Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary Near-field radiative heat transfer between a nanoparticle and a rough surface Svend-Age Biehs Design de matériaux à propriété radiatives fonctionalisées: de l’angstrom au millimètre Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
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Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Near-field radiative heat transfer between ananoparticle and a rough surface
Svend-Age Biehs
Design de matériaux à propriété radiatives fonctionalisées:de l’angstrom au millimètre
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Introducing the dipole model
Dipole model - LDOSI for λth � R and d � R:
PB→P =
∫ ∞0
dω 2ωα′′(ω)Θ(ω,TB)D(ω, rP)
I polarizability
α = 4πR3 εP − 1εP + 2
rP
x
z
d
TB
R
TP= 0 K
= 300 K
SiC
SiC
S(x)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Introducing the dipole model
Dipole model - LDOSI for λth � R and d � R:
PB→P =
∫ ∞0
dω 2ωα′′(ω)Θ(ω,TB)D(ω, rP)
I polarizability
α = 4πR3 εP − 1εP + 2
rP
x
z
d
TB
R
TP= 0 K
= 300 K
SiC
SiC
S(x)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Introducing the dipole model
Dipole model - LDOSI for λth � R and d � R:
PB→P =
∫ ∞0
dω 2ωα′′(ω)Θ(ω,TB)D(ω, rP)
I mean energy of oscillator TB
Θ(ω,TB) =~ω
e~ω/(kBT ) + 1
rP
x
z
d
TB
R
TP= 0 K
= 300 K
SiC
SiC
S(x)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Introducing the dipole model
Dipole model - LDOSI for λth � R and d � R:
PB→P =
∫ ∞0
dω 2ωα′′(ω)Θ(ω,TB)D(ω, rP)
I Local density of states (LDOS)
D(ω, rP) =ω
πc2 Im TrG(rP, rP)
rP
x
z
d
TB
R
TP= 0 K
= 300 K
SiC
SiC
S(x)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Energy transfer rate within dipole model
Flat surface
0.001
1
1000
1e+06
1e-08 1e-07 1e-06 1e-05
P /
PB
B
d / m
R = 5 nm, ev. modes
1e-18/x**3d
SiC
= 5nm= 0 KPT
R
BT SiC= 300 K,
I P � PBB
I P ∝ d−3
I dominated by SPhPI ⇒ p-modes
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Energy transfer rate within dipole model
Flat surface
0.001
1
1000
1e+06
1e-08 1e-07 1e-06 1e-05
P /
PB
B
d / m
R = 5 nm, ev. modes
1e-18/x**3d
SiC
= 5nm= 0 KPT
R
BT SiC= 300 K,
I P � PBB
I P ∝ d−3
I dominated by SPhPI ⇒ p-modes
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Energy transfer rate within dipole model
Flat surface
0.001
1
1000
1e+06
1e-08 1e-07 1e-06 1e-05
P /
PB
B
d / m
R = 5 nm, ev. modes
1e-18/x**3d
SiC
= 5nm= 0 KPT
R
BT SiC= 300 K,
I P � PBB
I P ∝ d−3
I dominated by SPhPI ⇒ p-modes
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Energy transfer rate within dipole model
Flat surface
0.001
1
1000
1e+06
1e-08 1e-07 1e-06 1e-05
P /
PB
B
d / m
R = 5 nm, ev. modes
1e-18/x**3d
SiC
= 5nm= 0 KPT
R
BT SiC= 300 K,
I P � PBB
I P ∝ d−3
I dominated by SPhPI ⇒ p-modes
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Energy transfer rate within dipole model
Flat surface
0.001
1
1000
1e+06
1e-08 1e-07 1e-06 1e-05
P /
PB
B
d / m
R = 5 nm, ev. modes
1e-18/x**3d
SiC
= 5nm= 0 KPT
R
BT SiC= 300 K,
I P � PBB
I P ∝ d−3
I dominated by SPhPI ⇒ p-modes
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Definining a rough surface
Stochastic surface profile
I gaussian profile S(x)
〈S̃(κ)〉 = 0,
〈S̃(κ)S̃(κ′)〉 = (2π)2δ(κ + κ′)δ2g(κ)
I power spectrum
g(κ) = πa2e−κ2a2
4
I root mean square (rms) δ,correlation length a
x
z
S(x)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Definining a rough surface
Stochastic surface profile
I gaussian profile S(x)
〈S̃(κ)〉 = 0,
〈S̃(κ)S̃(κ′)〉 = (2π)2δ(κ + κ′)δ2g(κ)
I power spectrum
g(κ) = πa2e−κ2a2
4
I root mean square (rms) δ,correlation length a
x
z
S(x)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Definining a rough surface
Stochastic surface profile
I gaussian profile S(x)
〈S̃(κ)〉 = 0,
〈S̃(κ)S̃(κ′)〉 = (2π)2δ(κ + κ′)δ2g(κ)
I power spectrum
g(κ) = πa2e−κ2a2
4
I root mean square (rms) δ,correlation length a
x
z
S(x)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Second-order perturbation theory
Mean LDOS above a rough surfaceI ensemble average
〈D(0)−(2)(ω,d)〉 = D(0) + 〈D(1)〉+ 〈D(2)〉
I reflection coefficient
〈r (0)−(2)p 〉 = r (0)
p (κ) + r (2)p (κa)
I main contribution for κ ≈ d−1
I 3 regimes for r (2)p :
I κa� 1, (a� d)I κa� 1, (a� d)I κa ≈ 1 , (a ≈ d)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Second-order perturbation theory
Mean LDOS above a rough surfaceI ensemble average
〈D(0)−(2)(ω,d)〉 = D(0) + 〈D(2)〉
I reflection coefficient
〈r (0)−(2)p 〉 = r (0)
p (κ) + r (2)p (κa)
I main contribution for κ ≈ d−1
I 3 regimes for r (2)p :
I κa� 1, (a� d)I κa� 1, (a� d)I κa ≈ 1 , (a ≈ d)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Second-order perturbation theory
Mean LDOS above a rough surfaceI ensemble average (ev. modes)
〈D(0)−(2)(ω,d)〉 ≈∫ ∞
0dκ
κ2e−2κd
4Im(〈r (0)−(2)
p 〉)
I reflection coefficient
〈r (0)−(2)p 〉 = r (0)
p (κ) + r (2)p (κa)
I main contribution for κ ≈ d−1
I 3 regimes for r (2)p :
I κa� 1, (a� d)I κa� 1, (a� d)I κa ≈ 1 , (a ≈ d)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Second-order perturbation theory
Mean LDOS above a rough surfaceI ensemble average (ev. modes)
〈D(0)−(2)(ω,d)〉 ≈∫ ∞
0dκ
κ2e−2κd
4Im(〈r (0)−(2)
p 〉)
I reflection coefficient
〈r (0)−(2)p 〉 = r (0)
p (κ) + r (2)p (κa)
I main contribution for κ ≈ d−1
I 3 regimes for r (2)p :
I κa� 1, (a� d)I κa� 1, (a� d)I κa ≈ 1 , (a ≈ d)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Second-order perturbation theory
Mean LDOS above a rough surfaceI ensemble average (ev. modes)
〈D(0)−(2)(ω,d)〉 ≈∫ ∞
0dκ
κ2e−2κd
4Im(〈r (0)−(2)
p 〉)
I reflection coefficient
〈r (0)−(2)p 〉 = r (0)
p (κ) + r (2)p (κa)
I main contribution for κ ≈ d−1
I 3 regimes for r (2)p :
I κa� 1, (a� d)I κa� 1, (a� d)I κa ≈ 1 , (a ≈ d)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Second-order perturbation theory
Mean LDOS above a rough surfaceI ensemble average (ev. modes)
〈D(0)−(2)(ω,d)〉 ≈∫ ∞
0dκ
κ2e−2κd
4Im(〈r (0)−(2)
p 〉)
I reflection coefficient
〈r (0)−(2)p 〉 = r (0)
p (κ) + r (2)p (κa)
I main contribution for κ ≈ d−1
I 3 regimes for r (2)p :
I κa� 1, (a� d)I κa� 1, (a� d)I κa ≈ 1 , (a ≈ d)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Second-order perturbation theory
Mean LDOS above a rough surfaceI ensemble average (ev. modes)
〈D(0)−(2)(ω,d)〉 ≈∫ ∞
0dκ
κ2e−2κd
4Im(〈r (0)−(2)
p 〉)
I reflection coefficient
〈r (0)−(2)p 〉 = r (0)
p (κ) + r (2)p (κa)
I main contribution for κ ≈ d−1
I 3 regimes for r (2)p :
I κa� 1, (a� d)I κa� 1, (a� d)I κa ≈ 1 , (a ≈ d)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Second-order perturbation theory
Mean LDOS above a rough surfaceI ensemble average (ev. modes)
〈D(0)−(2)(ω,d)〉 ≈∫ ∞
0dκ
κ2e−2κd
4Im(〈r (0)−(2)
p 〉)
I reflection coefficient
〈r (0)−(2)p 〉 = r (0)
p (κ) + r (2)p (κa)
I main contribution for κ ≈ d−1
I 3 regimes for r (2)p :
I κa� 1, (a� d)I κa� 1, (a� d)I κa ≈ 1 , (a ≈ d)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Second-order perturbation theory
Mean LDOS above a rough surfaceI ensemble average (ev. modes)
〈D(0)−(2)(ω,d)〉 ≈∫ ∞
0dκ
κ2e−2κd
4Im(〈r (0)−(2)
p 〉)
I reflection coefficient
〈r (0)−(2)p 〉 = r (0)
p (κ) + r (2)p (κa)
I main contribution for κ ≈ d−1
I 3 regimes for r (2)p :
I κa� 1, (a� d)I κa� 1, (a� d)I κa ≈ 1 , (a ≈ d)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Large distance regime
Large distance approximation (LDA), d � a
I approximating r (2)p for 1� κa
I r (2)p ∝ δ2
a , (a� ds)I effective layer (Maradudin and
Rahman)I 〈D(2)〉, 〈P(2)〉 ∝ δ2
a
0.5 ( + 1)ε
ε
ddeff
= δaL
2
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Large distance regime
Large distance approximation (LDA), d � a
I approximating r (2)p for 1� κa
I r (2)p ∝ δ2
a , (a� ds)I effective layer (Maradudin and
Rahman)I 〈D(2)〉, 〈P(2)〉 ∝ δ2
a
0.5 ( + 1)ε
ε
ddeff
= δaL
2
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Large distance regime
Large distance approximation (LDA), d � a
I approximating r (2)p for 1� κa
I r (2)p ∝ δ2
a , (a� ds)I effective layer (Maradudin and
Rahman)I 〈D(2)〉, 〈P(2)〉 ∝ δ2
a
0.5 ( + 1)ε
ε
ddeff
= δaL
2
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Large distance regime
Large distance approximation (LDA), d � a
I approximating r (2)p for 1� κa
I r (2)p ∝ δ2
a , (a� ds)I effective layer (Maradudin and
Rahman)I 〈D(2)〉, 〈P(2)〉 ∝ δ2
a
0.5 ( + 1)ε
ε
ddeff
= δaL
2
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Small distance regime
Proximity approximation (PA), d � a
I approximating r (2)p for 1� κa
Ir (2)p
r (0)p≈ 2(κδ)2 ⇒ ∆D = 〈D(2)〉
D(0) ≈ 6 δ2
d2
I〈D(2)〉D(0) ,
〈P(2)〉P(0) > 0, do not depend on a, ε, T
I PA for δ � d � a
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Small distance regime
Proximity approximation (PA), d � a
I approximating r (2)p for 1� κa
Ir (2)p
r (0)p≈ 2(κδ)2 ⇒ ∆D = 〈D(2)〉
D(0) ≈ 6 δ2
d2
I〈D(2)〉D(0) ,
〈P(2)〉P(0) > 0, do not depend on a, ε, T
I PA for δ � d � a
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Small distance regime
Proximity approximation (PA), d � a
I approximating r (2)p for 1� κa
Ir (2)p
r (0)p≈ 2(κδ)2 ⇒ ∆D = 〈D(2)〉
D(0) ≈ 6 δ2
d2
I〈D(2)〉D(0) ,
〈P(2)〉P(0) > 0, do not depend on a, ε, T
I PA for δ � d � a
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Small distance regime
Proximity approximation (PA), d � a
I approximating r (2)p for 1� κa
Ir (2)p
r (0)p≈ 2(κδ)2 ⇒ ∆D = 〈D(2)〉
D(0) ≈ 6 δ2
d2
I〈D(2)〉D(0) ,
〈P(2)〉P(0) > 0, do not depend on a, ε, T
I PA for δ � d � a
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Small distance regime
Proximity approximation (PA), d � aI approximating r (2)
p for 1� κa
Ir (2)p
r (0)p≈ 2(κδ)2 ⇒ ∆D = 〈D(2)〉
D(0) ≈ 6 δ2
d2
I〈D(2)〉D(0) ,
〈P(2)〉P(0) > 0, do not depend on a, ε, T
I PA for δ � d � a
〈D(d)〉 ≈ 〈D(0)(d − S(x))〉
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Small distance regime
Proximity approximation (PA), d � aI approximating r (2)
p for 1� κa
Ir (2)p
r (0)p≈ 2(κδ)2 ⇒ ∆D = 〈D(2)〉
D(0) ≈ 6 δ2
d2
I〈D(2)〉D(0) ,
〈P(2)〉P(0) > 0, do not depend on a, ε, T
I PA for δ � d � a
〈D(d)〉 ≈ 〈D(0)(d − S(x))〉
≈ D(0)(d) + 6δ2
d2 D(0)(d) + . . .
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Intermediate distance regime
Intermediate distances d ≈ a (κa ≈ 1)I Im(r (2)
p )/Im(r (0)p ) for SiC and ωt ≤ ω ≤ ωl
I a = 200 nm, δ = 5nm→ δa = 0.025
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Intermediate distance regime
Intermediate distances d ≈ a (κa ≈ 1)I Im(r (2)
p )/Im(r (0)p ) for SiC and ωt ≤ ω ≤ ωl
I a = 200 nm, δ = 5nm→ δa = 0.025
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Intermediate distance regime
Intermediate distances d ≈ a (κa ≈ 1)I Im(r (2)
p )/Im(r (0)p ) for SiC and ωt ≤ ω ≤ ωl
I a = 200 nm, δ = 5nm→ δa = 0.025
4
8
12
16
1.77 1.78 1.79 1.8
Im(r
p)
ω / 1014 s-1
flat
rough
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Resulting distance dependence
Distance dependence of LDOSI ∆D = D(2)
D(0) ; a = 200 nm, δ = 5nm→ δa = 0.025
0.01
0.1
1
10
100
1e-07 1e-06 1e-05
∆ D
E (
perc
ent)
d / m
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Resulting distance dependence
Distance dependence of LDOSI ∆D = D(2)
D(0) ; a = 200 nm, δ = 5nm→ δa = 0.025
0.01
0.1
1
10
100
1e-07 1e-06 1e-05
∆ D
E (
perc
ent)
d / m
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Resulting distance dependence
Distance dependence of LDOSI ∆D = D(2)
D(0) ; a = 200 nm, δ = 5nm→ δa = 0.025
0.01
0.1
1
10
100
1e-07 1e-06 1e-05
∆ D
E (
perc
ent)
d / m
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Resulting distance dependence
Distance dependence of LDOS
I ∆D = D(2)
D(0) ; a = 200 nm, δ = 5nm→ δa = 0.025
0.01
0.1
1
10
100
1e-07 1e-06 1e-05
∆ D
E (
perc
ent)
d / m
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Resulting distance dependence
Distance dependence of LDOS
I ∆D = D(2)
D(0) ; a = 200 nm, δ = 5nm→ δa = 0.025
0.01
0.1
1
10
100
1e-07 1e-06 1e-05
∆ D
E (
perc
ent)
d / m
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Roughness induced correction
Distance dependence of heat transfer rateI ∆P = P(2)
P(0) ; δ = 5nm
0.01
0.1
1
10
100
1e-08 1e-07 1e-06 1e-05
∆ P
(pe
rcen
t)
d / m
PA
d
SiC
= 5nm= 0 KPT
R
BT SiC= 300 K,
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Roughness induced correction
Distance dependence of heat transfer rateI ∆P = P(2)
P(0) ; δ = 5nm
0.01
0.1
1
10
100
1e-08 1e-07 1e-06 1e-05
∆ P
(pe
rcen
t)
d / m
PA
LDA
d
SiC
= 5nm= 0 KPT
R
BT SiC= 300 K,
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Roughness induced correction
Distance dependence of heat transfer rateI ∆P = P(2)
P(0) ; δ = 5nm
0.01
0.1
1
10
100
1e-08 1e-07 1e-06 1e-05
∆ P
(pe
rcen
t)
d / m
PA
LDA
a = 200nm
d
SiC
= 5nm= 0 KPT
R
BT SiC= 300 K,
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Roughness induced correction
Distance dependence of heat transfer rateI ∆P = P(2)
P(0) ; δ = 5nm
0.01
0.1
1
10
100
1e-08 1e-07 1e-06 1e-05
∆ P
(pe
rcen
t)
d / m
PA
LDA
a = 200nm
a = 100nmd
SiC
= 5nm= 0 KPT
R
BT SiC= 300 K,
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Roughness induced correction
Distance dependence of heat transfer rateI ∆P = P(2)
P(0) ; δ = 5nm
0.01
0.1
1
10
100
1e-08 1e-07 1e-06 1e-05
∆ P
(pe
rcen
t)
d / m
PA
LDA
a = 200nm
a = 100nm
a = 500nmd
SiC
= 5nm= 0 KPT
R
BT SiC= 300 K,
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Summary
SummaryI ∆P and ∆D non-monotonous (ev. modes)
I d � a:
∆D,∆P ≈ 6δ2
d2
I coincides with PAI independent of a, ε, T
I d � a (LDA):
∆D,∆P ∝ δ2
a
I effective layerI intermediate: ∆P,∆D negative due to SPhP
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Summary
SummaryI ∆P and ∆D non-monotonous (ev. modes)
I d � a:
∆D,∆P ≈ 6δ2
d2
I coincides with PAI independent of a, ε, T
I d � a (LDA):
∆D,∆P ∝ δ2
a
I effective layerI intermediate: ∆P,∆D negative due to SPhP
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Summary
SummaryI ∆P and ∆D non-monotonous (ev. modes)
I d � a:
∆D,∆P ≈ 6δ2
d2
I coincides with PAI independent of a, ε, T
I d � a (LDA):
∆D,∆P ∝ δ2
a
I effective layerI intermediate: ∆P,∆D negative due to SPhP
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Summary
SummaryI ∆P and ∆D non-monotonous (ev. modes)
I d � a:
∆D,∆P ≈ 6δ2
d2
I coincides with PAI independent of a, ε, T
I d � a (LDA):
∆D,∆P ∝ δ2
a
I effective layerI intermediate: ∆P,∆D negative due to SPhP
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Summary
SummaryI ∆P and ∆D non-monotonous (ev. modes)
I d � a:
∆D,∆P ≈ 6δ2
d2
I coincides with PAI independent of a, ε, T
I d � a (LDA):
∆D,∆P ∝ δ2
a
I effective layerI intermediate: ∆P,∆D negative due to SPhP
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Summary
SummaryI ∆P and ∆D non-monotonous (ev. modes)
I d � a:
∆D,∆P ≈ 6δ2
d2
I coincides with PAI independent of a, ε, T
I d � a (LDA):
∆D,∆P ∝ δ2
a
I effective layerI intermediate: ∆P,∆D negative due to SPhP
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Summary
SummaryI ∆P and ∆D non-monotonous (ev. modes)
I d � a:
∆D,∆P ≈ 6δ2
d2
I coincides with PAI independent of a, ε, T
I d � a (LDA):
∆D,∆P ∝ δ2
a
I effective layerI intermediate: ∆P,∆D negative due to SPhP
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Dipole Model Roughness Distance regimes LDOS Heat transfer rate Summary
Merci pour votre attention !!!
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Perturbation theory Prop. modes Evanescent s-polarized modes NSThM
Perturbation theory
Direct perturbation theory
I plane-wave representationof Ei,Er and Et
I bc via extinction theorem(Rayleigh hypothesis)
I expand with respect toS(x)
I determine Green’s dyadicup to second order
I calculate LDOS
x
zE E
E
i r
t
S(x)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Perturbation theory Prop. modes Evanescent s-polarized modes NSThM
Perturbation theory
Direct perturbation theory
I plane-wave representationof Ei,Er and Et
I bc via extinction theorem(Rayleigh hypothesis)
I expand with respect toS(x)
I determine Green’s dyadicup to second order
I calculate LDOS
x
zE E
E
i r
t
S(x)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Perturbation theory Prop. modes Evanescent s-polarized modes NSThM
Perturbation theory
Direct perturbation theory
I plane-wave representationof Ei,Er and Et
I bc via extinction theorem(Rayleigh hypothesis)
I expand with respect toS(x)
I determine Green’s dyadicup to second order
I calculate LDOS
x
zE E
E
i r
t
S(x)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Perturbation theory Prop. modes Evanescent s-polarized modes NSThM
Perturbation theory
Direct perturbation theory
I plane-wave representationof Ei,Er and Et
I bc via extinction theorem(Rayleigh hypothesis)
I expand with respect toS(x)
I determine Green’s dyadicup to second order
I calculate LDOS
x
zE E
E
i r
t
S(x)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Perturbation theory Prop. modes Evanescent s-polarized modes NSThM
Perturbation theory
Direct perturbation theory
I plane-wave representationof Ei,Er and Et
I bc via extinction theorem(Rayleigh hypothesis)
I expand with respect toS(x)
I determine Green’s dyadicup to second order
I calculate LDOS
x
zE E
E
i r
t
S(x)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Perturbation theory Prop. modes Evanescent s-polarized modes NSThM
s- and p-polarized modes
Propagating modes
I Im(r (2)s )/Im(r (0)
s ) and Im(r (2)p )/Im(r (0)
p )
0
-0.2
-0.4
κ / k0
ω /
1014
s-1
10.750.50.250
1.5
1.6
1.7
1.8
(a)0
-0.2
-0.4
-0.6
κ / k0
ω /
1014
s-1
0 0.25 0.5 0.75 1
1.8
1.7
1.6
1.5
(b)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Perturbation theory Prop. modes Evanescent s-polarized modes NSThM
coupling of s-polarized modes to SPhP
Coupling of prop. modes to SPhPI s-polarized wave with E in y-direction and κi
kx
ky
1
2
i
sp
κ
κ
k0
0
-0.2
-0.4
κ / k0
ω /
1014
s-1
10.750.50.250
1.5
1.6
1.7
1.8
(a)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Perturbation theory Prop. modes Evanescent s-polarized modes NSThM
s-polarized modes
Evanescent modesI Im(r (2)
s )/Im(r (0)s )
60
40
20
0
(κ - k0) a
ω /
1014
s-1
43210
1.8
1.7
1.6
1.5
(a)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau
Perturbation theory Prop. modes Evanescent s-polarized modes NSThM
Near-field Scanning Thermal Microscope
(a) (b)
1.3 3.3
50nm
1.0 3.5
50nm(a) (b)
Near-field radiative heat transfer between a nanoparticle and a rough surface LCFIO, Palaiseau