Analytical expression for effective thermal conductivity of superlattices

Analytical expression for
effective thermal
conductivity of superlattices
F. X. Alvarez, A. Lopeandia, J. Rodriguez-Viejo, D. Jou

Outline
• Motivation
• Introduction
• Boundary resistance
• Bulk conductivity modification
• Results

Motivation
Obtain an analytical and simple
expression to predict thermal
conductivity for superlattice

Phonon
• Mean free path
• Coherence length
• Wavelength
l c
l s
l w

Phonon
Boltzmann equation
1/kBT
Wave equation
Wave equation

Large Phonons (Continuum/wave approx)

Small Phonons (Boltzmann eq approx)

Intermediate size Phonons (?)

Different behaviours
• Ballistic - Diffusive
• Coherent – Incoherent
• Equilibrium – Non-equilibrium

Thermal conductivity of SL

Transmissivity
( ) 2
cos
cos
cos
cos
4
j
j
j
i
i
i
j
j
j
i
i
i
S
v
v
v
v
θ
ρ
θ
ρ
θ
ρ
θ
ρ
τ
+
=
Specular
Diffuse
j
j
i
i
i
i
D
v
C
C v
C v
+
=
τ

Transmission Coefficient
Γ
ijS
/ D
= ∫τ ( μ )
ijS / D i
μ
i
dμ
i
Γ = pΓ
+ (1 − p)
Γ
ij
ijS
ijD

Mean Transmission Coefficient
Γ =
Γ + Γ
12 21
2
Γ 12
Γ 21

Transmission Coefficient

TBR
R'
=
4
π
k
4
B
2
h
ΓT
3
v
2
i
3
i
∫
0
x Di
x
4
e
x
1
/( e
x
−1)
2
dx
l l
1
R = R ' −
1 −
2
κ
01
κ
02
l l
2
Can. J. Phys. 37, 334 (1959)

TBR

Boltzmann equation
J. Appl. Phys. 107, 084303 (2010)
Appl. Phys. Lett. 90, 083109 (2007)

Nonequilibrium effects on bulk
κ
2
⎛
⎜
⎜
⎝
L ⎛ 2πl
⎞
= κ
eff
1−
⎜ ⎟ −1
0
2πl
2
⎝ Leff
⎠
2
⎞
⎟
⎟
⎠
J. Appl. Phys. 107, 084303 (2010)
Appl. Phys. Lett. 90, 083109 (2007)

Effective length
L
N −2
N −1
eff
= ( 1− Γ)
L1
+ (1 − Γ)
ΓL2
+ ... + (1 − Γ)
Γ LN-1
+ Γ LN
J. Appl. Phys. 107, 084303 (2010)

Cross plane
κ
κ
CP
CP
=
L
L
κ
1
1
1
+
+
L
L
κ
κ
1
2
2
2
CP
=
+ R
2κ
1+
L +
CP R
L
2
J. Appl. Phys. 107, 084303 (2010)

In plane
κ
IP
=
κ
1
L
L
1
1
+
+
κ
L
2
2
L
2
J. Appl. Phys. 107, 084303 (2010)

SL effective thermal conductivity
J. Appl. Phys. 107, 084303 (2010)

SL effective thermal conductivity
J. Appl. Phys. 107, 084303 (2010)

Conclusions
• Non-Equilibrium Thermodynamics combined with
TBR models gives a good approximation for thermal
conductivity of superlattices
• Obtained expressions are simple and analytical

Thank you