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Radiative Heat Transfer at the Nanoscale Jean-Jacques Greffet Institut d’Optique, Université Paris Sud Institut Universitaire de France CNRS Outline of the lectures d 1. Heat radiation close to a surface d 2. Heat transfer between two planes 3. Mesoscopic approach, fundamental limits and Applications Outline 1. Some examples of radiative heat transfer at the nanoscale 2. Radiometric approach 3. Fluctuational electrodynamics point of view 4. Basic concepts in near-field optics 5. Local density of states. Contribution of polaritons to EM LDOS 6. Experimental evidences of enhanced thermal fields Heat radiation at the nanoscale : Introduction 0 d Casimir and heat transfer at the nanoscale: Similarities and differences Heat radiation at the nanoscale : Introduction 0 d Casimir and heat transfer at the nanoscale: Similarities and differences Fluctuational fields are responsible for heat transfer and forces between two parallel plates. Key difference: 1. no heat flux contribution for isothermal systems. 2. Spectral range: Casimir : visible frequencies. Heat transfer : IR frequencies. Heat radiation at the nanoscale : Introduction 1 h R DT T+DT d T 1 d2 The flux is several orders of magnitude larger than Stefan-Boltzmann law. Microscale Thermophysical Engineering 6, p 209 (2002) Heat radiation at the nanoscale : Introduction 2 d d=10 nm, T=300K Heat transfer can be quasimonochromatic Microscale Thermophysical Engineering 6, p 209 (2002) Heat radiation at the nanoscale : Introduction 3 1.0 Energy density Density of energy near a SiC-vacuum interface z=100 m 0.8 0.6 0.4 0.2 Energy density 0.0 z z=1 m 15 10 5 T=300 K Energy density close to surfaces is orders of magnitude larger than blackbody energy density and monochromatic. Energy density 0 20x10 3 z= 100 nm 15 10 5 0 0 100 200 300 (Hz) 400 500x10 12 PRL, 85 p 1548 (2000) Heat radiation at the nanoscale : Introduction 4 Thermally excited fields can be spatially coherent in the near field. de Wilde et al. Nature (2006) Proof of spatial coherence of thermal radiation Greffet et al., Nature 416, 61 (2002) Modelling Thermal Radiation The standard radiometric approach T+DT d T The standard radiometric approach q I T I T e BB The standard radiometric approach Isotropic black body emitter (=1) dQ BB BB I (T)cos q d I (T) 2 Q BB 4 I (T) d T 0 What is missing ? Where is the extra energy coming from ? Material with emissivity in front of a black body 1,T1 Kirchhoff’s law: a ,T2 Energy balance for lower medium: Q Qabs Qe aT T 4 1 4 2 T T 4 1 4 2 Modelling Thermal Radiation The fluctuational electrodynamics approach Rytov, Kravtsov, Tatarskii, Principles of radiophysics, Springer The fluctuational electrodynamics approach T 1) The medium is assumed to be at local thermodynamic equilibrium volume element = random electric dipole j(r,t) with j(r,t) 0 2) Radiation of random currents = thermal radiation E(r, ) i o Gr,r' , jr' , dr' The fluctuational electrodynamics approach 3) Spatial correlation (cross-spectral density) E j E *k r,r', o2 2 G r,r G r',r j r j r jm 1 * kn Ur, Er, 2 m 1 * n 2 dr1dr2 2 4) The current-density correlation function is given by the FD theorem jm (r1 ) j (r2 ) o Im mn (r1 r2 ) exp / kB T 1 * n Fluctuation Absorption Casimir and Heat transfer d Maxwell-stres tensor can be computed Poynting vector can be computed Dealing with non-equilibrium situation jm (r1 ) j (r2 ) o Im mn (r1 r2 ) exp / kB T 1 * n T2 d T2+DT T2 T1 What about the temperature gradient ? Can we assume the temperature to be uniform ? hT2 T1 T k z T h 1000 7 DT d d 10 104 K z k 1 Physical origin of Kirchhoff’s law Kirchhoff’s law: a Physical origin of Kirchhoff’s law Physical meaning of emissivity and absorptivity Kirchhoff’s law: a 1 2 T21=T12 Basic concepts of near field What is so special about near field ? Basic concepts of near field Back to basics : dipole radiation 1/r terms Radiation Near field: k0r<<1, k0=2 1/r2 and 1/r3 terms Near field Evanescent waves filtering Basic concepts of near field: evanescent waves exp ikR i R 2 k k c c k dk x dk y exp i(k x x k y y k z z ) kz kz ; kz i ; ; 2 c 2 k 2 2 c2 ; k (k x , ky ) k 2 k z ik c exp ik z z exp k z Take home message: large k are confined to distances 1/k. Evanescent waves filtering Energy density: in vacuum, close to nanoparticle, close to a surface Black body radiation in vacuum g( ) 2 3 c 2 U 2 3 e 0 c 2 kT 1 d Black body radiation close to a nanoparticle Questions to be answered : Is the field orders of magnitude larger close to particles ? If yes, why ? Is the thermal field quasi monochromatic ? If yes, why ? Black body radiation close to a nanoparticle pm (r1 ) p*n (r2 ) 2 o Ima mn (r1 r2 ) exp /kB T 1 Black body radiation close to a nanoparticle pm (r1 ) p (r2 ) * n 2 o Ima mn (r1 r2 ) exp /kB T 1 1. The particle is a random dipole. 2. The field diverges close to the particle: electrostatic field ! 3. The field may have a resonance, plasmon resonance. Where are these (virtual) modes coming from ? Polaritons Strong coupling between material modes and photons produces polaritons: Half a photon and half a phonon/exciton/electron. + + - + - - + + - - - + Where are the modes coming from ? Estimation of the number of electromagnetic modes in vacuum: N V 3 g( ')d ' 3 2c 3 0 N V 3 Estimation of the number of vibrational modes (phonons): V N 3 a The electromagnetic field inherits the DOS of matter degrees of freedom Can we define a (larger and local) density of states close to a particle or a surface ? LDOS in near field above a surface LDOS is also used to deal with spontaneous emission using Fermi golden rule Joulain et al., Surf Sci Rep. 57, 59 (2005), Phys.Rev.B 68, 245405 (2003) Local density of states close to an interface Lifetime of Europium 3+ above a silver mirror Drexhage, 1970 Electrical Engineering point of view • Interference effects • (microcavity type) Near-field contribution Drexhage, 1970 Quantum optics point of view: LDOS above a surface • Interference effects • (microcavity type) Near-field contribution Drexhage, 1970 Energy density above a SiC surface 1.0 Energy density Density of energy near a SiC-vacuum interface z=100 m 0.8 0.6 0.4 0.2 Energy density 0.0 z z=1 m 15 10 5 T=300 K Energy density close to surfaces is orders of magnitude larger than blackbody energy density and monochromatic. Energy density 0 20x10 3 z= 100 nm 15 10 5 0 0 100 200 300 (Hz) 400 500x10 12 PRL, 85 p 1548 (2000) Density of energy near a Glass-vacuum interface z T=300 K Physical mechanism The density of energy is the product of - the density of states, the energy hn the Bose Einstein distribution. The density of states can diverge due to the presence of surface waves : Surface phonon-polaritons. First picture of a surface plasmon E x exp ikx iz it Dawson, Phys.Rev.Lett. Surface plasmon excited by an optical fiber D Courtesy, A. Bouhelier Dispersion relation of a surface phonon-polariton k c 1 k It is seen that the number of modes diverges for a particular frequency. PRB, 55 p 10105 (1997) LDOS in near field above a surface Joulain et al., Surf Sci Rep. 57, 59 (2005), Phys.Rev.B 68, 245405 (2003) LDOS in near field above a surface Joulain et al., Surf Sci Rep. 57, 59 (2005), Phys.Rev.B 68, 245405 (2003) LDOS in the near field above a surface This can be computed close to an interface ! Joulain et al., Surf Sci Rep. 57, 59 (2005), Phys.Rev.B 68, 245405 (2003) LDOS in the near field above a surface Plasmon resonance Electrostatic enhancement Experimental evidence of thermally excited near fields Direct experimental evidence de Wilde et al. Nature 444, p 740 (2006) Imaging the LDOS of surface plasmons Thermally excited fields can be spatially coherent in the near field. de Wilde et al. Nature (2006) Spatial coherence of thermal radiation M P z r T=300 K PRL 82, 1660 (1999) Proof of spatial coherence of thermal radiation Greffet, Nature 416, 61 (2002) Spectrum of the thermal near field Babuty et al. Arxiv Lifetime of an atom in a magnetic trap Superconducting niobium Copper 1.8 and 6.24 MHz Lifetime in a magnetic trap versus distance to surface. Thermal blackbody is increased by ten orders of magnitude. D. Harber et al., J.Low Temp.Phys. 133, 229 (2003); C. Henkel et al. Appl.Phys.B 69, 379 (1999) C. Roux et al., EPL 87, 13002, (2009) Second Lecture Radiative heat transfer at the nanoscale Heat transfer between two nanoparticles PRL 94, 85901, (2005) PRL 94, 85901, 2005 Heat transfer between a particle and a dielectric (SiC) surface Energy transfer betwen a particle and a plane d=100nm T=300 K Take home message: For source at distance d, power dissipated within a distance 2 d Mulet et al. Appl.Phys.Lett. 78, p 2931 (2001) Energy transfer between a particle and a plane Key feature 1 d T=300 K Key feature 2 Microscopic resonance if +2=0 Take home message: The flux is dramatically enhanced by the surface waves resonances Mulet et al. Appl.Phys.Lett. 78, p 2931 (2001) Heat transfer between a particle and a metallic surface Radiative heat transfer between a particle and a surface B E j The magnetic field penetrates in the particle. It induces eddy currents that produce Joule losses. How do magnetic losses compare with electric losses ? It is often assumed that the magnetic dipole is negligible. Losses are given by the product of two terms : 0 | E | | B |2 PE 2 Im( a E ) 2 Im( a M ) 2 20 Chapuis, Phys.Rev.B 125402, 2008 2 Radiative heat transfer between a particle and a surface Magnetic losses versus electric losses a E 4R3 •Electric dipole • Magnetic polarisability 1 2 2 3 R 2 aM R 2 ( 1) 15 | a M || a E | for R 0 Ima) (m3) R=10 nm Im( a E ) | |1 Im(aM ) for a metal at low frequencies Im( a E ) Im(aM ) R=5 nm Chapuis, Phys.Rev.B 125402, 2008 Radiative heat transfer between a particle and a surface Electric and magnetic energy density above an interface : an unusual property z z E B kp kp k0 B y x s polarisation:magnetostatics propagating E B c k0 E evanescent E k p B 2 1 c k0 Magnetostatics 2 y x p polarisation: electrostatics propagating evanescent k p E cB 2 1 k0 2 E cB Chapuis, Phys.Rev.B 125402, 2008 Radiative heat transfer T1 T T 4 1 d 4 2 T2 Theme : influence of surface waves on the mesoscopic energy transfer Anomalous heat transfer Radiation shield Hargreaves, 1969 Anomalous heat transfer Radiative heat transfer at nanoscale : Experimental data 1) Measurements at low temperature, Hargreaves (69), 2) Measurements between metals in the infra red, Xu: inconclusive, 3) Measurements in the nm regime with metal (Oldenburg, 2005). Au Kittel et al. , PRL 95 p 224301 (2005) GaN Radiative heat transfer at the nanoscale Radiometric approach T1 d Radiative heat transfer at the nanoscale Theory : Polder, van Hove, PRB 1971 (Tien, Caren, Rytov, Pendry) T1 d Transmission factor Heat transfer between two SiC half spaces Enhancement due to surface waves Mulet et al., Microscale Thermophysical Engineering 6, 209, (2002) Mulet et al. Appl.Phys.Lett. 78, 2931 (2001) Analogy with Casimir force Henkel et al., Phys.Rev.A 69, 023808 (2004) Casimir force for aluminum 10 nm 100 nm 10 µm Henkel et al., Phys.Rev.A 69, 023808 (2004) Drude model is not accurate Gold Nordlander’s model 4 Lorentzian terms are used Silver Circle: Johnson and Christy Blue : Drude model Hao and Nordlander, Chem.Phys.Lett. 446, 115 (2007) It is even worse than that ! Optical properties depend on the deposition technique. Recent experimental results Radiative heat transfer at nanoscale Measurements with silica A. Narayanaswamy, G. Chen PRB 78, 115303 (2008) Detection scheme - Vertical configuration Optical fiber - Heating of the sample - Vacuum P~10-6 mbar E. Rousseau A. Siria, J. Chevrier Heater Nature Photonics 3, p 154 (2009) Comparison theory/data Nature Photonics 3, p 154 (2009) C. Otey and S. Fan, Phys.Rev.B 84, 245431 (2011) Textured surfaces Heat transfer between gold gratings d=1 µm d=2.5 µm d=10 µm PFA R. Guerout et al., Phys.Rev.B 85, 180301 (2012) Textured surfaces R. Guerout et al., Phys.Rev.B 85, 180301 (2012) Heat transfer between SiC gratings Heat transfer between two SiO2 gratings: Physical mechanism 8,75 µm SiO2 resonance L=25 nm a=500 nm d=1500 n J. Lussange et al., Phys.Rev.B 86, 085432 (2012) 9.15 µm Introduction to the concept of quantum conductance Mesoscopic analysis of heat transfer at the nanoscale Mesoscopic charge transport : a tutorial 1 Experimental observation 2 Landauer formalism I V 2e 2 Tn n h Hand wavy derivation of the quantum of cinductance Perfect conductors resists ! I= 2e/Dt (One electron at a time per state) Dt=h/DE DE=eV I= [2e2/h] V (R=12,9 k) Landauer formula: I n Tn 2e 2 h V 87 Fundamental limits to radiative heat transfer at the nanoscale Sum over modes A mesoscopic formulation of radiative heat transfer at nanoscale I Tn n 2e 2 h V 2 kB2 T d 2k T(k) DT 2 4 3 h Transmission factor Thermal quantum conductance d 2/L Biehs et al., Phys.Rev.Lett. 105, 234301 (2010) Fundamental limit: a new perspective on Stefan’s constant Are we there yet ? Biehs et al., Phys.Rev.Lett. 105, 234301 (2010) Quantized radiative thermal conductance L If L is on the order of the thermal wavelength, d 2k 2 4 n Biehs et al., Phys.Rev.Lett. 105, 234301 (2010) Applications Thermophotovoltaics Application : thermophotovoltaics Photovoltaics Thermophotovoltaics Near-field thermophotovoltaics T= 6000K thermal source T= 2000 K thermal source T= 2000 K d << rad TPV cell PV cell TPV cell T= 300 K T= 300 K T= 300 K Near-field output electric power tungsten source quasi-monochromatic source near field :15.105 W/m2 near field : 2.5.106 W/m2 50 3000 far field :3.104 W/m2 BB 2000 K BB 2000 K d (m) far field : 1.4.103 W/m2 d (m) output electric power enhanced by at least one order of magnitude Laroche et al. J. Appl.Phys. 100, 063704 (2006) Near-field TPV converter efficiency quasi-monochromatic source tungsten source near field : 35% (%) far field : 21 % BB 2000 K d (m) Pel Prad (%) near field : 27% significant increase of the efficiency Laroche et al. J. Appl.Phys. 100, 063704 (2006) far field : 8 % BB 2000 K d (m) Modulators Modulator with phase change material Van Zwol et al. , Phys.Rev.B 83, 201404 (R) (2011) Summary Energy density 1.0 z=100 m 0.8 0.6 0.4 0.2 Energy density 0.0 z=1 m 15 10 5 Energy density 0 20x10 3 z= 100 nm 15 10 5 0 0 100 200 300 (Hz) 400 500x10 12 0 2 kB2 T 3 h