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The Casimir Energy and Radiative Heat Transfer Between Nanostructured Surfaces Lussange Johann

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The Casimir energy and radiative heat transfer between nanostructured surfaces Lussange Johann To cite this version: Lussange Johann. The Casimir energy and radiative heat transfer between nanostructured surfaces. Quantum Physics [quant-ph]. Université Pierre et Marie Curie - Paris VI, 2012. English. tel-00879989 HAL Id: tel-00879989 https://tel.archives-ouvertes.fr/tel-00879989 Submitted on 5 Nov 2013 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. THESE DE DOCTORAT DE L’UNIVERSITE PIERRE ET MARIE CURIE Sp´ecialit´e: Physique Ecole doctorale ED 107 R´ealis´ee au Laboratoire Kastler Brossel (UPMC-ENS-CNRS) Pr´esent´ee par Johann LUSSANGE Pour obtenir le grade de : DOCTEUR de l’UNIVERSITE´ PIERRE ET MARIE CURIE Sujet de la th`ese : The Casimir energy and radiative heat transfer between nanostructured surfaces Soutenue le 10 Septembre 2012 devant le jury compos´ede : Mme. Astrid LAMBRECHT : Directrice de th`ese M. Karl JOULAIN : Rapporteur M. George PALASANTZAS : Rapporteur M. Mauro ANTEZZA : Examinateur M. Daniel BLOCH : Examinateur M. Jean-Marc FRIGERIO : Examinateur M. Cyriaque GENET : Examinateur M. Serge REYNAUD : Invit´e 2 blank 3 Contents I. Bref r´esum´een Fran¸cais 7 II. Short summary in English 9 III. Acknowledgments 11 IV. Introduction 13 V. Theory : Quantum fields and the vacuum state 19 A. Quantum theory and radiative heat transfer in the classical description 19 1. Quantization of light : Planck’s law, and Einstein’s generalization to photons 19 2. Laws of thermodynamics and Onsager’s reciprocal relations 21 3. Properties of radiative heat transfer 25 4. Thermal radiation through a medium 28 5. Conclusion 31 B. Quantum fields and electrodynamics 32 1. From quantum particles to relativistic fields 32 2. Noether’s theorem and the stress-energy tensor 35 3. Green’s functions and field interactions 37 4. QED equation of motion and quantization 41 5. The vacuum state, zero-point energy, and vacuum fluctuations 47 6. Conclusion 50 VI. Theory : Scattering theory applied to Casimir and near-field heat transfer 51 A. Casimir effect in the plane-plane geometries 51 1. Complex permittivity and fitting models 51 2. Fresnel-Stokes amplitudes and S-matrices 55 3. Quantization and Airy function 60 4. Derivation of the Casimir force between two planes 64 5. Analyticity conditions of the cavity function : causality, passivity, stability 67 6. The Casimir force over imaginary frequencies and Cauchy’s theorem 71 7. Matsubara frequencies and Casimir for non-zero temperatures 75 8. Derjaguin’s Proximity Approximation 79 9. Polariton coupling with surface-quasiparticles 80 10. Conclusion 84 B. Casimir effect in non-planar geometries 86 4 1. The RCWA method and associated reflection matrix for gratings 86 2. The Casimir energy for periodic gratings 95 3. The Casimir energy for arbitrary periodic gratings 98 C. Out-of-thermal equilibrium phenomena 101 1. The Casimir energy for periodic gratings 101 2. Near-field radiative heat transfer for periodic gratings 114 VII. Numerical evaluation : Casimir for zero temperatures 120 A. Casimir energy between planar surfaces 120 B. Casimir energy between corrugated gratings 121 1. Casimir energy between corrugated gratings as a function of the separation distance L 122 2. Casimir energy between corrugated gratings as a function of the grating period d 126 3. Casimir energy between corrugated gratings as a function of the filling factor p 129 4. Casimir energy between corrugated gratings as a function of the corrugation depth a 134 5. Casimir energy between corrugated gratings as a function of the lateral displacement δ 137 C. Casimir force between corrugated gratings 141 D. Lateral Casimir force between corrugated gratings 143 E. Casimir energy between arbitrary periodic gratings 147 1. Casimir energy between periodic profiles shaped as sawteeth 147 2. Casimir energy between periodic profiles shaped as barbed wires 149 3. Casimir energy between periodic profiles shaped as a sinusoid 151 4. Casimir energy between periodic profiles shaped as ellipsoids 154 5. Casimir energy for different compared arbitrary profiles 155 VIII. Numerical evaluation : Casimir for non-zero temperatures 158 A. Casimir at thermal equilibrium 158 1. Casimir energy as a function of the separation distance for T = 0 K and T = 300 K 158 2. Casimir energy as a function of the temperature for L = 100 nm, 500 nm, 1µm, and 10µm 161 B. Casimir out-of-thermal equilibrium 167 1. Comparison between equilibrium and out-of-equilibrium situations 167 2. Dependence on the temperature gradient and temperature average 170 3. Repulsivity and contribution of the non-equilibrium term 171 IX. Numerical evaluation : Radiative heat transfer in near-field 175 A. Radiative heat transfer between planes 175 B. Radiative heat transfer between gratings 179 C. A thermal modulator device for nanosystems 184 5 X. Discussion of the results and conclusion 189 References 205 6 blank 7 I. BREF RESUM´ E´ EN FRANC¸AIS Le sujet de cette th`ese porte sur les calculs num´eriques de deux observables quantiques influents `al’´echelle sub- microm´etrique : le premier ´etant la force de Casimir et le second ´etant le transfert thermique radiatif. En champ proche, ces deux grandeurs physiques sont `al’origine de nombreuses applications potentielles dans le domaine de la nano-ing´enierie. Elles sont th´eoriquement et exp´erimentalement bien ´evalu´ees dans le cas de g´eom´etries simples, comme des cavit´es de Fabry-P´erot form´ees par deux miroirs plans parall`eles. Mais dans le cas des g´eom´etries complexes invariablement rencontr´ees dans les applications nanotechnologiques r´eelles, les modes ´electromagn´etiques sur lesquels elles sont construites sont assujettis `ades processus de diffractions, rendant leur ´evaluation consid´erablement plus complexe. Ceci est le cas par exemple des NEMS ou MEMS, dont l’architecture est souvent non-triviale et hautement d´ependante de la force de Casimir et du flux thermique, avec par exemple le probl`eme de malfonctionnement courant dˆu `al’adh´erence des sous-composants de ces syst`emes venant de ces forces ou flux. Dans cette th`ese, je m’int´eresse princi- palement `ades profils p´eriodiques de forme corrugu´ee —c’est-`a-dire en forme de cr´eneaux— qui posent d’importantes contraintes sur la simplicit´ede calcul de ces observables. Apr`es une revue fondamentale et th´eorique jetant les bases math´ematiques d’une m´ethode exacte d’´evaluation de la force de Casimir et du flux thermique en champ proche centr´ee sur la th´eorie de diffusion, la seconde et principale partie de ma th`ese consiste en une pr´esentation des estimations num´eriques de ces grandeurs pour des profils corrugu´es de param`etres g´eom´etriques et de mat´eriaux diverses. En particulier, j’obtiens les tous premiers r´esultats exacts de la force de Casimir hors-´equilibre-thermique et du flux thermique radiatif entre des surfaces corrugu´ees. Je conclus par une proposition de conception d’un modulateur thermique pour nanosyst`emes bas´ee sur mes r´esultats. 8 blank 9 II. SHORT SUMMARY IN ENGLISH The subject of this thesis is about the numerical computations of two influent quantum observables at the nanoscale: the Casimir force and the radiative heat transfer. In near field, these two physical quantities are at the origin of numerous potential applications in the field of nano-engineering. They are theoretically and experimentally well evaluated in the case of simple geometries such as Fabry-P´erot cavities, which consist in two parallel plane mirrors separated by vacuum. But in the case of the more complex geometries which are unavoidably encountered in practical nanotechnological applications, the electromagnetic modes from which they are derived are subject to scattering processes which make their evaluation considerably more complex. This is for instance the case of NEMS and MEMS, whose general architecture is often non-trivial and highly dependent on the Casimir force and radiative heat flux, with for example the often encountered problem of stiction in these nano-devices. In this thesis I mainly focus on corrugated periodic profiles, which bring important constraints on the simplicity of the computations associated with these observables. After a fundamental review of the mathematical foundations of an exact method of computation of the Casimir force and of the heat flux based on scattering theory, I present in the second part of this thesis the results of the numerical calculations of these quantities for corrugated profiles of various geometrical parameters and for different materials. In particular, I obtain the very first exact results of the out-of-thermal equilibrium Casimir force and of the radiative heat flux between corrugated surfaces. I conclude with a proposal for the design of a thermal modulator device for nanosystems based on my results. 10 blank 11 III. ACKNOWLEDGMENTS Je voudrais tout d’abord remercier chaleureusement Romain, sans qui la plus grande partie de ce travail n’aurait pas pu ˆetre fait. Un grand merci ´egalement `aAstrid, qui m’a guid´ependant ces trois ann´ees de th`ese et qui a toujours su m’encourager dans les diff´erents sujets de recherche, ainsi qu’`aSerge pour tous ses conseils et son aide. Merci ´egalement `aDiego Dalvit, Jean-Jacques Greffet, Felipe Rosa, et Jean-Paul Hugonin pour tout leur travail, et pour leur fructueuse collaboration.
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