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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  aT  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   Gr,r' ,   jr' ,  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
Ur,   Er,
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 ?
hT2  T1
T
 k
z
T
h
1000 7
DT 
d
d
10  104 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 Ima  mn (r1  r2 )

exp   /kB T  1
Black body radiation close to a
nanoparticle
pm (r1 ) p (r2 ) 
*
n

2

o Ima  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  iz  it 

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
20
Chapuis, Phys.Rev.B 125402, 2008

2
Radiative heat transfer between a particle and a surface
Magnetic losses versus electric losses
a E  4R3
•Electric dipole
• Magnetic polarisability
 1
2
2 3 R 2
aM 
R 2 ( 1)
15

| a M || a E | for R 0


Ima) (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