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