (a) 0;=7

m=3 n=3

01.=5

01.=4 L m=2 n=2 01.=3 m=1 n=1 <0 a 1 yI n]1 III a .~ ri:)~ -l3 t .. . t x m=O n=O

(b) 01.=9 (c) 0;=10 ot=8 01.=9 m-3 ··· .. ···· .. ······u<=,········· .. ····· n-3 m-3 ·················W=8········ .. ··· .. · n-3

01.=6 01.=7 01.=6 0;=5 m-2 n=2 m=2 ········· .. ······a=5··· .. ······ .... · n=2 01.=4 ot=4 0;=3 m=1 n=1 m=1 01.=3 n=1 ·················(j,=2················ ...... ii;;:ï ...... 0;=1 0;=1 m=O 01.=0 n=O m=O ot=O n=O

Figure 3.10: Schematic of the subband threshold energy (cutoff frequency) for the leads and the stub region with different ratios: (a) ~ = 2; (b) ~ = 2.5; (c) ~ = 2.8. The left column is the subband energies of the left lead; the middle column is for the stub-region; and the right column is for the right lead. The T -shaped wire is also shown to remind the geometry.

First, let's discuss the situation when ~ = J is an integer. Then, from Eq.(3.26), we expect a large transmission when ~ = J. Because Ct is an integer, ~ = Jean indeed be satisfied for any m. Renee, for any subband m inside the left lead, there is always a subband Ct = m x J inside the stub-region. Fig.3.1O(a) gives a specifie Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 73

example of !:..a = 1 = 2. Here, for instance, when m = 1, there is an a = 2 so that ~ = 2; while for m = 3 there is an a = 6 so that ~ = 2; etc.. A similar discussion holds for the right lead. Therefore, for any integer ratio ~, the matching of subbands occurs and we expect large transmission coefficients at (w - wm ) -t o. This explains why in Figs.3.2,3.5, and 3.7, transmission coefficient Tm = 1 at (w - wm) -t 0 for integer ratios of ~.

Next, let's consider ~ = (21 + 1)/2, Le., a half-integer. Again, to get large transmission we require ~ = (21 + 1)/2 as well, which cannot be satisfied unless a = (21 + 1 )m/2. Since a is itself an integer, we conclude that large transmission oc curs for even m. This is illustrated in Fig.3.1O(b) where ~ = 5/2. Here, for instance, for m = 2 subband (even number), there is the a = 5 subband which matches. On the other hand, for m = 3 (odd number), no subbands in the stub-region can match the incoming wave, and Tm is small (actually zero). This explains why in Figs.3.3, 3.6, and 3.9 when ~ being a half-integer, transmission coefficient Tm = 1 at (w - wm) -t 0 for even m.

Then, when ~ equals to any irrational number, it is impossible to satisfy Eq.(3.26), hence the transmission is small. Finally, when ~ is neither an integer nor a half integer, but is a rational number p/ q, it is possible to satisfy ~ = ~. In this situ ation, incoming subband m = q can match the stub-subband a = p to give a large transmission. But for most values of m, Eq.(3.26) cannot be satisfied and those Tm are small.

Of course, the m = 0 mode (massless mode) is an exception: it can always match the a = 0 mode for all situations, i.e., the massless mode is not affected by the shape of device structure. Because the massless mode always has zero cutoff frequency, Eq.(3.26) is always satisfied and the matching situation is drawn in Fig.3.1O(a)-

3.1O(c). Hence, transmission coefficient To(w -t 0) -t 1, as shown in Fig.3.2-3.9. This is also different from electron transport. For electrons, all transmission coeffi- Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 74

cients start with zero (at zero energy). From Fig.3.2-3.9, we observe that the oscillation of Tm(w) intensifies but its am

plitude decreases as w is increased. Tm (w) ---+ 1 for large w. This is understandable since an incident wave with high energy hw can easily traverse the scattering region.

3.4 Features of transmission co eHicients: periodicity

In Fig.3.2-3.9, we observe that the m = 0 mode shows a quasi-periodic characteristic

in the frequency range (w-wm )/ /j.w < 1 (where /j.w = (Wm+l -wm ) is the difference of frequencies between neighboring modes), while it becomes rather irregular for higher frequencies. This behavior is shown more clearly by re-plotting1 aIl the 10 (w) of Figs.3.2-3.9. into Fig.3.11. This clearly shows that 1O(w) consists of two behaviors: periodicity and non-periodicity. In the periodic part, the period of the oscillation reduces when the stub length L increases. In the non-periodic part, transmission coefficient has an irregular oscillation. We now investigate these behaviors in more detail. The periodic behavior of 10 (w) at w/ /j.w < 1 can be understood as due to wave interference. For this small range of incoming phonon energy, the out-going phon ons must also stay at the massless mode. Then, when the incoming wave enters the stub region from left lead, it can excite modes inside the stub with indices a > 0 because the stub length Lis wider than the wire width (by Eq.(3.7), the wider the width, the closer the modes). These different modes will interfere causing a periodic behavior of Ta (w ). Afterwards, the waves will exit to the massless mode of the right lead. When L is increased further, the mode spacing of the stub-region becomes smaller, and more modes will be excited by the incoming wave, thereby reducing the transmission lNote, sinee Fig.3.U is for m = 0 modes for which Wo = 0, the horizontal axis is shawn simply as wj6.w. Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 75

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

T. 0 0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

~~~~~~~~~~~~~~~~~~~~~~~~~o 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

00/"'-00

Figure 3.11: Transmission coefficient Ta versus the incident phonon frequency w with different stub lengths L. The geometrical parameters are: a = lOnm, b = lOnm.(a) L = 20nm, (b) L = 25nm, (c) L = 28nm, (d) L = 50nm. periodicity. The period of the oscillation can be roughly estimated as aj(L - a). When energy of the incoming phonons is large, i.e., when w j fJ.w > 1, in the out going lead the modes with m > 0 can be excited. In other words, although the incoming mode is massless, the out-going modes can be "massive" (m > 0). This causes transmission coefficient Ta (w) to lose the regular periodic feature, as shown in Fig.3.11. Indeed, when more modes are excited in the right lead, according to Eq.(3.23) Ta(w) is a summation of many terms Tao(w), 11o(w), .... Adding many oscillatory functions results to an irregular behavior. ln the following, an example will be worked out to illustrate the details of the individu al mode contribution to Ta(w). In this example, the length ratio! = 3 is chosen. The mode matching situation (as discussed in the last section) at the stub lead interface is drawn in Fig.3.12. Ta(w) and its components Tao, 110, 720, 130 are plotted in Fig.3.13. Chapter 3. Thermal Transport in a T -Shaped Quantum Wire 76

Figure 3.12: Mode mat ching profile of the lead and the stub at the stub-lead interface. w is the incoming frequency and k is the wave vector. The length ratio ~ = 3.

As shown in Fig.3.l3, when w/!::"w < 1, Le., w 1, i.e., w > !::"w, 1i.o begins to contribute to Ta(w), as the dotted line in Fig.3.l3 shows. Because energy is conserved, when 1i.o becomes nonzero, Tao must reduce, and Ta(w) = Tao+1i.o. When w/!::,.w > 2,

Le., w > 2!::"w, 720 starts to contribute so that Ta(w) = Tao + 1i.o + 720' FinaIly, when w/!::"w > 3, 730 begins to contribute. For clarity, the individual mode-to-mode transmission coefficients are also separately shown in the four panels of Fig.3.l4. Figs. 3.13 and 3.14 clearly show that it is the summation of the individu al mode-to-mode transmission coefficients which produced the irregular behavior when w/!::,.w > 1. 80 far we have discussed the periodic features with frequency. It turns out that there is a periodic behavior against the stub-Iength L as weIl. 8uch a behavior shows even clearer the wave interference phenomenon. For our example system, Fig.3.l5 plots Ta versus L for four fixed frequencies below the lst cutoff frequency. A very reasonable periodic behavior is clearly seen. When w increases, the period of the oscillation decreases. Examining Fig.3.l5, it is evident that the period in T(L) is Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 77

0.8

0.6 To , 0.4 , , ~

0.2 ~ .:f\:, H I,.,.:ll\" ,..,. ~ 1: t!'c." i i If ',\ : \.. ,1! \J\, ,i "l.I"/\\ i f'=.!J ...... ~::\...... \ .: 0 0 2 3 4 ro/i1ro

Figure 3.13: Transmission coefficient Ta(w) and its components, Tao, 'lio,12o,and 130 versus the reduced frequency wl6.w. 6.w = Wm+1 - Wm = v;. Here, v = 5000mlsee, a = lOnm, b = lOnm, and L = 30nm. Solid Hne: total transmission coefficient Ta (w). Dashed Hne: Tao. For w1 6.w < 1, only Tao has contribution to the total transmission Ta, therefore the dashed Hne coincides with the soHd Hne. Dotted Hne: 'lio; long dashed line: 120; and dash-dot Hne: 130'

0.8 0.8 (b)

0.6 0.6

0.4 0.4

(a) 0.2 0.2

T no 0 0 2 3 4 0.8 0.8

(c) (d) 0.6 0.6

0.4 0.4

0.2 0.2

0 0 0 2 3 ro/Ôro

Figure 3.14: (a) Tao versus wl6.w. Tao is finite for any w. (b) 'lio versus wl6.w. 'lio becomes nonzero when wl6.w = 1. (c) 120 versus wl6.w. 120 becomes nonzero when wl6.w = 2. (d) 130 versus wl6.w. 130 becomes nonzero when w / 6.w = 3. Other parameters are the same as in Fig.3.13 Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 78

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2

To 0 0

0.8 0.8

0.6 0.6

0.4 0.4

0.2 0.2 (c) (dl

L-L,,(nm)

Figure 3.15: Transmission coefficient To(w) versus stub length L with fixed incident frequency. (a) ZL!!. (b) 27rV (c) W 37rV (d) 47rV b and w = 5a ' w = 5a ' = 5a ' w = 5a . a = = 10nm " Lü = lOnm v = 5000m/sec•

0.9 0.9

0.8 0.8

0.7 0.7

(a) (b)

T o 0.6 0.6

0.9 0.9

0.8 0.8

0.7 0.7 (c) (d)

L-Lo(nm)

Figure 3.16: Transmission coefficient To(w) versus stub length L with fixed incident frequency. (a) w = 6;av, (b) w = 7:av, (c) w = 8;:, (d) w = 9:av. a = b = 10nm, Lü = lOnm, and v = 500Om/sec. Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 79

roughly f where k = w j v. This clear periodic behavior is due to interference. When the incoming wave enters the stub-region, it partly goes into the stub and get reflected back at the far end of the stub. This reflected wave interferes with the incoming wave coherently, producing the periodic change of Ta as L is increased. Finally, in Fig.3.16 we plot waves having frequency higher than the lst cutoff, where T(L) loses the periodic features. This is, again, due to the summation of the mode-to-mode transmission coefficients as discussed above.

3.5 Thermal conductance

80 far we have discussed the transmission coefficient properties in the T -shaped di electric junction. In this section, based upon the above results, we investigate thermal conductance in the T -shaped junction. For convenience of discussion, we rewrite the formula of thermal flux (2.51) by which the energy flux Q from the left lead to the right lead can be expressed as:

(3.27) where n(L/R)(W) = (exp(k ;w )- 1)-1 is the Bose-Einstein distribution function B (L/R) of the phonons in the leftjright reservoir, and Wm is the cutoff frequency for the mode m in the leads. In the following, we assume that the temperature difference b..T = TL - TR is very small, therefore the thermal conductance can be obtained straightforwardly from Eq.(3.27):

. Q 00 dw dn(w) G = hm AT = L 1 -2 fiwTm(w)-dT (3.28) LlT-+O ~ m Wrn 1r

In terms of Eq.(3.28), we will study and discuss thermal conductance under the influence of external factors, e.g., temperature and structure geometry. We emphasize that Eq.(3.28) is a general result for any two-probe dielectric system. Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 80

6 0.8

0.6 ::2 5 Nt!2 .J' 0.4 N.!:- 4 ~ al 0.2 0 c 3 s0 :J -0 0.2 0.4 0.6 0.8 C 0 () 2 lii E al .<:: 1- 1

0 0 1 2 3 4 5 Temperature (K)

Figure 3.17: Thermal conductance G versus temperature T. Here a = lOnm, b = lOnm. Solid line: L = 30nm; dotted line: L = 50nm; dashed line: L = 70nm; long dashed line: L = 100nm; dot-dashed line: L = 120nm. Inset: thermal conductance G versus temperature for T < 1K.

First, we consider how does thermal conductance G change with temperature T. Since d2n(w)/dT2 > 0 for any w and T, and Tm(w) must certainly be non-negative, we conclude from Eq.(3.28) that d~fJ) ~ O. This means that the thermal conductance must be a monotonically increasing function of temperature for any two-probe devices. For our T-shaped junction example, Fig.3.17 confirms such a behavior. Although the transmission coefficient Tm(w) has many oscillations as we presented in the last section, these oscillations do not appear in thermal conductance. This is because of the wide energy integration range in Eq.(3.28) which smears out the oscillations in Tm(w). It is also worth noting that this monotonie behavior is different from that of the electric conductance, which does not have to change monotonically with both temperature and bias voltage[88]. NumericaIly, Fig.3.17 plots G as a function of temperature T for the T-shaped junction, for a fixed stub width b = lOnm and several values of stub length L as weIl as terminal wire width a. At higher temperatures, e.g., T > 2K, the thermal Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 81

1.2

;;:;- 1.1 N-'" { 1 ~ 1\ li lSc: al 0.9 f1\ t5 :l '1\ "0 ~ 1\ § 11\ () 0.8 ~ 1i ëii § Il'. \. Cl) ..c: 0.7 ~ t.,~::-- 0.6 1 2 3 4 5 ° Temperature (K)

Figure 3.18: Thermal conductance GfT versus temperature T. Here a = lOnm, b = 10nm. The soHd Hne: L = 30nm; dotted Hne: L = 50nm; dashed Hne: L = 70nm; long dashed Hne: L = lOOnm; dot-dashed Hne: L = 120nm.

2 conductance G is proportional to T . At low temperatures, e.g., T < O.lK, G is linear function of T. These results are similar to those of previous works on two probe devices[22]. In certain temperature range, e.g., near T t'V O.5K, G versus T may exhibit a T~ behavior (x > 1) where G increases very slowly with temperature. The change of stub length L has sorne effect on G in the range T < 1K, as shown in the inset of Fig.3.17, but this effect diminishes for T > 1K. As discussed previously, when T < 1K only the massless phonon mode can propagate in the lead of the T shaped wire. The change of L causes transmission coefficient to vary periodically for w < !:J..w, as seen in Fig.3.11. Perhaps this has led to the slight change in G. When T> 1K, although an irregular behavior appears in the transmission coefficients (see last section), G has litt le change when Lis varied. Again, it is the wide energy range involved in thermal conductance calculation which smears out aIl oscillations in the transmission coefficient. Can we observe the universal quantum, Go =1r2k~T/3h, of thermal conductance Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 82

in the T-shaped junction? To see this, we plot G/T versus temperature T in Fig.3.18 for sever al values of L. We found that G /T exhibits a non-monotonie behavior and has a minimum at T =1- O. Similar behavior has been observed in the experiments of Schwab et. al[ll]. Importantly, at zero temperature limit, our calculated G/T, for aIl values of L, tends to unit y in terms of the universal quantum Go, as Fig.3.18 shows. For such a low T, only the O-th mode is open for transport, and one can easily confirm that occupation number for the first mode at T = O.lK is merely

27 n(wcut) = 8.8 x 10- rv 0 (again, Wcut = 7raV). Hence, only the massless mode whose transmission Ta (w) rv 1, contributes to G. The transport is therefore in the universal regime as our calculations show. The univers al quantum obtained at T -t 0 limit, is also independent of other deviee parameters: width b of the stub and width a of leads. Fig.3.19 and 3.20 show Gand G/T versus T for different values of a for two different

L's. It is obvious that at low T, G = 7r2k~T/3h so that G/T = 7r2k~/3h, for an the cases. Clearly, thermal conductance G increases with a: a wider wire transport more energy. FinaIly, we plot G versus stub length L for the T -shaped junction for three stub widths b in Fig.3.21. When L is equal to the wire width a, Le., no stub at aIl (see

Fig.3.1), transmission coefficient Tm(w) = 1 for aIl m, and G reaches its maximum value NoGo where No is the total propagating modes in the wire. G reduces for L > a due to wave scattering. When L reaches up to 3a or more, G exhibits an irregular fluctuation. Such a behavior is related to modes interference and these interference has not been completely smeared out by the energy integration of Eq.(3.27). We also found that at T > lK, G with L = a is larger than the universal value. This is because of the contribution due to massive modes in addition to the massless mode, as discussed ab ove , see Eq.(2.54). Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 83

10 1 9 1 a=20nm 1 1 .....~ 8 1 { 7 (a) / 1 0 1 œ 6 c0 Il a=10nm s 5 1 / ... 0 l ,// ~ c 4 ,,1 / .... "0 () ii 3 ",," ~.~"""., ..••. /"""/' a=5nm ~ œ ~" J: 2 ",." .. 1- ..,.,,-:,...... 1 ;r..::::.:::.:::.. - .. ..- 0 0 2 3 4 5 Temperalure T(1<) TerI'!leralure T(1<)

Figure 3.19: (a) Thermal conductance Gand (b) GjT versus temperature T with stub length L = 30nm and width b = lOnm, for different wire widths, a = 5nm, a = lOnm, and a = 20nm.

10 1 9 a=20nm 1 1 z 8 1 .....'" 1 J (a) 1 .l:- 7 1 1 0 6 1 8 l a=10nm ... c I / 5 1 ,/ ~J 1 /' c 4 Il / ... "0 () ,,/ ./'" a=5nm // ii 3 ./ " ...... E .",." ...... œ 2 J: " /." l- ..".~...... " l ~~~...... " 0 0 2 3 4 5 Temperature T(1<) Temperature T(K)

Figure 3.20: (a) Thermal conductance Gand (b) GjT versus temperature T with stub length L = 100nm and width b = lOnm, for different wire widths, a = 5nm, a = lOnm, and a = 20nm. Chapter 3. Thermal Transport in a T -Shaped Quantum Wire 84

~ (a) T=1 K == ~:~~~~ \ ---~~~ ~ 0.8 \\ Cl \\ CD \\ \\ i_ 0.6 .... ~ "CJ c '""'c.::..~-~...... _...... 8 "- (ij \ ...... \ ,;' -- ...... ~ 0.4 / ~----,---." ~ '-,;'

0.2 '------'~--'-~-----'--~--'----'---'-~~~~-~-----' 10 20 30 40 50 60 70 80 90 100 110 L(nm)

1.2 (b) T=2K -- b=10nm "...... ~20nm --- b=50nm

\ ~'------~'1\ - \~.'-...... ".;;.. ~ ...... , ...... _/~', (ij '/', ,~/-~ ~-----~-~, ~ 0.4

0.2 '--~~~~~~~~-~~~~~~~~~~---' 10 20 30 40 50 60 70 80 90 100 110 L(nm)

1.4 (c) T=4K -- b=10nm ...... b=20nm --- b=~nm \. ~ ...... "\~\ ...... ~ 0.8 "CJ ,/-/,-~~,_ __~~/'~J~ c ~ -~~-,~~~ o ~ 0.6 ! 0.4

0.2 ~---'-~--'----'-~-'--~-'----'--'---'---'-~--'---' 10 20 30 40 50 60 70 80 90 100 110 L(nm)

Figure 3.21: Thermal conductance G versus the stub length L with different b = lOnm, 20nm, and 50nm. The wire width is a = lOnm. Temperature is (a) 1K, (b) 2K, and (c) 4K. Chapter 3. Thermal Transport in a T-Shaped Quantum Wire 85

3.6 Summary

In this chapter, we have investigated phonon transmission properties and thermal conductance for a T -shaped dielectric wire, by a mode matching numerical technique. Due to quantum interference, transmission coefficients of phonons become quite com plicated. We found the value of Tm(w ~ 0) may be unity or zero depending on the geometrical ratio~. Tm(w) has an oscillation behavior with quasi-periodicity and irregularity. For the thermal conductance of such a system, we deduced that it increases monotonically with the temperature - a result that is generally true for any two-probe device. We confirmed the existence of universal quantum of thermal conductance which exists at the low temperature limit, and such a quantum is robust against all the system parameters. 4

Thermal Transport in a Four-Terminal Deviee

4.1 Introduction

In the last chapter, phonon transport and thermal conductance in a T-shaped di electric wire have been studied. One of the main findings was that at the limiting temperature T -7 0 where the phonon number associated with transport tends to zero, thermal conductance of the T -shaped wire has a universal behavior - it tends T to zero as G = Go =7[2:: where Go is the universal quantum, indicating that the thermal resistance tends to infinity. This behavior is different from electric transport where for two-terminal devices the electric conductance may take any finite value depending on the scattering region. Another very interesting problem of thermal transport is to understand properties offour-terminal devices, i.e., the quantity Gij,kl, and in particular, ifthe four-terminal thermal conductance also has a univers al tendency at the T -7 0 limit. For a meso scopic dielectric system, four-terminal measurements of thermal conductance Gij,kl should be experimentally feasible[89], in which two terminaIs i, j carry thermal flux to and from the scattering region while temperatures are measured at the other two terminaIs k, l which are at local equilibrium with the dielectric device. In fact, the experimental setup of Fig.1.3 is a four terminal system, although Gij,kl has not yet been measured. A four-terminal measurement is interesting as it can, in principle,

86 Chapter 4. Thermal Transport in a Four-Terminal Deviee 87

exclude effects of contact thermal resistance. Theoretically, Gij,kl provides physical understanding of how energy flux is partitioned among the multiple leads. The un derstanding will provide useful information for application of the basic idea to novel devices with heat effect[87, 90]. One of the most important and useful works of mesoscopic physics is the work by Marcus Büttiker[71] who derived a four-terminal phase coherent electrical con ductance formula. This four-terminal formula has been wide spread applications in electrical quantum transport in multi-terminal devices[91, 92, 93, 94]. On the other hand, unlike the Landauer-Büttiker electrical conductance formula[70, 71, 95], to the best of our knowledge the physics associated with the four-terminal thermal quantity

Gij,kl has not been investigated before, and it is the subject of this chapter. In this chapter, in terms of the formula derived before, Eqs.(2.51) and (2.53), we study the general properties of thermal conductance. We will also analyze a special four-terminal device by calculating explicitly Gij,kl and compare this quantity with that of the two-terminal situation. Our investigation suggests that the four-terminal quantity Gij,kl is non-monotonic in temperature T, unlike its two-terminal counterpart

G ij . Gij,kl indeed has a universal behavior in the zero temperature limit, but it tends to infinity instead of tending to zero, and the Onsager relations Gij,kl = Gk1,ij and 1 G-:-11'k G-:- , = G~lkl2J, + 2 ,J + 2 k,Jl °hold.

4.2 Four-terminal thermal conductance

Considering a four-terminal dielectric system schematically shown in Fig.4.1. This plot is similar to Fig.2.4 but the number of terminaIs is now fixed to four. The terminaIs are connected to thermal reservoirs at equilibrium with temperatures Ti where i = 1,2,3,4, respectively. The terminal wires are assumed to be perfect and phonons coming from reservoirs are not scattered inside the wires, i.e., there are Chapter 4. Thermal Transport in a Four-Terminal Deviee 88

Scattering Region

Figure 4.1: Schematic diagram for the four-terminal system with an arbitrary scattering region. no interactions between phonons. Phonon scattering only occurs in the scattering region which may have an arbitrary shape involving defects, rough surfaces, etc .. Considering coherent transport, we define phonon transmission coefficient Tji,mn(W) for the pro cess where an incident phonon with energy 1iw from terminal i at phonon mode n is scattered to terminal j at mode m. From time reversaI invariance, the transmission coefficient has the property 7}i,mn(W) = 7ij,nm(w), In Chapter 2, we have derived the multi-terminal expression for energy flux Qi, by quantizing the classical energy flux, see Eq.(2.51). Renee the flux Qi from terminal i flowing into the central scattering region is:

(4.1) where ~(w) = [exp(k~~) - 1]-1 is the Bose-Einstein distribution function of the phonons in the i-th reservoir, and Win is the cutoff frequency of mode n in terminal wire i. For W < Wjm, the incident phonon can not be scattered to mode jm; and for W < Win, there does not exist incident phonons from the mode in. Therefore the integration st arts from max(win, Wjm). Chapter 4. Thermal Transport in a Four-Terminal Deviee 89

Let's assume that each terminal has a different temperatures Ii. In order to conveniently measure temperature, we introduce a reference temperature To , which could be sm aller or equal to the lowest of the four temperatures Ii, here we take the lowest of the four temperatures Ii as the reference To. Below To , the thermal fluxes flowing in negative and positive directions are equal, hence there is zero net flux in each of the terminaIs. We need only to consider the temperature range D..Ii = Ii - To above To. We further assume that the differences between Ii 's are so small that only linear thermal conductance needs to be considered. Then the distribution function ni (w) can be expanded as:

(4.2)

Putting (4.2) into (4.1), the thermal flux can now be rewritten as

XJ L L t dW 1iw [D..Ii - D..Tj]dn(w)'1ji,mn(W) '( '..J-') m n Jmax(Win,Wjm) 27r J JT~ , dT L Gij(b.Ii - b.Tj) . (4.3) j(#i)

Eq.(4.3) represents the total energy flux in terminal i coming from other terminaIs. From (4.3), we obtain the thermal conductance Gij between an arbitrary pair of terminaIs i,j among the multi-terminals: '" 100 dw ic.wrr. , dn( w ) L.J lb .J.iJ,nm m,n maX(WimWjm) 27r dT [00 dw 1i 'L, dn(w) (4.4) Jo 27r w ~J dT ' where Tù(w) = L B(w - win)B(W - wjmYIij,nm(w) is the total transmission coefficient m,n from terminal i to terminal j. From time-reversaI invariance, we must have Tij (w) =

'1ji(W) and hence Gij(T) = Gji(T). Similar to the results of Chapter 3, Eq.(4.4) allows us to exactly prove that the two terminal thermal conductance Gij(T) in a multi-terminal setup is still a monotonie function of temperature T. At the low Chapter 4. Thermal Transport in a Four-Terminal Deviee 90

temperature limit T ~ 0, thermal conductance Gij(T) still tends to zero as rv T. These results are similar to those of the last chapter which discussed the two-terminal conductance in a two-terminal device.

4.2.1 Thermal conductance measurement among multi-reservoirs

The measurement of Gij needs to know the temperature difference between the mea suring points. For a two-terminal device such as the thing studied in the last chapter, we measure the temperature difference and thermal flux from the same two terminaIs. In Eq.(4.4), the temperature difference and thermal flux are also measured from two thermal reservoirs although there are more terminaIs in the device. Thermal conduc tance measured in this way will therefore be influenced by the measuring contacts, i.e., the contact thermal resistance is mixed inside the final measured value. Similar situation is true for electrical measurements. To avoid this contact resistance, one should make four-terminal measurements. For example, we can consider that a net thermal flux Qi flows from terminal i to terminal j through the scattering region, and we caU these terminaIs "flux terminaIs". Then, we measure temperature difference /:lTkl = Tk - Tz at two other terminaIs k and l by keeping a zero thermal flux in terminaIs k and l[70] , which are caUed "temperature terminaIs". Then, a four-terminal thermal conductance, Gij,kl, can be defined through

(4.5)

We now derive an expression for the four-terminal quantity Gij,kl. From (4.3), we obtain

Qk = Gki(/:lTk - /:l'li) + Gkj(/:lTk - /:lTj) + Gkl(/:lTk - /:lTz), (4.6)

QI = Gli(/:lTz - /:l'li) + G1j(/:lTz - /:lTj) + G1k(/:lTz - /:lTk) . (4.7)

Here /:l'li - 'li - To is the temperature measured from the reference as discussed in the last subsection. Because terminaIs k, lare temperature terminaIs, we have Chapter 4. Thermal Transport in a Four-Terminal Deviee 91

Qk = QI = o. Therefore Eqs. (4.6, 4.7) becomes:

Gki i::l.1i, + Gkji::l.Tj = Gkii::l.Tk + Gkl(i::l.Tk - i::l.Tz) + Gkji::l.Tk ,

Gli i::l.1i, + Glji::l.Tj = Glii::l.Tz + Glk(i::l.Tz - i::l.Tk) + Glji::l.Tz

From these equations we solve for the temperatures and obtain:

i::l.1i, [(GikGZj + GklG1j + GkjGlj + GkIGkj)i::l.Tk - (GklGlj + GilGkj

+ GklGkj + GkjGlj)i::l.Tzl!(GikGlj - GilGkj ) , (4.8)

i::l.Tj -[(GikGil + GilGkl + GilGkj + GikGkl)i::l.Tk - (GikGkl + GikGil

+ GikGkl + GikGlj)i::l.Tzl!(GikGlj - GilGkj ) . (4.9)

From (4.8) and (4.9), a temperature difference ratio, Œij,kl, can be defined:

Œij,kl i::l.1i,j/i::l.Tkl = [Gkl(Gik + Gil + Gkj + Glj )

+ (Gik + Gjk)(Gil + Gjl)l! D . (4.10)

By Eq.(4.5), we obtain

Gij,kl - QïI i::l.Tkl = [Gkl ( Gik + Gil )( Gjk + Gjl)

+ Gij(Gik + Gjk)(Gil + Gjl)

+ GijGkl(Gik + Gil + Gkj + Glj) + GikGilGkjGlj(Gii/ + G;/ + G"k} + Gi/)l! D , (4.11) where D =GikG lj - GilGkj . This is one of the main results of this chapter. The four-terminal quality Gij,kl in Eq.(4.11) has several simple but general features. First, from Eq.(4.11) by renaming subscripts, we can easily confirm:

Gij,kl = -Gij,lk = -Gji,kl = Gji,lk .

This result means the value of the four terminal thermal conductance does not change if we switch the flowing direction of flux. Chapter 4. Thermal Transport in a Four-Terminal Deviee 92

Second, from time-reversal invariance we have a reciprocity relation,

1 1 1 Gij,kl = Gkl,ij , G-:-. + G:- ' + G:- · = 0 . (4.12) ZJ, kl Z1 ,J k ~ k,J1

The second expression of (4.12) can be trivially proved: by Eq.(4.5) we have

G- 1 + G-1 + G-1 ~J,.. kl ~'1 ,J'k ~ 'k ,Jl' (b..Tkl + b..Tjk + b..llj)/Qi

(Tk - Il + T j - Tk + Il - T j ) / Qi o .

The reciprocity relation indicates that if one exchanges the roles of the flux terminaIs and temperature terminaIs, the four-terminal thermal conductance is the same. On the other hand, from Eq.(4.1O) we can easily confirm that the temperature difference ratio Û'.ij,kl does not satisfy a similar relation.

In (4.11), when the denominator D becomes zero, Le., GikGjl = GilGjk, the four terminal "thermal bridge" reaches equilibrium by analogy to the electric bridge problem[96]. When this happens, the temperature difference D.Tkl between the two temperature terminaIs k and 1 is always zero regardless how large a thermal flux passing through the other two terminaIs i and j. In such a situation, quantities Gij,kl and Û'.ij,kl will tend to infinity.

4.3 An example: four-terminal mesoscopic dielectric system

In this section, we investigate the four-terminal thermal conductance of a specific two dimension al four-terminal mesoscopic dielectric system, shown in FigA.2, in which four semi-infinite wires parallel to the x axis are coupled to the center window region with width band height L. The 2D system has no obvious symmetry as the terminal Chapter 4. Thermal Transport in a Four-Terminal Deviee 93 wires are made to have different widths, so that the characteristics of thermal con ductance Gij,kl are not caused by any special geometric symmetry. Here our concern is still on the general properties of the four-terminal features of transmission coeffi cient which allows us to discuss thermal transport properties of energy flux as well as energy flux partition in different terminaIs.

: al 1 : : IV a4 V

~b-.. L

III a3 a2 II y t x

Figure 4.2: Schematic view of a 2D four-terminal device where region V is the scattering region. The terminal wires have different widths as indicated, the system has no obvious geometric symmetry. The incoming phonon waves is in terminal wire 1 with width al. Horizontal direction is along the x-axis, the vertical direction is y.

We apply a mode matching numerical method, similar to that discussed in Chapter 3, on the scalar model of elasticity[62, 80] where the displacement field u(x, y) satisfies the wave equation:

(4.13) where v is the sound velo city. If the coupling between three different components of the displacement vector field is small enough, the scalar model is a good model. We assume that incident phonons at mode n cornes from terminal i = 1 (see Fig.4.2). We again assume that the system size is smaller than the mean free path of the phonons Chapter 4. Thermal Transport in a Four-Terminal Deviee 94 so that phonon scattering is only due to geometric factors, i.e., the region V. The free boundary condition for solving Eq.(4.13) is the same as that in Chapter 3, Eq.(3.2). The wave fun ction u(x, y) in the four-terminal wires and in region V are:

,T, () iklnX ,T, () -iklmX Ur ( x, y ) = 'J'ln Y e + '"'~ rn,mn 'J'lm Y e , m

m uv(x,y) = L [aanWa(y)eikaX + bamWa(y)e-ikax] , a where, wim(y) (mode index m = 0,1,2,···) and wa(y) (mode index Œ = 0,1,2,···) are orthonormal transverse wave functions in terminal i and in center region V, respec tively. They have the function form: Wim(y) = j!cos(r;::)y, wa(y) = y1cos(Ï",)y, due to the boundary condition Eq.(3.2). kim and ka are the corresponding wave vectors, satisfying the following dispersion relation:

in which Wim = ~:v and Wa = alv are cutoff frequencies of modes im and Œ, re spectively. ru,mn and tjl,mn (j = 2,3,4) are refiection and transmission amplitudes. The subscripts of tjl,mn means that incoming wave from mode n in terminal-wire 1 is transmitting to mode m in terminal-wire j. Similar interpretation is true for the suhscripts of rn,mn. aan and ban are constants to he determined. Chapter 4. Thermal Transport in a Four-Terminal Deviee 95

We now apply boundary conditions Eqs.(3.8) to (3.10) of Chapter 3 at the four boundaries between the terminal-wires and the region V. Matching boundaries be tween regions 1 and V, regions II and V at x = 0 (see Fig.4.2), we obtain:

k1nWln(y) - Lk1mrn,mnWlm(Y) = L [kaaanwa(y) - kabanwa(y)] , m a

Next, matching boundaries between regions 1 and V, regions II and V at x = b (see Fig.4.2), we obtain:

Lt31,mnW3m(y)eik3rnb = L [aanWa(y)eikctb + banWa(y)e-ikctb] , 0 ~ y ~ a3 m a Lt41,mnW4m(y)eik4rnb = L [aanWa(y)eikab + banWa(y)e-ikctb] , m a L - a4 ~ Y ~ L

Lk3mt31,mnW3m(y)eik3rnb = L [kaaanWa(y)eikctb - kabanWa(y)e-ikctb] , m a

L [kaaan'lla(y)eikctb - kaban'lla(y)e-ikctb] = 0 , a Lk4mt41,mnW4m(y)eik4rnb = L [kaaanWa(y)eikctb - kabanWa(y)e-ikctb] . m a L - a4 ~ Y ~ L

Letting i31 ,mn = t31,mneik3mb, i41 ,mn = t41,mneik4mb, multiply the above equations by transverse conjugate wave-functions 'lIim(Y), w2m (y), 'lI;m(y), w4m (y), and 'lI~(y), integrate over coordinate y, we derive the following expressions: Chapter 4. Thermal Transport in a Four-Terminal Deviee 96

61n,lm + ru,mn = L Œlk~l k1nAQ,lnAQ,lm - L L Œlk~l kllAQ,llAQ,lmrll,ln Q Q 1 - L L Œlk~l k2IAQ,2lAQ,lmt21,ln - L L Œ2k~1 k3lAQ,3lAQ,lmi31,ln Q l Q l - L L Œ2k~lk4lAQ,4lAQ,lmi41,ln , Q l

t 21 ,mn = LŒlk~lklnAQ,lnAQ,2m - LLŒlk~lkllAQ,llAQ,2mrll,ln Q Q l - L L Œlk~l k2lAQ,2IAQ,2mt21,ln - L L Œ2k~1 k3lAQ,3lAQ,2mi31,ln Q l Q l - L L Œ2k~1 k4lAQ,4lAQ,2mi41,ln , Q l

i 31 ,mn = L Œ2k~1 k1n AQ,lnAQ,3m - L L Œ2k~1 kll AQ,llAQ,3mrl1,ln Q Q l - L L Œ2k~lk2lAQ,2lAQ,3mt21,ln - L L Œlk~lk3lAQ,3lAQ,3mi31,ln Q l Q l - LLŒlk~lk4lAQ,4lAQ,3mi4l,ln , Q l

i41 ,mn = L a2k~1 k1n AQ,lnAQ,4m - L L a2k~1 kll AQ,llAQ,4mrll,ln Q Q l - L L a2k~1 k2lAQ,2lAQ,4mt21,ln - L L Œlk~l k3lAQ,3lAQ,4mi31,ln Q l Q l - L L alk~l k4lAQ,4lAQ,4mi41,ln , Q l where AQ,lm Jf-al WQ(y)wlm(y)dy, AQ,4m - Jf-a4 WQ(y)w4m(y)dy, AQ,2m = JOa2W Q(y)w2m(y)dy, AQ,3m = Joas WQ(y)w3m(y)dy. Chapter 4. Thermal Transport in a Four-Terminal Deviee 97

Re-arranging the above equations, we obtain:

y (6ml + ~(~>:)(lk~lkllAa,llAa,lm) rll,ln + y (~alk~lk2IAa,2IAa,lm) t2l ,ln + y (~a2k~lk3IAa,3IAa,lm) t3l ,ln + y (~a2k~lk4IAa,4IAa,lm) t4l,ln

= -6ln,lm + E alk~lklnAa,lnAa,lm , a

y (~ alk~l kll Aa,llAa,2m ) rn,ln + y (6ml + ~ alk~l k2IAa,2IAa,2m ) t2l ,ln

+ y (~ a2k~l k3IAa,3IAa,2m ) t3l ,ln + y (~ a2k~ l k4IAa,4IAa,2m ) t4l,ln

= E alk~lklnAa,lnAa,2m , a (4.14) y (~ a2k~1 kllAa,llAa,3m ) rn,ln + y (~a2k~l k2I Aa,2I Aa,3m ) t21,ln + y (6ml + ~alk~lk3IAa,3IAa,3m) t31 ,ln + y (~alk~lk4IAa,4IAa,3m) t4l,ln

= E a2k~1 klnAa,lnAa,3m , a

y (~a2k~lkllAa,llAa,4m) rn,ln + y (~a2k~lk2IAa,2IAa,4m) t2l ,ln+

y (~ al k~ l k3IAa,3IAa,4m ) t3l,ln + y (6ml + ~ al k~ 1 k4IAa,4IAa,4m ) t4l,ln

= E a2k~l klnAa,lnAa,4m . a This is a set of linear aigebraic equations for the transmission and refiection ampli tudes which can be soived by a standard matrix technique. After obtaining rll,mn and tjl,mn (j = 2,3,4), the refiection and transmission coefficient can be obtained as

Rn,mn(w) B(w - wlm)lrn,mn(w)12klm/kln

Tjl,mn(W) B(w - Wjm)ltjl,mn(w)12kjm/kln where the step function B(w - Wjm) is because the outgoing wave vector kjm wouid be imaginary at W < Wjm so that this wave can not propagate. If the incident phonons come from other terminaIs 2, 3, or 4, the corresponding transmission coefficients Chapter 4. Thermal Transport in a Four-Terminal Deviee 98 can be solved in exactly the same fashion. Here the multi-terminal transmission coefficient Tji,mn has the physical meaning of probability for an incident wave in mode n in terminal-wire i to transmit to mode m at terminal-wire j. Since these are mode-to-mode probability, Tji,mn must be no greater than unity. According to flux conservation, we must also have:

(4.15)

The relation expresses the fact that the total probability for four waves to go anywhere must be unity.

4.3.1 'fransmission coefficient Due to strong scattering in the coupling region V, the transmission coefficient is very complicated with interesting features. Figs.4.3-4.6 plot the transmission coefficients as a function of phonon frequency by fixing aH the terminal-wire widths to be the same. The total transmission coefficient Tjl(W) can be greater than unity due to the opening of higher modes at larger frequency w. Since the incoming wave is in terminal-wire 1, from the device structure Fig.4.2 we expect nI to be the largest, which is indeed seen in Figs.4.3(b)-4.6(b). Similar situation is true for the mode-to-mode transmission Tji,OO in Figs.4.3(a)-4.6(a). This is especially true when incident ph on on frequency w is smaH, because it is very difficult for long wavelength waves to navigate the corners in the window region V of the structure. This is why coefficients 121,00 and 131,00 are aH similarly smaH. With incoming frequency wjv increased to wjv > 0.1, for this structure the O-th mode in terminal 1 can couple with more modes in region V. This way 121,00 and 131,00 start to increase. The total transmission Tji = L L Tji,mn has an m n overall monotonie increase as w is increased: more energetic phonons can go through the device easier. We found that the window width b to be an important factor which affects energy flux propagation into the different terminaIs. When b becomes larger, Chapter 4. Thermal Transport in a Four-Terminal Deviee 99 more modes can "turn the corner" from terminal-wire 1 into the scattering region V, see Fig.4.2. Renee coefficients 121,00 and 131,00 become larger for devices with larger b, while 141,00 becomes smaller as the expense due to energy conservation Eq.(4.15). These features are shown in Figs.4.3-4.6. For the results shown in Figs.4.3-4.6, the deviees all have the same terminal-wire width. Therefore, when a wave cornes down from region V and splits leftjright into wires 2 and 3 (see Fig.4.2), symmetry suggests 121,00 ~ 131,00, which is what we observe in Figs.4.3-4.6. Such a leftjright symmetry is broken if we use different terminal-wire widths al, a2, a3 and a4. The results for different ai's are plotted in Figs.4.7-4.1O. lndeed, 121,00 is no longer as close as 131,00 for the asymmetrical structures. We have checked that 121,00 changes to 131,00 and viee versa, if we exchanges a2 with a3. We have done further calculations by varying the window height L (see Fig.4.2), ex eept the expected more transmission oscillations, the basic behavior does not change. N amely, L has little effect on the total transmission at small w for all the device structures we studied. This can be easily understood as follows. From Eq.(4.14), when w --7 0 one can deduce

(4.16) and refiection coefficient

4 where a = Lai. Rence these coefficients only depend on the width of the terminal- i=l wires and not on L. This suggests that in the small w limit, thermal conduction becomes independent of the scattering region: a somewhat unexpected outcome. From the point of view of phonon wavelength, namely when the wavelength becomes much larger than the linear size of the scattering region, we do expect that the displaeement field u(x, y) becomes essentially equal throughout the scattering region. Chapter 4. Thermal Transport in a Four-Terminal Deviee 100

0.8 (b) T21 --- T31 "._.-- T41-- 0.6 & r:':. 0.4 T21 .00 ---- (a) TII.00---" 0.2 \ 1\ 1 T41.00-- . . \ 1 V~I 0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 1 -1 oYv(nm- ) oYv(nm )

Figure 4.3: Transmission coefficient '1jl,OO (a) and '1jl (b) versus incident phonon frequency w. The parameters: al = a2 = a3 = a4 = 30nm, L = lOOnm, b = 20nm, and v = 5000m/sec.

0.8 (b) T21 --- T31-'-'

T41-- 0.6

0 q ~ 0.4 T21 .00 ---- (a) T31.00"·- 0.2 T41.00 --

° 0.4 0.5 0.2 0.4 0.6 0.8 ° -1 OJ/v (nm )

Figure 4.4: Transmission coefficient '1jl,OO (a) and '1jl (b) versus incident phonon frequency w. The parameters: al = a2 = a3 = a4 = 30nm, L = lOOnm, b = 30nm, and v = 5000m/ sec. Chapter 4. Thermal Transport in a Four-Terminal Deviee 101

0.8 (h) T21 --- Tai .... _ ... -

TII-- 0.6

°0, r:':. 0.4 T21 .00---- (a) T31.00--" 0.2 T41.00--

0 l'- 0 0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 -1 -1 lfiv(nm) lfiv (nm )

Figure 4.5: Transmission coefficient 1jl,OO (a) and 1jl (b) versus incident phonon frequency w. The parameters: al = a2 = a3 = a4 = 30nm, L = lOOnm, b = 40nm, and v = 5000m/ sec.

0.8 (b) T21 --- TSI'-" T41-- 0.6 ~ r:':. 0.4 T21.00 ---- (a) T31.00·-· 0.2 T41.00--

0 0 0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 -1 -1 lfiv (nm ) OlIV(nm )

Figure 4.6: Transmission coefficient 1jl,OO (a) and 1jl (b) versus incident phonon frequency w. The parameters: al = a2 = a3 = a4 = 30nm, L = lOOnm, b = 50nm, and v = 5000m/ sec. Chapter 4. Thermal Transport in a Four-Terminal Deviee 102

0.8 (b) T21 --- T31-- T41-- 0.6

0 q

~ 0.4 . T~.oo ---. (a) \ T31.00"'-" 0.2 ~\.J T41.00-- '-" \\\"::. ° 0.1 0.2 0.3 0.4 0.5 0.2 0.4 0.6 0.8 ° ·1 1 oYv(nm ) mtv (nm- )

Figure 4.7: Transmission coefficient '1jl,OO (a) and '1jl (b) versus incident phonon frequency w. The parameters: al = 40nm, a2 = 20nm,a3 = 50nm, a4 = 30nm, L = lOOnm, b = 20nm, and v = 5000m / sec.

0.8 (b) T21 --- T31 -- T41-- 0.6 ~ ~

T21.00 ---- (a) T31.00"-" T41 .00 --

° 0.1 0.2 0.3 0.4 0.5 0.2 0.8 ° -1 oYv(nm )

Figure 4.8: Transmission coefficient '1jl,OO (a) and '1jl (b) versus incident phonon frequency w. The parameters: al = 40nm, a2 = 20nm, a3 = 50nm, a4 = 30nm, L = lOOnm, b = 30nm, and v = 5000m / sec. Chapter 4. Thermal Transport in a Four-Terminal Deviee 103

0.8 (b) T21 --- T31'- T41 -- 0.6 ~ ~ 0.4 T21.oo ---- (a) T31.00·--"

0.2 T41 .00 --

0 0 0.1 0.4 0.5 0.2 0.4 0.6 0.8 1 œ/v(nm- )

Figure 4.9: Transmission coefficient 1jl,OO (a) and 1jl (b) versus incident phonon frequency w. The parameters: al = 40nm, a2 = 20nm, a3 = 50nm, a4 = 30nm, L = lOOnm, b = 40nm, and v = 5000m / sec.

0.8 (b) T21 ---- T31'--'- T41-- 0.6

0 q 1-"- 1-"-

\ T21 .00 ---- (a) \ T31.oo"·-" 0.2 T41.00-- - \. Ii \ \ Vi . ili i \A~, ~ 0 0 0 0.1 0.2 0.3 0.4 0.5 0 0.2 0.4 0.6 0.8 -1 -1 oYv (nm ) OJ!V (nm )

Figure 4.10: Transmission coefficient 1jl,OO (a) and 1jl (b) versus incident phonon frequency w. The parameters: al = 40nm, a2 = 20nm, a3 = 50nm, a4 = 30nm, L = 100nm, b = 50nm, and v = 5000m / sec. Chapter 4. Thermal Transport in a Four-Terminal Deviee 104

This conclusion is actuaIly true for two-terminal system as weIl. By setting a3 = a4 = 0, the four-terminal device is converted into a two-terminal one and we obtain

~ _ 4a1a2 21 -a 2 ' where a = al + a2. Assuming al = a2, we get '121 = 1 and R2l = o.

4.3.2 Thermal conductance in two-terminal measurements

In this subsection, we discuss thermal conductance Cij between any two terminaIs i,j of the four-terminal device. Cij is the linear response thermal conductance obtained by any two-terminal measurement, plotted in Fig.4.11 as a function of temperature

T. Cij is a monotonicaIly increasing function of temperature T. At high temperature,

2 e.g., T > lK, Cij f'.J T . At low temperature, e.g., T < O.IK, Cij f'.J T. In certain temperature range, Cij may exhibit a TO scaling with 8 < 1, e.g., C 13 in the range of 0.2K < T < 0.5K. These results are consistent with previous literature for two terminal systems[12, 24, 25] as weIl as our results in Chapter 3. In Fig.4.11(b), Cij/T versus T exhibits a non-monotonie behavior. Cij/T can have a minimum at T =1=- 0, for instance C 12 /T and C 13 /T in Fig.4.11(b). Similar results have been seen in the experiments of Schwab et. al. [11]. Although the two-terminal quantity Cij is actuaIly measured between two ter minaIs of a four-terminal device, it has similar properties as that of a two-terminal device [11 , 12, 24, 25]. This suggests that we may be able to look for sorne universal behavior. For instance, for two-terminal devices as that analyzed in Chapter 3, there 2 2 1f k T is the universal quantum of thermal conductance Co =~. Does Cij of the four- terminal device have any universallimits? We now examine this question. In thermal conductance formula Eq.(4.4), letting x = 1iw/kB T, we have

(4.17) Chapter 4. Thermal Transport in a Four-Terminal Deviee 105

(a)

~~ 0.6

3c ~ 0.4 :l 'Ilc o ~ 0.2 E œ ------~ 3

Temperature T(K) Temperature T(K)

Figure 4.11: Two-terminal thermal conductance Gij (a) and Gij/T (b) versus temperature T. The parameters: al = 40nm, a2 = 20nm, a3 = 50nm, a4 = 30nm, L = lOOnm, b = 30nm, and v = 5000m / sec.

As seen already, this Gij is a monotonically increasing function of T. In the limit T -+ 0, only the O-th mode (massless) is excited and the wavelength of incident phonons is much larger than the dimension of the scattering region. Therefore 7;,j,nm(w) -+

7;,j,nm(O). Using 7;,j = L 7;,j,nm, Eq.(4.17) reduces to mn

'" T , where 7;,j(O) = (4aiaj)/(a2) (see Eq.(4.16)) and a = al + a2 + a3 + a4. This allows us to conclude: Gij T (4.18)

Therefore, for the two terminal quantity Gij in four terminal device, Gij/T is equal Chapter 4. Thermal Transport in a Four-Terminal Deviee 106 to the universal quantum multiplied by the ratio of the terminal-wire widths at low temperature and w -+ 0 limit, and it does not depend on the shape of the scattering region. The result Eq.(4.18) should be testable experimentally: if the linear size of the scattering region is 100nm and phonon wavelength is ten times larger, the corre sponding temperature sc ale T = ~ = k:\ ~ 0.2K. This temperature is accessible to experiments.

4.3.3 Thermal conductance in four-terminal measurements

As discussed before, in a four-terminal measurement, the flux flows from terminal i to j, while we measure temperature difference between another pair of terminais k and 1. This way, the influence contact resistance can be, in principle, eliminated. We now investigate the four-terminal device shown in Fig.4.2: we take terminais 1,4 as flux terminais, 2,3 as temperature terminais. The net thermal flux in terminais 2 and 3 is assumed to be zero.

200~------~ 180 :a 160 :{"" 140 - 120 100 80 60 40

20 GI2,,. o ...-=r. _____~ ...... '=' __ '='_._=_ __ '_=:' __ _=.::::.=_::::.::::.::::_=_::__=.=.=_::.=.= ... _

-20 GI 3.4'

-40 '---~---~~--'---~~---~~---~~-----' o 2 3 Temperature T (K)

Figure 4.12: Four-terminal thermal conductance Gl4,23 versus temperature T. The parameters: al = 40nm, a2 = 20nm, a3 = 50nm, a4 = 30nm, L = lOOnm, b = 30nm, and v = 5000m/sec.

The results of the four-terminal thermal conductance G14,23 is plotted in Fig.4.12 against T. Different from the two-terminal quantity G14 which is monotonie, G14,23 Chapter 4. Thermal Transport in a Four-Terminal Deviee 107

2 is a non-monotonic function of T. At high temperatures, G14,23 is proportional to T similar to G14 . However, at the low temperature limit (T ---70), G14,23 does not tend to zero as G14 did, it tends to 00 as T-1, i.e., thermal resistance tends to vanish. This is a quite surprising result. In terms of relation (4.11), G14,23 should be in proportion to the two-terminal thermal conductance Gij, therefore it should vary apparently with 1 T rather than T-1. Since the Gi/s in Eq. (4.11) an have finite value, T- behavior must derive from the denominator D of Eq.(4.11). When T ---7 0, G14 f'.J 4d,2a4T (see

Eq.(4.18)), we have G12G34 = G13G24 , similar to the relation satisfied by electric conductances of an electrical bridge at equilibrium. This gives rise to D ---7 0 in

Eq.(4.11), thereby Gij,kl ---7 00. If we think the four-terminal device of Fig.4.2 as a thermal bridge, the relation

G12G34 = G13G24 can be considered as the condition that this thermal bridge reaches equilibrium (at T ---7 0). When this happens, the temperature measured at the terminaIs 2 and 3 will be almost equal. To see this, we plot the temperature difference ratio 0:14,23 = ~R: as a function of temperature T in Fig.4.13. It clearly exhibits that 2 when T ---7 0, 0:14,23 ---7 T- . In other words, at very low temperature, if we keep ~T14 finite so as to have a thermal flux flowing between terminaIs 1 and 4, the temperature 2 difference ~T23 of temperature terminaIs 2 and 3 tends to vanish as f'.J T . As a numerical example, if temperature is high, e.g., T = 3K, the corresponding ratio

0:14,23 ~ 25, and we obtain ~T23 ~ 20mK which is an appreciable temperature difference when maintaining ~T14 f'.J 0.5K. At lower temperature T = 100mK, the corresponding ratio 0:14,23 f'.J 1500; when keeping ~T14 f'.J 500mK, the temperature difference ~T23 is merely 0.33mK. Because the temperature difference is so small, we could think T2 ~ T3 . Chapter 4. Thermal Transport in a Four-Terminal Deviee 108

1000

800 '""~ ~ t 600 J ,g <0 Cl: 400 .a~ e(1) c- 200 E ~ (l14,23

00 2 3

Temperature T (K)

Figure 4.13: Temperature ratio Œ14,23 versus temperature T. The parameters: al = 40nm, a2 = 20nm, a3 = 50nm, a4 = 30nm, L = lOOnm, b = 30nm, and v = 5000m/ sec.

Clearly, if we use different pairs of the terminaIs as fiuxjtemperature terminaIs, similar conclusions will be obtained as shown by the curves of G13,42 and G12,34 in FigA.12. Therefore, we can make a general statement for the four-terminal quantity: at equilibrium GikGlj = GilGkj , !Gij,kl! ---+ 00 when T ---+ O.

200 180 ;d' 160 ~:J."" ~-=- 140 "1 120 f!J" 100 c:2l 80 ~:::J c: 8" 60 ta 40 ê .c:(1) 20 1- °0L------~--~------2~------~3

Temperature T (K)

Figure 4.14: Four-terminal thermal conductance G14,23 with different width b versus temperature T. Other parameters: al = 40nm, a2 = 20nm, a3 = 50nm, a4 = 30nm, L = lOOnm, and v = 5000m / sec. Chapter 4. Thermal Transport in a Four-Terminal Deviee 109

600 ! ~, 500 ! b=50nm "JI \ :::! b=40nm E-<" 400 \ b=30nm t b=20nm ~~ 300 ~ l!1 200 ;3 ~ QI Q. E 100 QI f-

2 3

Temperature T (K)

Figure 4.15: Temperature Ratio a14,23 with different width b versus temperature T. Other param eters: al = 40nm, a2 = 20nm, a3 = 50nm, a4 = 30nm, L = lOOnm, and v = 5000m/sec.

Although the scattering region becomes irrelevant at w --+ 0 in the low temperature limit, it do es change result for higher T. FigsA.14 and 4.15 plot thermal conductance

G14,23 and Œ14,23 with different width b of the scattering region, respectively. Substan tial changes are seen as b is varied at finite temperatures T. Since G 14,23 measures energy flow from terminaIs 1 to 4 (going straight in FigA.2), we get larger values of

G14 ,23 for devices with smaller b which makes it more difficult for the wave to turn into the coupling region. This is consistent with the behavior of transmission coefficient '141 (FigsA.7-4.10).

4.4 Summary

In this chapter, we have investigated the physical behavior of four-terminal ther mal conductance for mesoscopic dielectric system with arbitrary shapes of scattering region. If we make a two-terminal measurement in the four-terminal device, the two-terminal quantity Gij is a monotonically increasing function of temperature. If we make a four-terminal measurement, the four-terminal quantity Gij,kl has a non monotonic change with temperature. In the low temperature limit T --+ 0, we predict Chapter 4. Thermal Transport in a Four-Terminal Deviee 110

1 1Gij,kl 1 ---+ 00 as rv T- while G ij ---+ 0 as rv T. Furthermore, at low T, the two terminal quantity Gij IT = Ago where A is a geometrical parameter related to the 2 ratio of the terminal-wire widths, and go = 7r k1/3h is the universal constant. Fi nally, in a four-terminal measurement, the temperature difference between the two

2 temperature terminaIs approaches to zero as rv T • 5

Generalized Discussion for Thermal Conductance

5.1 Introduction

Up to this point, we have investigated thermal transport in mesoscopic dielectric systems whose dimensions are smaller than the thermal phonon wavelength. Such a situation may occur at low temperature. We derived a four-terminal thermal con ductance formula and numerical investigated its features for a specific structure. An important outcome of low temperature thermal transport is the universal quantum

'Jr2k2 T of thermal conductance Go = ~ for two-terminal devices. We have also shown that Go shows up naturally in two-terminal measurements of four-terminal devices. While phonons satisfy Bose-Einstein statistics at equilibrium, recent theoretical in vestigations showed that the ballistic thermal conductance of ideal electron gas in one dimension [97, 98, 99, 100], and of inter acting electrons that form chiral[101] and nor mal Luttinger liquids[102] should all be quantized in integer multiples of Go. Hence the amount of heat carried by fermion modes is the same as that of bosons. Indeed, it has been reported that ropes of single wall carbon nanotubes conduct heat in amounts proportional to Go [103], where both electrons and phonons contribute to hear flux. Hence it is rather interesting to ask the following theoretical question: is the universal quantum Go depending on the nature of statistics of the carriers? Such a question has recently been discussed in the context of two-terminal ballis-

111 Chapter 5. Generalized Discussion for Thermal Conductance 112 tic conductor[64] in 1D (a straight wire), and found that Go is statistics-independent and thus truly univers al. However for a more complex system, i.e., systems with a scattering region and/or systems with multi-thermal reservoirs, this question has not been addressed and will be the content of this chapter. In the following, with the help of fractional exclusion statistics FES[104], we will study the quantum phenomena of thermal conductance in a multi-terminal system, and discuss the behavior of ther mal conductance in bosons, fermions, and quasi-particles with fraction al statistical property in an unified fashion. Statistics is a distinctive property of particles that plays a fundamental role in determining macroscopic thermodynamic properties of a quantum many-body sys tem. Bose-Einstein and Fermi-Dirac statistics are the two well-established statistics distinguishing bosons and fermions. FES is less well-known, but is a mathemati cal construct about those particles whose statistics interpolates between bosons and fermions. According to Haldane's work[104], one can formulate a quantum statistical mechanics of generalized ideal gas (with no interaction between particles) for particles having fractional statistics[105]. According to these theories[104, 105], for a system with identical ideal gas particles, the FES is written as: 1 n = =-:----:--- (5.1) g W(x,g) + 9 , where x (3(e - p,), (3 = l/(kB T), and W(x,g) is given by the equation

wg(x,g)[l + W(x,g)p-g = eX (5.2)

The number 9 is a statistics parameter. p, is the chemical potential. For 9 = 0 or

9 = 1, Eq.(5.1) becomes the Bose-Einstein or the Fermi-Dirac distribution function respectively. Whether or not there exists particles in nature which satisfy the distri bution (5.1), is not the topic of research in this thesis. Nevertheless, we note that there may exist peculiar elementary excitations in 2D strongly correlated electrons who are neither bosons nor fermions, called "anyons", which do satisfy a FES[106]. Chapter 5. Generalized Discussion for Thermal Conductance 113

Our purpose of this work is to examine the consequence of thermal transport if heat carriers satisfy Eq.(5.1). As mentioned above, Rego and Kirczenow were the first to tackle this problem for ID systems[64]. Our work follows that of Ref.[64] but we focus on multi-terminal devices. In the following, we start by reviewing Rego and Kirczenow's work[64]. We will then investigate the consequence of FES for a four terminal device studied in the last chapter. We must emphasize that the work reported in this chapter, at this moment, is purely theoretical and no experiments exist.

5.2 Thermal transport in iD with FES

In the model of Rego and Kirczenow [64], a two-terminal device consists of two thermal reservoirs adiabatically connecting to an ID thermal conductor in between. Each reservoir is characterized by independent variables of a temperature T and a chemical potential/1. Assuming that the temperature differences between reservoirs are small, the change of energy flux Q in the linear response regime can be written[64] as

(5.3) where oT = TR - TL and 0/1 = /1R - /1L, with subscripts R, L representing the right and left reservoirs. From Eq.(1.36) of Chapter 1, the thermal flux between two reservoirs is

(5.4)

In the above, the sum is over n independent propagating modes in the ID conduc tor. én(k) and vn(k) are the energy and velocity of the particle with wave vector k. nR, nL represent the statistical distribution function in the reservoirs, and Tn(k) is the particle transmission probability through the channel. For the ID situation, the Chapter 5. Generalized Discussion for Thermal Conductance 114

1 velocity Vn = n- (Dên/Dk) is canceled by the ID density of states D(ên) = Dk/Dên, (5.4) can be simplified to

. 1 00 Q = -h I: 1 dêê[nR - nL]Tn(ê) (5.5) n ên(O) Substituting the expression (5.5) into (5.3), taking the limit 8T -+ 0, and noting that for ID Tn(k) = 1, one obtains the desired thermal transport coefficient. With the fractional statistics distributions (5.1) and (5.2), we obtain:

kB (OO 2 G = DaiDT = h(3 I: J, dx(x + xJ-t(3)F(x, g) , (5.6) p, n Xn(O) where xn(O) - (3(ên(O) - J-t), and

F( ) = W(x, g)[W(x, g) 1] + (5.7) x,g [W(x,g) + g]3 For FES, using Eq.(5.2) we have:

x(W,g) = ln(W + 1) + [ln(W) -ln(W + l)]g (5.8)

Using Eq.(5.8) and Eq.(5.7), we have dW F(x, g)dx = (W + g)2 (5.9)

Substituting expression (5.9) into Eq.(5.6), the thermal transport coefficient G can be worked out for arbitrary statistical parameter g. When 9 ~ 0: X2 G = M k1T {'Xl dW (W,g) + J-t(3x(w,g) = M1r2k1T (5.10) h Jo (W + g)2 3h ' where M is an integer, Le., the total number of occupied modes. Therefore, for any 2 T parameter g, one obtains G = MGo where Go = 7r :k the familiar universal quantum of thermal conductance. This suggests[64] that for a two-terminal ID system, the low temperature thermal transport has the universal quantum independent of carrier statistics. In other words, Go produced by fermions is the same as by bosons or anyons. The same, however, cannot be said to ID ballistic electrical conductance which has been known to depend on g. Chapter 5. Generalized Discussion for Thermal Conductance 115

5.3 Thermal transport in 2D with FES

The 1D conductor situation diseussed ab ove [64] has no scattering center. Here we investigate a more complicated situation in 2D where a scattering junction exists. 2D systems also present an opportunity for investigating multi-terminal situation with

FES. The quantities we are interested in are G ij and Gij,kl for multi-terminal systems, these were studied in the last chapter with Bose-Einstein statistics.

5.3.1 Two-terminal device with FES Even with FES, the two-terminal thermal conductance G can be written in a similar fashion as that for phonons as in (4.4). All we need to do is to replaee the Bose Einstein distribution function n(w) of (4.4) by the FES distribution ng(w) of (5.1).

Then G beeomes 1

where ê = nw, Tij is the energy dependent transmission coefficient between the two terminaIs. Here we have assumed that thermal flux is only driven by a temperature bias. The temperature dependence of thermal conductance G for any 2D two-terminal device ean be worked out. In partieular, to see if G is a monotonie function of temperature T, we consider the quantity tTG:

(5.11)

To be convenient for further discussions, we ean rewrite Eq. (5.11) into the following lSince we are studying two-terminal system, we can omit the subscripts i,j in Gij without causing any confusion. Chapter 5. Generalized Discussion for Thermal Conductance 116 form:

(5.12)

To obtain d?:;~ê) in (5.12), we first deal with dn1,;,ê) , dng(E) dng dW dx dT dW dx dT 1 W(W + 1) dx (W+g)2 W+g dT W(W + 1) dx (W + g)3 dT dx -P(x) dT ' (5.13) where x - (3(E - J-l), (3 - l/(kBT), and P(x) in (5.13) is the same as (5.7), i.e., 2 - W(W +1) Th c d n g (ê) h th f . P( x ) - (W +g)3 . erelore ----;j'f2 as e orm.

_ dP dW( dX)2 _ pd2X] E [ dW dx dT dT2

(xkBT + J-l)~:~;~1 {[W(W + 2) - (2W + l)g]x - 2(W + g)2} xW(W + 1) (xkBT + J-l) (W + g)5T2 B(E) .

With this expression, Eq.(5.11) can be reduced to:

(5.14) where

B(E) = [W(W + 2) - (2W + l)g]x - 2(W + g)2 . (5.15)

We are now ready to study the temperature dependence of thermal conductance G. First, we consider the special situation that the second term in Eq.(5.12) vanishes, i.e.:

!!:. (XJ d 7:. dng(E) = 0 (5.16) h Jo E ~J dT . Chapter 5. Generalized Discussion for Thermal Conductance 117

This can happen if 7;,j (E) is a constant independent of energy E, a case was studied in Ref.[64] and in the last subsection. This can also happen if the chemical potential p, = 0, true for phonons. With this, Eq.(5.14) becomes

(5.17)

In order to see if G increases monotonically with T or not, we need to examine the sign of the function B(E) in the above equation. Note that W, g, T ~ 0, 1ij(ê) ~ 0, and (~~~~) are aIl non-negative. We discuss different situations .

• When 9 = 0, e.g., for bosons or a phonon system, according to (5.2), W = eX -1

and x = ~;!f ~ O. Renee

B(E) W(W + 2)x - 2W 2 (eX - l)[(eX+ l)x - 2(eX - 1)]

It is easy to prove that (eX - 1) ~ 0, [(eX + l)x - 2(eX - 1)] ~ O. Therefore B(E) ~ O. The result is :TG is always positive. We therefore conclude that for any 2D boson system, G is always a monotonically increasing function of temperature T.

• When 9 > 0, since W changes from zero to +00 with respect to E, and since we only need to find out the allowed sign of function B(E) of Eq.(5.15), we only need to choose a special situation for W. Therefore, let's fix W = 9 to prove that B(E) can actually be negative, Le., from Eq.(5.15) we now try to prove

2 B(E) = g(l - g)x - 8g < ° (5.18)

In term of (5.2), we have

l.e., x = glng + (1 - g)ln(l + g) . (5.19) Chapter 5. Generalized Discussion for Thermal Conductance 118

Sinee 9 2:: 0, if Eq.(5.18) is true, with the help of Eq.(5.19) we must have

(1- g)x < 8g , plug in x : (1 - g)glng + (1 - g)2ln(1 + g) < 8g , (g - l)[(g - l)ln(g + 1) - glng] < 8g (5.20)

Hence the proof of Eq.(5.18) is reduced to the proof of Eq.(5.20). Two cases need to be analyzed. First, when 9 2:: 1, letting

F(g) =glng - (g - l)ln(g + 1) ,

then F(l) = 0 , (5.21) 2 F'(g) = lng -ln(g + 1) +- (5.22) g+l N ow we need to prove F' (g) 2:: 0 for 9 2:: 1. Due to

F'(oo) = 0 , F"( ) _ ~ __1_ _ 2 l-g 9 - 9 9 + 1 (g + 1)2 g(g + 1)2 ' we have F"(g) :::; 0 for 9 2:: 1, henee F'(g) 2:: O. Then, according to Eqs.(5.21) and (5.22), we conclude that F(g) is a monotonically increasing function. There fore, the assumption (5.18) is correct.

Next we consider the case where 0 < 9 :::; 1. Letting y = ~ (clearly y 2:: 1). Eq.(5.20) becomes:

(~ - 1) [(~ - 1) ln (~+ 1) - ~ln (~) 1< ~ , which can be rewritten as, (y - 1) [(y - l)ln(y + 1) - ylny] < 8y (5.23)

Eq.(5.23) has exactly the same form as Eq.(5.20). Then following the same logic we can prove B(c) < O. Henee, for any 9 > 0, we have B(c) < O. Note, Chapter 5. Generalized Discussion for Thermal Conductance 119

although the demonstration that B(E) can be negative is made for W = g, this already suffie es to show that the possibility of B(E) < 0 exists.

Back to Eq.(5.17), since B(E) can be negative, tTG may therefore be negative. Renee the two-terminal thermal conductance G for 2D conductors is in general a non-monotonie function of T for 9 > 0 when Eq.(5.16) is satisfied. It is worth mentioning that the sm aller the g, the less likely for a non-monotonie behavior of G to occur.

Next, if Eq.(5.16) is not satisfied, we have to go back to (5.14). In (5.14), the argument (xkBT + J-l)(:~~t~~ may be negative, therefore :rG is more likely to be negative. We conclude that any 2D thermal conductance G with fractional statistics (g > 0) is a non-monotonie function of temperature. Another general result we can work out is the low temperature limit of G. Thermal transport involves relevant energy sc ales roughly from J-l- kBT to J-l+ kBT. Therefore for very smaU T we can safely assume that the transmission coefficient Tij (E) ~ Tij (J-l), i.e., independent of energy E. This becomes exact at T ~ O. Then the two-terminal thermal conductance G can be calculated analytically with the help of Eq.(5.1O). We obtain,

(5.24)

This shows that the two-terminal thermal conductance G for any 2D device with FES tends to zero as rv T.

5.3.2 Four-terminal device with FES In this subsection we attempt to make sorne general statements about the four terminal thermal conductance for FES. Importantly, even with fractional exclusion statistics, formula (4.11) for the four-terminal thermal conductance Gij,kl is still valid. Chapter 5. Generalized Discussion for Thermal Conductance 120

This is because in Eq.(4.11), the statisticai distribution function do es not appear ex plicitIy. We copy Eq.(4.11) for clarity of discussion here:

Gij,kl Qd!:::..Tkl = [Gkl(Gik + Gil)(Gjk + Gjl )

+ Gij(Gik + Gjk)(Gil + Gjl)

+ GijGkl(Gik + Gil + Gkj + Glj ) + GikGilGkjGlj(G;;.1 + G::/ + G-;;] + Glj1)] ID , (5.25) where D = GikGlj - GilGkj , and !:::..~j = !:::..~ - !:::..Tj = ~ - Tj is the temperature difference between the temperature terminaIs. In addition, the Onsager relations Eq.(4.12) still hold for for 9 > 0 as no distribution function explicitly appears in them.

In the low temperature limit (T -+ 0), apply Eq.(5.24) to (5.25), we obtain

7f2k~T Gij,kl = 3h {'lkl (j-t) ['lik (j-t) + 'lil (j-t)] ['1jk (j-t) + '1j1 (j-t)] +'lij(j-t) ['lik(j-t) + '1jk(j-t)] ['lil(j-t) + '1j1(j-t)] + 'lij (j-t )'lkl (j-t) ['lik (j-t) + 'lil (j-t) + 'lkj (j-t) + '1lj (j-t)] + 'lik(j-t )'lil (j-t )'lkj (j-t )'1lj (j-t) [1;-;;1 (j-t) + 1;z1 (j-t) + 7,.-/ (j-t) + 1l;l (j-t)]} 1H , (5.26) where H ='lik(j-t)'1lj(j-t) -'lil(j-t)'lkj(j-t). For FES, Le., for 9 > 0, due to the fractional exclusion rule, the chemical potential j-t is generally not equal to energy 1ïw. Hence 'lik(j-t)'1lj(j-t) - 'lil(j-t)'lkj(j-t) can take any value. In particular, when 'lik(j-t)'1lj(j-t) -

'lil(j-t)'lkj(j-t) i= 0, Gij,kl tends to zero at T -+ 0 by Eq.(5.26). If 'lik(j-t)'1lj(j-t) -

'lil(j-t)'lkj(j-t) = 0, Gij,kl may tend to 00 or any finite value at T -+ 0, which depends on the behavior of transmission coefficient 'lij(c). Since Gij,kl may tend to 00, Gij,kl may be a non-monotonie function of temperature T. Chapter 5. Generalized Discussion for Thermal Conductance 121

5.4 Summary

In this chapter, an interesting theoretical problem has been discussed which concerns sorne general behavior of thermal conductances G ij and Gij,kl for multi-terminal sys tems when thermal carries satisfy fraction al exclusion statistics. We can only make a general discussion because any practieal calculations require knowledge of the trans mission coefficient which must be calculated explicitly. In general, we found that the Onsager relations are still valid for FES. It can be proved that the monotonic behavior of two-terminal conductance is only true for 9 = 0, i.e., for ph on ons. For

9 > 0, both G ij and Gij,kl can be non-monotonie or monotonic function of tempera ture T, depending on peculiarities of the device. At the low temperature limit, Gij tends to 1[2:~T7Ù(f.-l) for aIl g. For 9 > 0, a four-terminal thermal bridge may be in a non-equilibrium state, in that case Gij,kl tends to zero as G ij does. When it is at equilibrium, Gij,kl may tend to any value for 9 > O. Therefore, the results for FES are quite different from the Bose-Einstein statistics in whieh Gij,kl always tends to 00 at very low temperature T. 6

Conclusion

In this thesis, theoretical investigations on sever al aspects of thermal transport in mesoscopic dielectric systems are presented. The research reported here focuses on situations where the system linear size is smaller than the thermal phonon wavelength thereby wave phenomena dominate thermal energy transport. Such situations occur in mesoscopie and nanoscopic se ale dielectric structures which can now be fabricated by several laboratories. The main theoretical result is the derivation, for dieleetric materials, of a formula for thermal energy flux in devices having multi-terminals each connected to a thermal reservoir at local equilibrium. The energy flux is driven by a temperature bias and traverses the system by virtue of phonon wave scattering. A multi-terminal thermal conductance formula is then derived in terms of phonon transmission coefficient. Us ing our theoretieal formulation, we investigated thermal transport properties of both two-terminal and four-terminal dielectric devices by solving the quantum scattering problem. In order to solve such scattering problems, a mode matching numerical technique has been developed and implemented. For thermal transport in a T-shaped dielectric nanostructure with two-terminals at low temperature, due to quantum interference the transmission coefficient of phonon becomes quite complicated. We found that the value of phonon transmission coeffi cients at small energy may be unity or zero depending on a geometrical ratio of the

122 Chapter 6. Conclusion 123

nanostructure, ~, where L is the stub width of the T-shaped wire, and a is the termi nal wire width. This reflects the fact that geometry has an influence on phonon wave transmission. The transmission has an oscillation behavior with quasi-periodicity and irregularity. The thermal conductance is found to increase monotonically with temperature - a result that we conclude to be generally true for any two-terminal device, and is markedly different from electron transport. We confirm the existence of universal quantum of thermal conductance which exists at the low temperature limit, and such a quantum is robust against all the system parameters. The physical behavior of four-terminal thermal conductance for mesoscopic di electric systems with arbitrary shapes of scattering region is also investigated in detail. If we make a two-terminal measurement in the four-terminal device, the two-terminal conductance is a monotonically increasing function of temperature, and is equal to the universal quantum of thermal conductance masked by a geometric factor. If we make a four-terminal measurement, the four-terminal conductance has a non-monotonic dependence. In the low temperature limit, we predict that the four-terminal conductance diverges inversely proportional to temperature. For a gen eral four-terminal thermal conductance Gij,kl, a set of relations have been proven:

Gij,kl = -Gij,lk = -Gji,kl = Gji,lk; Gij,kl = Gkl,ij; and Gij;kl + Gii,Jk + Gik~lj = o. Finally, we have discussed an interesting but purely theoretical problem, namely the general behavior of thermal conductance for multi-terminal systems when thermal carries satisfy fraction al exclusion statistics. Our analysis allows us to conclude that for two-terminal devices, the universal quantum of thermal conductance in lD ballistic transport is indeed statistics independent. But for four-terminal devices, the results for fractional exclusion statistics are quite different from those of the Bose-Einstein statistics.

Our work presented here points to the inter~sting nature of thermal transport in the quantum regime where wave phenomena of the phonons must be taken into Chapter 6. Conclusion 124 account. Such a scenario should be experimentally testable as the device setup of Roukes[ll] was already four-terminal. Furthermore, cross-lined carbon nanotubes have been reported in literature which may provide another ballistic thermal conduc tor for interesting experimental measurements. On the theoretical side, we have only investigated 2D devices for its mathematical simplicity, but full 3D models should be possible to solve by the mode mat ching methods. Another important and difficult theoretical issue is how to cross over from the quantum regime studied in Chapt ers 3-5 to the classical regime discussed in Chapter 2: one needs to introduce decoherence effects for the phonon waves for such a study. These and many other problems of thermal transport at the mesoscopic and nanoscopic dimension present interesting future research directions. Appendices

A.l A relation for transmission amplitudes In this Appendix, a detailed derivation of the relation, Eq.(2.39), for transmission amplitudes in multi-terminal system will be given. Starting from the classical energy current Q for displacement field u, Eq:(2.29): Q(x, t) LPxdydz âUjâUl -Cx'kl ----dydz J JA ât âXk -Cxjkl LdydzâtUjâkU l (A.1) where subscripts i,j, l, m indicate x, y, z coordinates (i,j, l, m = x, y, z). The dis placement field U in (A.1) has been made real by Eq.(2.25): 1 U = 2(u + u*) , (A.2) where u* is the conjugate of u. In terms of Eq.(2.24) in Chapter 2,

0"( 0' (t)tO"O'() u r, t) = ~anqUn,q~ a' -a' ( r, t) + ~~~anqUn',_q'~ ~ ~ -a' r, n'n W , (A.3) n 0' n n' where we represent the wave number with q. After putting (A.3) into (A.2), we obtain the displacement Ua' in the terminal wire c/,

Ua' = ~ (ua' (r, t) + ua' * (r, t))

1 a' - a' ( ) 0' - a' ( ) 0"0' ( ) 2 (~~ anqUn,q r, t + ~ ~ ~ anqUn',_q' r, t tn'n W n 0' n n' (A.4)

The displacement Ua' is divided into two parts, U(1) and U(2). After assuming that energy flux propagation is in the x direction, we have

(l) = 1 a' -a' ( ) 0"* -0"*( )) U 2 (~~anqUn,q r, t + ~anq~ Un,q r, t , (A.5)

125 Appendices 126

(A.6)

Similarly, we divide energy current Q(x, t) into two parts in terms of Eqs.(A.4), (A.5) and (A.6). We then take time average to energy current Q(x, t) in (A.1), namely (A.7)

1. Derivation for (Q(l)(X,t)): Because wave propagation direction is in the x-direction, the displacement in terminal a' is written as: _ , 1.( u' t ) , uO' .(r t) = --e-~ W nq -qx \Ji0' ,(y z) n,q,J' V'2'ii n,q,J" (A.8) where q is the wave number. Putting Eq.(A.4) into (A.1), we obtain the energy flux component Q(l)(X, t) by Eq.(A.5)-(A.8):

Q(l)(X, t) = -Cxjki LdydzotU?)Ok U ?)

{ d d 1 0 0" - 0" ( ) 0" * - 0" * ( )] - C xjki } A y z"'2 8t [~~ anqUn,q,j r, t + ~ a nq Un,q,j r, t

1 0 [~ 0" -0" ( ) ~ 0"* -0"* ( )] x "'2 OXk L....J an' q' un' ,q',1 r, t + L....J an' q' un' ,q' ,I r, t n' n' xjki _ C {d d ~ [~ 0" _1_ e -i(w:::~t-qxhTrO".( ) 4 } A Y Z 8t ~ a nq V'2'ii 'le' n,q,J y, Z

U ~ 0"* 1 i(w ' t-qX),TrO" ( )] 0 [~ 0" -0" ( ) + ~anq V'2'ii e nq 'le'n,q,j y,z X OXk ~an'q,Un',q,,1 r,t

+ L....J~ an'q,Un',q',10"* -0"* ( r, t )] n'

(A.9) Appendices 127

Now we take time average to Q(l). Note that the first two terms in (A.9), ü~:q,jak Ü~;~q',l and ü~:~ jakÜ~;,q"l' are independent of time t in terms of Eq.(A.8). However, using Eq.(2.22J, i.e.:

W~q W~;q' = W , (A. 10) the 1 a tt er t wo t erms ln. E q. (A .9) , Un,q,j-a' akUn',q',l -a' and Un,q,j-a'* akUn',q',l' -a'* con t·am fac t or 2' u' t 2' U't e- tWnq or e tWnq Therefore these terms time-average to zero: (A.11) and then,

(A.12)

(A.13)

The time average to (A.9) is reduced to:

xjkl /Q. (1) ()) C {d d [""' ""' a' a'* ( .) a' -a' a -a'* \ x, t = --4- lA y Z L.J L.J anqan'q' -1, WnqUn,q,j kUn',q',l n n'

""' ""' a' a'*· a' - a'* -a' + L.J L.J an'q,anq 1,WnqU n,q,j akUn',q',l 1 (A.14) n n' After exchanging indices of the second term in the right side of (A.14): n, q ? n', q', and using the following orthogonality condition[24]: (A.15) where V~,q = aW~,q/aq is the group velocity, and p is the mass density. Eq.(A.14) becomes

(A.16) Appendices 128

2. Derivation of (Q(2) (x, t)). This derivation is the same as we did ab ove.

Q(2) (x, t) -Cxjk1 LdydzatU;2)akU~2)

xjk1 ' --4-C 1d y d Z at ('"'"~~~anqUnl,-ql,jtnln '"'" '"'" U -u u'u A u n n'

(A.17) where ( ... ) is the content of the last line. Taking time average to (A.17), using (A.12) and (A.13), we have

(Q(2) (x, t))

(A.18)

After exchanging indices of the second term, (1, n, n', q, q' relations (A.lO) and (A.15), Eq.(A.18) becomes

(Q(2) (x, t)) Appendices 129

(A.19)

With expressions (A.16) and (A.19), the time-average of Q(11 is:

(A.20)

In terms of energy conservation, the sum of energy current from aH terminaIs in Eq.(A.20) should be zero, Le., L::(11 (Q(1/) = o. This requires:

(A.21)

We rewrite Eq.(A.21) into a matrix form expressed by matrices and vectors a, v, and t:

1 vOq o 1 o V 1q o o

The relation (A.21) is then written as:

atva = atttvta Appendices 130

Binee the coefficient a= (a~q) is arbitrary, we have

ttyt = y , and its component form is:

(A.22)

It is not difficult to verify that for a two-terminal device, the result (A.22) leads to what was derived by Blencowe (see Eqs.(8,9,1O) in Blencowe's paper[24]). From Eq.(A.22), one can further obtain

ler (tOn''* VOqll1 tlerOn + tler'*ln' Vlqll1 tlerln + . . . ) + (t2er'*On' VOqll2 t2erOn + t2er'*ln' Vlqll2 t2erln + . . . ) + . ..

= ~JV~:q,8nn,8erer' ,

and then

er" ,n"

"II nllqll Note that V " ter"ern"n lS. a h erml·t· lan mat· flX. H ence we h ave Vnq

and then er' "\;' V n' q' ter' er ter' er* = 1 L...J er n'n n'n . (A.23) er,n Vnq This is the expression (2.39) in Chapter 2, which is a relation of transmission ampli tude between modes in a multi-terminal system. This relation is used in quantizing the energy flux in Chapter.2.4.5. Appendices 131

A.2 Energy current formula In this Appendix, we will derive the energy flux formula of multi-terminal device by quantizing the classical energy flux Eq.(A.l). This derivation gives Eq.(2.51) of Chapter 2. The displacement field operator in a terminal wire can be written as[lO, 24, 81]:

(A.24) where â~q and â~! are phonon annihilation and creation operators, they satisfy the commutation relation:

(A.25) and the average value of â~:!,â~q is

(A.26) where nu is the Bose-Einstein distribution function. In terminal wire (j', the field operator Û(r, t) can be expanded by the basis fi~:q(r, t) of Eq.(A.3). We obtain

In terms of (A.l), we define thermal energy current operator of terminal wire (j'as

2/' (x, t) = -~CXjkl Ldydz (OtÛjOkÛI + OkÛIOtÛj) . (A.28) Putting the field operator (A.27) into energy current operator (A.28), we get the en ergy current operator in terminal (j' expressed by the basis fi~~(r, t). To be convenient, we first deal with the first term on the right hand side of Eq.(A.28): Appendices 132

(A.29)

A (j' Next, we take time-average to Q (x, t):

(A.30) Appendices 133

(A.31)

In the above derivation of (A.3I), we have applied the relations (A.25) and (A.26), and the following known relations (these can be obtained from Eqs.(A.ll), (A.12), and (A.13) of the last Appendix): '

( L (... )) = ( L (... )) = 0 . n,al ,nl ,n~ a,n,n' ,nl

Now we write LL(' .. ) in (A.3I) as ( ... )(1). Another part, L L (... ) in n nl a,n,n' al,nl,n~ (A.31), is written as (- .. )(2). We have Appendices 134

(A.32)

We change the first term's indices in (A.32), i.e., (n, q) ~ (nI, ql), and use (A.lO), (A.15), (A.25), and (A.26). (A.32) becomes

00 00 iCXjkl L 10 dq 10 dql dYdZ;p [(â~~tâ~:ql + OnnlO(q - ql)) ii~:,ql,j8kii~:~,1 ( n,n1 L

A(T'tA(T' -(T'*!:) -(T' ]) -anq an1q1 Un,q,jUkUnl,q1,1 00 00 iCxjkl L 10 dq 10 dql LdYdZ;p [( (â~~tâ~:q1) + Onn1 0(q - ql)) ii~:m,j8kii~:~,1 n,ni

(A.33) where we use (A.15) at the last second step. Next, the second part E E (... ) (T,n,n' (Tl ,nl,ni in (A.31) is: Appendices 135

(A.34)

We change the second term's indices in (A.34), Le., (0', n, q) ~ (A.lO), (A.15), (A.25), and (A.26). (A.34) becomes

(A.35) Appendices 136

(A.36)

We first eonsider the first term at right hand side of (A.36). Sinee the group velo city v~~ = aw~~/aq is eaneeled by the ID density of states g(w~~) = aq/aw~~, the first term transforms to

(A.37) where w~;o is the eutoff frequeney of mode n, Le., w~;lq'=o, and (A.23) has been used. Next, the third term in (A.36) is eonverted into:

ii.

(A.38)

Putting Eqs.(A.37) and (A.38) into (A.36), we obtain Appendices 137

(A.39)

Relation (A.39) is the time-average of the first part of energy current operator (A.28). Using exactly the same method, we can also obtain time-average for the second term in (A.28). The result is:

2. \ -Cxjkl LdydZOkÛlOtÛ j )

L L: dw 2~ W~;q' V;;q' t~:~t~:~* [n<7,(w~;q') - n<7(w~q)] <7,n,n' n'O nq 'C ,,{':x; d {d d li -<7'* ~ -<7' -1, xjkl ~ Jo q JA Y Z 2p Un,q,jUk Un,q,l

00 'C d d d li -<7'* 0 -<7' <7'<7 <7'<7* -1, xjkl L 1 q 1 Y Z-2 Un' -q' J' kUn, -q'ltn'ntn'n (A.40) , ,0 ApI' l' " 1 u,n,n ,nl

Combining relations (A.39) with (A.40), we derive the energy flux ( 2/') going from terminal (JI to the scattering region. Namely, Eq.(A.28) becomes:

+ iCxjkl " {oo d {d dz~fi<7' ,0 fi<7'* 2 ~ Jo q JA y 2p n,q,J k n,q,l

00 iCxjkl d d d li -<7' 0 -<7'* <7'a <7'<7* +-2- L 1 q 1 y z-2Un' -q'J' kUn, -q' ltn'ntn'n <7,n,n , ,nI,0 A P" l' l' 1

_ iCxjkl " {oo d {d dz~fi<7'* ,0 fi<7' 2 ~ Jo q JA y 2p n,q,J k n,q,l xjkl 00 - --iC L 1 d q 1d y d Z-Uli -<7'*n' -q' J' 0k Un' -<7' -q' ltn'ntn'<7'a <7'<7* n 2 <7,n,n , ,nI,0 A 2p l' l' " 1 Appendices 138

Using the orthogonality condition (A.lO) and (A.15), the dispersion relation V~,q = aW~,q/aq, the above becomes :

Using the (A.23), one finally get :

(A.41)

We define a transmission coefficient T:;':: (w): u rru u ( ) _ Vn' ' q' uu ( ) u u* ( ) .Ln'n' W = -u-tn'n' W tnl' n W (A.42) Vnq It represents the average probability that a phonon at mode n in terminal Cl will transmIt. to anot h er mo d'e n m. termma . l Cl. ' Because W = wnqU = Wn'qlu' = Wn"q",u" we obtain finally the following energy flux formula for multi-terminal devices from Eqs.(A.41) and (A.42):

A~) 1 Q = LLL 100

(A.43) is the Eq.(2.51) of Chapter 2, which is an important result of this work. It is used in Chapter 2.4.6 to derive thermal conductance of multi-terminal devices.

A.3 Universal quantum of thermal conductance In this Appendix, we derive the universal quantum of thermal conductance Eq.(2.55) of Chapter 2. We write the part corresponding to the massless mode in Eq.(2.54) as:

00 G - -1 Na 1 dwnw-1 (1 - 1) - 21T L 0 enw/kBT2 - 1 enw/kBTl - 1 Cl< ~T · 1ïw h Lettmg X = k ' t en B

00 1 Na k~ 1 G = - L 1 -xdx [Illx - ---:;:-x- 21T Cl< 0 1ï e T2 - 1 e Tl - 1 T2 - Tl

· X X h Lettmg YI =T ' Y2 = T ' t en I 2

G = ~ ~ [k~Ti {OO Y2 dY2 _ k~Tf {OO yidYI 1 1 . Y2 21T Cl< 1ï Jo e - 1 1ï Jo eY1 - 1 T2 - Tl

00 00 Due to 1 f(YI)dYI = 1 f(Y2)dY2, YI, Y2 can be replaced with y. Then

G = ~ ~ k~ Ti - Tf {OO ydy 21T Cl< 1ï T2 - Tl Jo eY - 1 2 00 ydy 1T Because 1 = -6 ' then 1o eY -

G = k~1T2 (Tl + T2 ) N 3h 2 Cl<

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