radiative heat transfer at the nanoscale jean-jacques greffet institut d’optique, université...
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Radiative Heat Transfer at the Nanoscale
Jean-Jacques Greffet
Institut d’Optique, Université Paris Sud Institut Universitaire de France
CNRS
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Outline of the lecturesOutline of the lectures
1. Heat radiation close to a surface
2. Heat transfer between two planes
3. Mesoscopic approach, fundamental limits and Applications
d
d
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OutlineOutline
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
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d
Heat radiation at the nanoscale : Introduction 0
Casimir and heat transfer at the nanoscale:Similarities and differences
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d
Heat radiation at the nanoscale : Introduction 0
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.
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T+T
T
d
Heat radiation at the nanoscale : Introduction 1
€
Φ = hR ΔT
The flux is several orders of magnitude larger than Stefan-Boltzmann law.
€
1
d2
Microscale Thermophysical Engineering 6, p 209 (2002)
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d
Microscale Thermophysical Engineering 6, p 209 (2002)
Heat radiation at the nanoscale : Introduction 2
d=10 nm, T=300K
Heat transfer can be quasimonochromatic
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20x103
15
10
5
0500x10
124003002001000
ω ( )Hz
1.0
0.8
0.6
0.4
0.2
0.0
=100 z μm
=1 z μm
= 100 z nm
15
10
5
0
T=300 K
z
Density of energy near a SiC-vacuum interface
PRL, 85 p 1548 (2000)
Heat radiation at the nanoscale : Introduction 3
Energy density close to surfaces is orders of magnitude larger thanblackbody energy density and monochromatic.
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de Wilde et al. Nature (2006)
Heat radiation at the nanoscale : Introduction 4
Thermally excited fields can bespatially coherent in the near field.
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Proof of spatial coherence of thermal radiation
Greffet et al., Nature 416, 61 (2002)
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Modelling Thermal Modelling Thermal RadiationRadiation
The standard The standard radiometric radiometric approachapproach
T+T
T
d
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The standard radiometric The standard radiometric approachapproach
€
Iωe T( ) = εIω
BB T( )
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The standard radiometric The standard radiometric approachapproach
€
dQω = IωBB
2π
∫ (T)cosθdΩ = πIωBB (T)
Q = π IωBB (T) dω = σT 4
0
∞
∫
Isotropic black body emitter (=1)
What is missing ?Where is the extra energy coming from ?
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Material with emissivityMaterial with emissivity in front of a in front of a black bodyblack body
1 1
2
Kirchhoff’s law:
Energy balance for lower medium:
€
Q = Qabs − Qe = ασT14 −εσT2
4 = εσ T14 − T2
4( )
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The fluctuational electrodynamics approach
Modelling Thermal Radiation
Rytov, Kravtsov, Tatarskii, Principles of radiophysics, Springer
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The fluctuational electrodynamics approach
1) The medium is assumed to be at local thermodynamic equilibrium volume element = random electric dipole
€
j(r, t) with j(r, t) = 0
E(r,ω)=i μoω G r,r',ω( )∫ •j r',ω( )dr'
2) Radiation of random currents = thermal radiation
T
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3) Spatial correlation (cross-spectral density)
€
U r,ω( )∝ E r,ω( )2
4) The current-density correlation function is given by the FD theorem
jm(r1) jn
*(r2) = ωπ
εo Imε ω( )[ ]δmnδ(r1 −r2)hω
exphω/kBT( ) −1
AbsorptionFluctuation
€
E j E *k r,r ',ω( ) = μo2 ω2 G jm∫ r,r1( )Gkn
* r ',r2( ) jm r1( ) jn* r2( ) dr1dr2
The fluctuational electrodynamics approach
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d
Casimir and Heat transfer
Maxwell-stres tensor can be computedPoynting vector can be computed
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Dealing with non-equilibrium situation
T2
T1
d
jm(r1) jn
*(r2) = ωπ
εo Imε ω( )[ ]δmnδ(r1 −r2)hω
exphω/kBT( ) −1
What about the temperature gradient ?Can we assume the temperature to be uniform ?
€
h T2 − T1( ) = − k∂T
∂z
ΔT ≈∂T
∂z d ≈
h
k d ≈
1000
1 10−7 ≈10−4 K
T2
T2+T
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Physical origin of Kirchhoff’s law
Kirchhoff’s law:
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Physical origin of Kirchhoff’s law
Kirchhoff’s law:
T21=T121
2
Physical meaning of emissivity and absorptivity
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Basic concepts of near Basic concepts of near fieldfield
What is so special about near field ?
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Basic concepts of near fieldBasic concepts of near field
Back to basics : dipole radiation
1/r termsRadiation
1/r2 and 1/r3 termsNear field
Near field: k0r<<1, k0=2
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Evanescent waves filtering
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€
exp ikR[ ]R
=i
2π
dkxdky
kz
∫∫ exp i(kx x + ky y + kz z )[ ]
k⊥ <ω
c ; kz =
ω2
c 2−k
⊥
2 ; k⊥ = (kx, ky )
k⊥ >ω
c ; kz = i
ω2
c 2−k
⊥
2
k⊥ >>ω
c ; kz ≈ ik⊥
exp ikz z[ ] ≈ exp −k⊥z[ ]
Basic concepts of near field: evanescent Basic concepts of near field: evanescent waveswaves
Take home message: large k are confined to distances 1/k.
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Evanescent waves filtering
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Energy density:
in vacuum, close to nanoparticle, close to a surface
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Black body radiation in Black body radiation in vacuumvacuum
€
g(ω) = ω2
π 2c 3
U = ω2
π 2c 30
∞
∫ hω
ehω
kT −1 dω
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Black body radiation close to a Black body radiation close to a nanoparticlenanoparticle
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 ?
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Black body radiation close to a Black body radiation close to a nanoparticlenanoparticle
€
pm (r1)pn* (r2) =
2
πωεo Im α ω( )[ ]δmn δ(r1 − r2)
hω
exp hω /kBT( ) −1
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Black body radiation close to a Black body radiation close to a nanoparticlenanoparticle
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.
€
pm (r1)pn* (r2) =
2
πωεo Im α ω( )[ ]δmn δ(r1 − r2)
hω
exp hω /kBT( ) −1
Where are these (virtual) modes coming from ?
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+
+
+
+
+
+-
--
-
--
-
Polaritons
Strong coupling between material modes and photons produces
polaritons: Half a photon and half a phonon/exciton/electron.
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Estimation of the number of electromagnetic modes in vacuum:
€
N
V= g(ω')
0
ω
∫ dω'=ω3
3π 2c 3
€
N ≈V
λ3
Estimation of the number of vibrational modes (phonons):
€
N ≈V
a3
The electromagnetic field inherits the DOS of matter degrees of freedom
Where are the modes coming from ?
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Can we define a (larger and local) density of states close to a particle or a surface ?
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LDOS in near field above a surface
Joulain et al., Surf Sci Rep. 57, 59 (2005), Phys.Rev.B 68, 245405 (2003)
LDOS is also used to deal with spontaneous emission using Fermi golden rule
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Local density of states close to an interface
Lifetime of Europium 3+ above a silver mirror
Drexhage, 1970
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• Interference effects
• (microcavity type)
Electrical Engineering point of view
Near-field contribution
Drexhage, 1970
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• Interference effects
• (microcavity type)
Quantum optics point of view:
LDOS above a surface
Near-field contribution
Drexhage, 1970
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20x103
15
10
5
0500x10
124003002001000
ω ( )Hz
1.0
0.8
0.6
0.4
0.2
0.0
=100 z μm
=1 z μm
= 100 z nm
15
10
5
0
T=300 K
z
Density of energy near a SiC-vacuum interface
PRL, 85 p 1548 (2000)
Energy density above a SiC surface
Energy density close to surfaces is orders of magnitude larger than blackbody energy density and monochromatic.
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T=300 K
z
Density of energy near a Glass-vacuum interface
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Physical mechanism
The density of energy is the product of
- the density of states, - the energy h- the Bose Einstein distribution.
The density of states can diverge due to the presence of surface waves :Surface phonon-polaritons.
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First picture of a surface First picture of a surface plasmonplasmon
€
Ex exp ikx − iγz − iωt[ ]
Dawson, Phys.Rev.Lett.
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D
Surface plasmon excited by an optical fiber
Courtesy, A. Bouhelier
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Dispersion relation of a surface phonon-polariton
It is seen that the number of modes diverges for a particular frequency.
PRB, 55 p 10105 (1997)
€
k =ω
c
ε ω( )ε ω( ) +1
ω
k
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LDOS in near field above a surface
Joulain et al., Surf Sci Rep. 57, 59 (2005), Phys.Rev.B 68, 245405 (2003)
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LDOS in near field above a surface
Joulain et al., Surf Sci Rep. 57, 59 (2005), Phys.Rev.B 68, 245405 (2003)
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LDOS in the near field above a surface
Joulain et al., Surf Sci Rep. 57, 59 (2005), Phys.Rev.B 68, 245405 (2003)
This can be computed close to an interface !
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LDOS in the near field above a surface
Plasmon resonance Electrostatic enhancement
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Experimental evidence of
thermally excited near fields
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Direct experimental evidence
de Wilde et al. Nature 444, p 740 (2006)
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de Wilde et al. Nature (2006)
Imaging the LDOS of surface plasmons
Thermally excited fields can bespatially coherent in the near field.
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Spatial coherence of thermal radiationSpatial coherence of thermal radiation
T=300 K
z
M P
PRL 82, 1660 (1999)
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Proof of spatial coherence of thermal radiation
Greffet, Nature 416, 61 (2002)
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Spectrum of the thermal near Spectrum of the thermal near fieldfield
Babuty et al. Arxiv
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Lifetime of an atom in a magnetic Lifetime of an atom in a magnetic traptrap
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)
Superconducting niobium
Copper 1.8 and 6.24 MHz
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Second Lecture
Radiative heat transfer at the nanoscale
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Heat transfer between two nanoparticles
PRL 94, 85901, (2005)
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PRL 94, 85901, 2005
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Heat transfer between a particle and a dielectric (SiC) surface
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Energy transfer betwen a particle and a Energy transfer betwen a particle and a planeplane
Mulet et al. Appl.Phys.Lett. 78, p 2931 (2001)
T=300 K
d=100nm
Take home message:For source at distance d, power dissipated within a distance 2 d
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Energy transfer between a particle and a Energy transfer between a particle and a planeplane
Mulet et al. Appl.Phys.Lett. 78, p 2931 (2001)
T=300 K
d
Microscopic resonance if ω+2=0
Key feature 2
Key feature 1
Take home message: The flux is dramatically enhanced by the surface waves resonances
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Heat transfer between a particle and a metallic surface
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Bj
E
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 :
€
PE = 2ω Im(α E )< ε0 | E
→
|2>
2+ 2ω Im(α M )
<| B→
|2>
2μ0
Radiative heat transfer between a particle and a Radiative heat transfer between a particle and a surface surface
Chapuis, Phys.Rev.B 125402, 2008
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€
Im(α M )
€
Im(α E )
€
Im(α E )
€
Im(α M )
Magnetic losses versus electric losses
•Electric dipole
• Magnetic polarisability
€
E = 4πR3 ε −1
ε + 2
€
|α M |<<|α E |
R=5 nm
R=10 nm
Im
) (m
3 )
€
|ε |>>1
€
M =2π
15R3 R2
λ2(ε −1)
Radiative heat transfer between a particle and a surface
for R 0
for a metal at low frequencies
Chapuis, Phys.Rev.B 125402, 2008
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Electric and magnetic energy density above an interface :
an unusual property
p polarisation: electrostatics
z
x
kp
E
k0
B
propagating evanescent
z
x
kp
B
k0
E
propagating evanescent
yy
Radiative heat transfer between a particle and a surface
s polarisation:magnetostatics
€
B =E
c
€
B =E
c2
kp
k0
⎛
⎝ ⎜
⎞
⎠ ⎟
2
−1
€
E = cB
€
E = cB 2kp
k0
⎛
⎝ ⎜
⎞
⎠ ⎟
2
−1
Chapuis, Phys.Rev.B 125402, 2008Magnetostatics
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Radiative heat transferRadiative heat transfer
T1
T2
d
€
Φ < σ T14 − T2
4( )
Theme : influence of surface waves on the mesoscopic energy transfer
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Anomalous heat transferAnomalous heat transfer
Radiation shield
Hargreaves, 1969 Anomalous heat transfer
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Au GaN
Kittel et al. , PRL 95 p 224301 (2005)
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).
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Radiative heat transfer at Radiative heat transfer at the nanoscalethe nanoscale
T1
d
Radiometric approach
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Radiative heat transfer at Radiative heat transfer at the nanoscalethe nanoscaleTheory : Polder, van Hove, PRB 1971
(Tien, Caren, Rytov, Pendry)
T1
d
Transmission factor
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Heat transfer between two SiC half spaces
Mulet et al., Microscale Thermophysical Engineering 6, 209, (2002)Mulet et al. Appl.Phys.Lett. 78, 2931 (2001)
Enhancement due to surface waves
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Analogy with Casimir force
Henkel et al., Phys.Rev.A 69, 023808 (2004)
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Casimir force for aluminum
Henkel et al., Phys.Rev.A 69, 023808 (2004)
10 nm
100 nm
10 µm
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Drude model is not accurate
Hao and Nordlander, Chem.Phys.Lett. 446, 115 (2007)
Circle: Johnson and ChristyBlue : Drude model
Gold
Silver
Nordlander’s model
4 Lorentzian terms are used
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It is even worse than that !
Optical properties depend on the deposition technique.
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Recent experimental results
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A. Narayanaswamy, G. Chen PRB 78, 115303 (2008)
Radiative heat transfer at nanoscale
Measurements with silica
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- Vertical configuration
- Heating of the sample
- Vacuum P~10-6 mbar
Optical fiber
Heater
E. RousseauA. Siria, J. Chevrier
Detection scheme
Nature Photonics 3, p 154 (2009)
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Comparison theory/dataComparison theory/data
Nature Photonics 3, p 154 (2009)
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C. Otey and S. Fan, Phys.Rev.B 84, 245431 (2011)
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Textured surfaces
R. Guerout et al., Phys.Rev.B 85, 180301 (2012)
Heat transfer between gold gratings
d=1 µm
d=2.5 µm
d=10 µm
PFA
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R. Guerout et al., Phys.Rev.B 85, 180301 (2012)
Textured surfaces
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Heat transfer between SiC gratings
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Heat transfer between two SiO2 gratings:Physical mechanism
8,75 µmSiO2 resonance
9.15 µm
J. Lussange et al., Phys.Rev.B 86, 085432 (2012)
L=25 nm
d=1500 n
a=500 nm
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Introduction to the concept of quantum conductanceMesoscopic analysis of heat transfer at the nanoscale
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Mesoscopic charge transport : a tutorial
1 Experimental observation
2 Landauer formalism
€
Γ = 2e2
h
⎡
⎣ ⎢
⎤
⎦ ⎥
n
∑ Tn
€
I = Γ V
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Hand wavy derivation of the Hand wavy derivation of the quantum of cinductancequantum of cinductance
87
Perfect conductors resists !
I= 2e/t (One electron at a time per state)
t=h/E
E=eV
I= [2e2/h] V (R=12,9 k)
€
I = Tn 2e2
h
⎡
⎣ ⎢
⎤
⎦ ⎥
n
∑ VLandauer formula:
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Fundamental limits to radiative heat transfer at the nanoscale
Sum over modes
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A mesoscopic formulation of radiative heat transfer at nanoscale
€
Φ = d2k
4π 2∫∫ T(k) π 2
3
kB2T
h
⎡
⎣ ⎢
⎤
⎦ ⎥ ΔT
Thermal quantumconductance
Transmissionfactor
€
I = Tn 2e2
h
⎡
⎣ ⎢
⎤
⎦ ⎥
n
∑ V
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2/L
d
Biehs et al., Phys.Rev.Lett. 105, 234301 (2010)
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Fundamental limit: a new perspective on Stefan’s constant
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Are we there yet ?
Biehs et al., Phys.Rev.Lett. 105, 234301 (2010)
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Quantized radiative thermal conductanceL
If L is on the order of the thermal wavelength,
€
d2k
4π 2∫∫ → n
∑Biehs et al., Phys.Rev.Lett. 105, 234301 (2010)
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Applications
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Thermophotovoltaics
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thermal source
T= 2000 K
TPV cell
T= 300 K
d << rad
thermal source
T= 2000 K
TPV cell
T= 300 K
PV cell
T= 300 K
Photovoltaics Thermophotovoltaics Near-fieldthermophotovoltaics
T= 6000K
Application : thermophotovoltaics
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Near-field output electric powerNear-field output electric power
output electric power enhanced by at least one order of magnitude
tungsten source quasi-monochromatic source
d (m)
50
far field :3.104 W/m2
near field :15.105 W/m2
Pe
l (W
. m
- 2)
d (m)
near field : 2.5.106 W/m2
far field : 1.4.103 W/m2
3000
Pe
l (W
. m
- 2)
BB 2000 KBB 2000 K
Laroche et al. J. Appl.Phys. 100, 063704 (2006)
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Near-field TPV converter efficiencyNear-field TPV converter efficiency
€
η =Pel
Pradη (%
)
d (m) d (m)
η (%
)
near field : 27%
far field : 21 %
near field : 35%
significant increase of the efficiency
far field : 8 %
tungsten source quasi-monochromatic source
BB 2000 K BB 2000 K
Laroche et al. J. Appl.Phys. 100, 063704 (2006)
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Modulators
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Modulator with phase change material
Van Zwol et al. , Phys.Rev.B 83, 201404 (R) (2011)
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SummarySummary
€
Γ0 = π 2
3
kB2T
h
20x103
15
10
5
0500x10
124003002001000
ω ( )Hz
1.0
0.8
0.6
0.4
0.2
0.0
=100 z μm
=1 z μm
= 100 z nm
15
10
5
0