To pics in Ap plied Physics Volume 107 Topics in Applied Physics is p art of the SpringerLink service. For all customers with standing orders for Topics in Applied Physics we offer the full text in electronic form via SpringerLink free of c harge. P lease c ontact your librarian who can receive a password for free access to the full articles by registration at: springerlink.com → Orders I f you do not have a standing order you can nevertheless browse thro ugh the table of co nt ents of the volumes and the abstracts of each article at: springerlink.com → Browse Publications To pics in Ap plied Physics Topics in Applied Physics is a well-established series of review books, each of which presents a com- prehensive sur vey of a selected topic within the broad area of applied physics. Edited and written by leading research scientists in the field concerned, each volume contains review contributions cover- ing the various aspects of the topic. Together these provide an overv iew of the state of the art in the respective field, extending from an introduction to the subject right up to the frontiers of contempo- rary research. Topics in Applied Physics is addressed to all scientists at universities and in industry who wish to obtain an overview and to keep abreast of advances in applied physics. The series also provides easy but comprehensive access to the fields for newcomers starting research. Contributions are specially commissioned. The Managing Editors are open to any suggestions for topics coming from the community of applied physicists no matter w hat the field and encourage prospective editors to approach them with ideas. Managing Editors Dr. Claus E. Ascheron Springer-Verlag GmbH Tiergartenstr. 17 69121 Heidelberg Germany Email: claus.ascheron@springer.com Assistant Editor Adelheid H. Duhm Springer-Verlag GmbH Tiergartenstr. 17 69121 Heidelberg Germany Email: adelheid.duhm@springer.com Sebastian Volz (Ed.) Microscale and Nanoscale Heat Transfer With 144 Figures and 7 Tables In Collaboration with Rémi Carmina ti, Patrice Chantrenne, Stefan Dilhaire, Séverine Gomez, Nathalie Trannoy, and Gilles Tessier 123 Dr. Sebastian Vo lz Labortoire d’Enerégtique Moléculaire et Macroscopique, Combustion Ecole Central Paris Grande Voie des Vignes 92295 Châtenay Malabry, France volz@em2c.ecp.fr Library of Congress Control Number: 2006934584 Physics and A stronomy Classification S cheme (PA CS): 65.80.+n, 82.53.Mj, 81.16 c, 44.10.+i, 44.40.+a, 82.80.Kq ISSN print edition: 0303-4216 ISSN electronic edition: 1437-0859 ISBN-10 3-540-36056-5 Springer Berlin Heidelberg New York ISBN-13 978-3-540-36056-8 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part o f the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduct ion on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is per- mitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use m u st always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media springer.com © Springer-Verlag Berlin Heidelberg 2007 The use of gener al descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws a nd regulations and therefore free for general use. Typesetting: DA-T E X · Gerd Blumenstein · www.da-t ex.de Production: LE-T E X Jelonek, Schmidt & Vöckler GbR, Leipzig Cover design: WMXDesign GmbH, Heidelberg Printed on acid-free paper 57/3100/YL 5 4 3 2 1 0 Preface The development of micro- and nanoscale fabrication techniques has triggered a broad scientific and technical revolution. A prime example is provided by microelectronics, which has now become nanoelectronics. Other evolutionary breakthroughs are now clearly established in the fields of optoelectronics, materials, the production and conversion of energy, and techniques for data processing and communications. A remarkable feature of this trend is the way it has brought together physicists and engineers. On the one hand, the classical laws used to model macroscopic systems are generally unsuitable when system sizes approach characteristic microscopic scales, such as the mean free path or the length of carriers. The physical description of the individual or collective behaviour of the basic elements must then be reassessed. On the other hand, the de- velopment and integration of physical ideas exploiting very small structures, such as ultrathin films, superlattices, nanowires, and nanoparticles, in order to improve an industrial system, requires the physicist to understand some of the more technical aspects of engineering. The international community of thermal scientists, whether in research or engineering, base their approach on the mass, momentum and energy conser- vation equations associated with the laws of diffusion for conduction (Fourier) and for mass transfer (Fick), and Newton’s law for conduction–convection. For radiation, the radiative transfer equation is widely used to treat semi- transparent media, grey or otherwise. But this theoretical framework can no longer describe the conductive and conductive–convective transfer regimes on very small space and time scales, simply because the carriers undergo too few collisions. As the radiated ther- mal wavelengths are of the order of a few microns, the radiative transfer equation, and even the whole notion of luminance, become quite inappropri- ate on submicron scales. One does not even need to approach the limits of macroscopic models to observe that the phenomenology of heat transfer is quite different on the micron and centimeter length scales. Whilst heat transfer is generally felt to be a slow process – the time scale for heat conduction in macroscopic systems (∼ 50 cm) is a few minutes – the propagation of heat is an extremely efficient process on the microscale (∼ 10 ns). Indeed, the diffusion time is proportional to the square of the length. Moreover the thermal resistances of VI Preface microscale structures are so small that they become of the same order as the interface resistances between such structures. Microscale heat transfer thus occurs practically without inertia, and is essentially equivalent to interface heat transfer. Naturally, this is even more true for nanoscale heat transfer. From the experimental standpoint, very weak and highly localised contri- butions must be detected in order to measure the conductive flux in nano- structures. For example, the methods used must not introduce high con- tact thermal resistances. Ultrafast optical methods (nano- to picosecond) and near-field microscopy are best suited to satisfy these criteria. It is therefore clear that the study of heat transfer on micro- and nanoscales requires a quite new approach on the part of the thermal sci- ence community. The task here is to integrate the new physical models and also the novel experimental devices now available to treat energy exchanges in micro- and nanostructures. There are many consequences for industry: • In housing, superinsulating nanoporous materials can limit heat losses whilst increasing the ground surface, and their conductivity in vacuum is smaller than that of air. • Nanofluids, i.e., heat-carrying liquids transporting nanoparticles, have conductivities 10–40% higher than those of the base fluid and hence a greatly enhanced transfer efficiency. • In the nanoelectronics of processors, heating problems have led manufac- turers to slow down the miniaturisation trend by switching to multi-unit structures in which several computing units are integrated into the same chip. • Data storage will for its part be heat-assisted. Heating can activate or in- hibit magnetisation reversal. It can also change the phase or the geometry of a storage medium, and this over nanoscale areas. • Thermoelectric energy conversion is currently undergoing a revolution through manipulation of the thermophysical properties of nanostructured materials. In 2002, certain superlattice alloys were able to produce an intrinsic performance coefficient twice as high as had ever been measured for a bulk solid material. This breakthrough was achieved by improving thermal properties. In all these fields of application, our understanding of the relevant heat mech- anisms and the associated modelling tools remains poor or at best imperfect. The present book brings together for the first time the physical ideas and formalism as well as the experimental tools making up this new field of thermal science. Although these are usually considered to be the jurisdiction of the physicist, the aim of the book remains quite concrete, since it seeks to solve the problems of heat transfer in micro- and nanostructured mate- rials. The book itself results from a collaborative network in France known as the Groupement de Recherche Micro et Nanothermique (GDR), bringing Preface VII together teams organised by a unit of the Centre national de la recherche sci- entifique (CNRS) 1 and a unit of the department 2 of Sciences pour l’Ing´enieur. This group combines research centres involved in thermal science, solid state physics, optics and microsystems. Each chapter has been written by one or several authors – sometimes belonging to different research teams – and then edited by experts and non-experts in the GDR. The first part of the book is theoretical, making the connection between the fundamental approaches to energy transfer and the quantities describing heat transfer. Chapter 1 considers the limits of classical models on small scales. Chapters 2, 3 and 4 then treat the physical models describing heat transfer in gases, conduction, and radiation, respectively, all on these small scales. The second part of the book covers the numerical tools that can be imple- mented to solve the previously formulated equations in concrete situations. Chapters 5 and 6 examine solutions of the Boltzmann and Maxwell equations, respectively. Having discussed continuum models, microscopic simulations are tackled in Chap. 7 via the Monte Carlo method and in Chap. 8 via the tech- nique of molecular dynamics simulations. In each chapter, it is shown how to calculate a heat flux or conductivity explicitly through various examples. The last part of the book deals with experimental approaches. Chapter 9 introduces different forms of near-field microscopy and discusses their appli- cations in thermal science. A thermal microscope is presented in some detail with example applications. Chapter 10 discusses optical techniques as pro- vided by the photothermal microscope and reflectometry, whilst Chap. 11 brings together optical and near-field microscopy in a single hybrid system. This series of chapters on microscopy is followed by two chapters presenting the thermal applications of femtosecond lasers in pump–probe configurations. Chapter 12 deals with the electron–photon interaction on ultrashort time scales and Chap. 13 treats of thermal–acoustic coupling in various types of structure. The book thus constitutes a particularly complete and original collection of ideas, models, numerical methods and experimental tools that will prove invaluable in the study of micro- and nano-heat transfer. It should be of interest to research scientists and thermal engineers who wish to carry out theoretical research or metrology in this field, but also to physicists concerned with the problems of heat transfer, or teachers requiring a solid foundation for an undergraduate university course in this area. 1 The French National Research Institute. 2 Science for Engineering. VIII Preface Acknowledgements The strength of this book mostly relies on the collaborative effort of my dear colleagues. I am glad to express my deep thanks to R´emi Carminati, Patrice Chantrenne, Bernard Cretin, Stefan Dilhaire, Dani`ele Fournier, S´everine Gomez, Jean-Jacques Greffet, Karl Joulain, Denis Lemonnier, Bernard Perrin, Nathalie Trannoy, Gilles Tessier, Fabrice Vall´ee and Pascal Vairac for pro- viding a work of highest quality in their field of expertise. Paris, Sebastian Volz November, 2005 Contents Laws of Macroscopic Heat Transfer and Their Limits Jean-Jacques Greffet 1 1 Heat Conduction in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 MacroscopicApproach 1 1.2 Characteristic Length and Time Scales . . . . . . . . . . . . . . . . . . . . 2 1.3 Short-ScaleTransfer 5 2 Conduction in Fluids. Convection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.1 MacroscopicApproach 5 2.2 Short-Scale Transfer. Ballistic Transport . . . . . . . . . . . . . . . . . . . 7 3 Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 3.1 MacroscopicApproach 7 3.2 Characteristic Length and Time Scales . . . . . . . . . . . . . . . . . . . . 9 4 Conclusion 12 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 Transport in Dilute Media R´emi Carminati 15 1 Distribution Function and Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.1 Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.2 Averages 16 1.3 Conductive Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2 Thermodynamic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 2.2 Equilibrium Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . 18 3 Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.1 Dynamical Equation for the Distribution Function . . . . . . . . . . 19 3.2 The Relaxation Time Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4 Local Thermodynamic Equilibrium. Perturbation Method . . . . . . . . 21 4.1 Dimensionless Boltzmann Equation . . . . . . . . . . . . . . . . . . . . . . . 21 4.2 Mean Free Path. Collision Time. Knudsen Number . . . . . . . . . . 21 4.3 Local Thermodynamic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . 23 4.4 Perturbation Method. Linear Response . . . . . . . . . . . . . . . . . . . . 24 4.5 Fourier Law and Thermal Conductivity . . . . . . . . . . . . . . . . . . . . 24 X Contents 5 Example of a Non-LTE System. Short-Scale Conduction in a Gas . . 25 5.1 Can One Speak of Temperature on Short Scales? . . . . . . . . . . . 26 5.2 Calculating the Conductive Flux in the Ballistic Regime . . . . . 27 5.3 Transitions Between Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 6 Conclusion 30 A Equilibrium Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 B Dynamical Evolution of the Distribution Function forFreeParticles 32 C Calculating the Constants A and B for the Flux in the Ballistic Regime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 Electrons and Phonons Jean-Jacques Greffet 37 1 Electrons 38 1.1 Free Electrons 38 1.2 Electrons in a Periodic Potential . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.3 Electrical Conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.4 Semi-ClassicalApproach 43 1.5 Electrical Conductivity in the Collisional Regime . . . . . . . . . . . 45 1.6 Electrical Conduction in the Ballistic Regime . . . . . . . . . . . . . . . 46 2 Phonons 47 2.1 Vibrational Modes in a Lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 2.2 PhononEnergy 49 2.3 Density of States. Optical and Acoustic Modes . . . . . . . . . . . . . 50 2.4 Calculating the Heat Flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.5 Calculating the Thermal Conductivity . . . . . . . . . . . . . . . . . . . . . 52 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 Introduction to Radiative Transfer R´emi Carminati 55 1 Radiative Transfer Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 1.1 Specific Intensity, Flux, Energy Density. . . . . . . . . . . . . . . . . . . . 55 1.2 Absorption, Scattering and Thermal Emission . . . . . . . . . . . . . . 56 1.3 Establishing the RTE. Radiative Energy Balance . . . . . . . . . . . 59 1.4 Discussion 60 2 From the RTE to the Diffusion Approximation . . . . . . . . . . . . . . . . . . 60 2.1 From the P 1 Approximation to the Diffusion Equation . . . . . . . 61 2.2 Discussion 64 2.3 Rosseland Approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3 Transport Regimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 3.1 Static Transmission. Ohmic Conductance and Short-Scale Deviations . . . . . . . . . . . . 66 [...]... l’Ecole Centrale Paris Topics in Applied Physics Volume 107 Topics in Applied Physics is part of the SpringerLink service For all customers with standing orders for Topics in Applied Physics we offer the full text in electronic form via SpringerLink free of charge Please contact your librarian who can receive a password for free access to the full articles by registration at: springerlink.com → Orders If... here is to integrate the new physical models and also the novel experimental devices now available to treat energy exchanges in micro- and nanostructures There are many consequences for industry: • In housing, superinsulating nanoporous materials can limit heat losses whilst increasing the ground surface, and their conductivity in vacuum is smaller than that of air • Nanofluids, i.e., heat- carrying liquids... a standing order you can nevertheless browse through the table of contents of the volumes and the abstracts of each article at: springerlink.com → Browse Publications Topics in Applied Physics Topics in Applied Physics is a well-established series of review books, each of which presents a comprehensive survey of a selected topic within the broad area of applied physics Edited and written by leading... pressure Inserting the Fourier law into this expression and assuming that the thermal conductivity is homogeneous, we obtain a diffusion equation for the temperature field, viz., ρcp ∂T = k∇2 T ∂t Defining the thermal diffusivity by a = k/ρcp , (3) becomes ρcp ∇2 T = 1 ∂T a ∂t (3) (4) S Volz (Ed.): Microscale and Nanoscale Heat Transfer, Topics Appl Physics 107, 1–13 (2007) © Springer-Verlag Berlin Heidelberg... resistances of VI Preface microscale structures are so small that they become of the same order as the interface resistances between such structures Microscale heat transfer thus occurs practically without inertia, and is essentially equivalent to interface heat transfer Naturally, this is even more true for nanoscale heat transfer From the experimental standpoint, very weak and highly localised contributions... scientists in the field concerned, each volume contains review contributions covering the various aspects of the topic Together these provide an overview of the state of the art in the respective field, extending from an introduction to the subject right up to the frontiers of contemporary research Topics in Applied Physics is addressed to all scientists at universities and in industry who wish to obtain an... micro- and nano -heat transfer It should be of interest to research scientists and thermal engineers who wish to carry out theoretical research or metrology in this field, but also to physicists concerned with the problems of heat transfer, or teachers requiring a solid foundation for an undergraduate university course in this area 1 2 The French National Research Institute Science for Engineering VIII... Abstract In this introductory text, we examine the three mechanisms of heat transfer For each one, we review the main ideas used in the traditional macroscopic description of heat transfer This is followed by a discussion of the length and time scales characterising these transfer mechanisms We then study the hypotheses underlying these models in order to determine their field of validity We outline transfer... of the individual or collective behaviour of the basic elements must then be reassessed On the other hand, the development and integration of physical ideas exploiting very small structures, such as ultrathin films, superlattices, nanowires, and nanoparticles, in order to improve an industrial system, requires the physicist to understand some of the more technical aspects of engineering The international... specific intensity and the theory of geometric optics 8 Jean-Jacques Greffet Specific Intensity A radiative energy flux crossing a surface of area dS in the direction u, in a solid angle dΩ and a frequency band [ν, ν + dν] is expressed in the form dφν = Lν (u, r) dS cos θ dΩ dν (17) The quantity L is the specific intensity It depends on the frequency, the direction and the point considered It can be interpreted . To pics in Ap plied Physics Volume 107 Topics in Applied Physics is p art of the SpringerLink service. For all customers with standing orders for Topics in Applied Physics we offer. Macroscopique, Combustion de l’Ecole Centrale Paris To pics in Ap plied Physics Volu me 107 Topics in Applied Physics is p art of the SpringerLink service. For all customers with standing orders. Browse Publications To pics in Ap plied Physics Topics in Applied Physics is a well-established series of review books, each of which presents a com- prehensive sur vey of a selected topic within