f xavier alvarez , andr es cantarero, carla de tom as, daniel m oller, aitor...
TRANSCRIPT
Coherent behaviour in thermal transportNonequilibrium approach
F Xavier Alvarez, Andres Cantarero, Carla de Tomas, DanielMoller, Aitor Lopeandia, Pablo Ferrando
Universitat Autonoma de Barcelona (UAB). Physics Department
Universitat de Valencia (UV)
November 18, 2013
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Coherent behaviour in thermal transport
Motivation
Motivation
Study the limits of classical approaches in thermal transportproblems
Understand the role of normal scattering and its relation withnonequilibrium
Determine the region where local equilibrium hypothesis isbroken.
Obtain an expression for thermal conductivity valid at allranges of sizes and temperature
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Coherent behaviour in thermal transport
Motivation
Motivation
It is difficult to fit (Si) thermal conductivity at all ranges of sizeand temperature.
x BulkPhys. Rev. 134, 4A A1058
Phys. Stat. Sol. C, 1(11) 1610
xx MicroscaleAppl. Phys. Lett. 71 (13), 1798
Appl. Phys. Lett. 84 (19), 3819
x NanoscaleAppl. Phys. Lett. 83, 2934
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Coherent behaviour in thermal transport
Motivation
Motivation
Proposal 1:Modification of DOS by confinement effectsPhys. Rev. B, 68, 113308 - Nano Lett., 3, 1713
DOS of 115nm wire is different from bulk (?)
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Coherent behaviour in thermal transport
Motivation
Motivation
Proposal 2:Modification of Dispersion Relations and velocitiesJ. Appl. Phys, 107, 08350 - J. Appl. Phys, 89, 2932
Double projection of velocity (?)
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Coherent behaviour in thermal transport
Boltzmann Transport Equation (BTE)
Boltzmann Transport Equation (BTE)
∂fq∂t
∣∣∣∣drift
=∂fq∂t
∣∣∣∣scatt
. (1)
����0
∂f
∂t+ ~vg · ∇f +���
�:0~a · ∇v f =
∂fq∂t
∣∣∣∣scatt
(2)
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Coherent behaviour in thermal transport
Boltzmann Transport Equation (BTE)
Equilibrium and local equilibrium forms
Equilibrium
f 0q =
1
e~ωq/kBT − 1, (3)
Out of equilibrium
fq ' f 0q +
f 0q (f 0
q + 1)
kBTΦq , (4)
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Coherent behaviour in thermal transport
Boltzmann Transport Equation (BTE)
Collisions
In Silicon we have Impurities or mass-defect, boundary andAnharminicity
q + q′ = q′′ + G, (5)
G = 0 (Non-resistive)
κ→∞
G <> 0 (Resistive)
κ <∞
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Coherent behaviour in thermal transport
Boltzmann Transport Equation (BTE)
Solutions of the BTE
The solution depends on the dominant scattering mechanismmResistive scattering dominant
fq =1
e~ωq/kB(T+δT ) − 1(6)
Φq = ~ωqδT
T. (7)
Normal scattering dominant
fq =1
e(~ωq−u·q)/kBT − 1(8)
Φq = u · q (9)
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Coherent behaviour in thermal transport
Kinetic and Collective Regimes
Variational Method and Entropy production
Variational method can be postulated from the second law
sq
kB= −fq ln fq + (fq + 1) ln(fq + 1) . (10)
Taking only linear terms in Φq
∂sq
∂t
∣∣∣∣scat
=Φq
T
∂fq∂t
∣∣∣∣scat
(11)
Macroscopically (Thermodynamically)
∂sq
∂t
∣∣∣∣drift
= jq · ∇(
1
T
)=
j2qκqT 2
(12)
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Coherent behaviour in thermal transport
Kinetic and Collective Regimes
Kinetic Regime
sq|scat=
Φq
T
∂fq∂t
∣∣∣∣scat
sq|drift=
j2qκqT 2
κq =j2q
TΦq∂fq∂t
∣∣∣scat
(13)
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Coherent behaviour in thermal transport
Kinetic and Collective Regimes
Collective Regime
stot|scat=
∫sq|scat
dq
stot|scat=
∫Φq
T
∂fq∂t
∣∣∣∣scat
dq
stot|drift=
j2tot
κT 2
stot|drift=
[∫~ωqvg f 0
q (f 0q + 1)
Φq
kBTdq]2
κT 2
κcoll =
[∫~ωqvg f 0
q (f 0q + 1)
Φq
kBTdq]2
T 2∫ Φq
T∂fq∂t
∣∣∣scat
dq(14)
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Coherent behaviour in thermal transport
Kinetic and Collective Regimes
Combination of regimes - Guyer Krumhansl model
Matrix treatment of operators
Df = Cf (15)
where D and C are respectively the drift and collision operators
D ≡ ~v · ∇ (16)
C ≡ N + R (17)
- In the collision term normal N and resistive R processes areseparated
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Coherent behaviour in thermal transport
Kinetic and Collective Regimes
Combined solution (Kinetic - Collective)
κ =1
3Cvc
2[〈τRB〉 (1− Σ) +
⟨τ−1R
⟩−1G (R)Σ
](18)
Σ =1
1 + 〈τN〉 / 〈τR〉(19)
with
〈α〉 =2
3kBT
∫τCvD(ω)dω∫CvD(ω)dω
(20)
Submitted - arXiv:1310.7127 [cond-mat.mes-hall]
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Coherent behaviour in thermal transport
Kinetic and Collective Regimes
Extended Irreversible Thermodynamics (EIT)
κ =1
3Cvc
2[〈τRB〉 (1− Σ) +
⟨τ−1R
⟩−1F (R)Σ
](21)
Inclusion of higher order fluxes (EIT) (T ,q,Q, . . . , , Jn)
∇T−1 − α1q + β1∇ ·Q = µ1q (22)
βn−1J(n−1) − αnJ
(n) + βn∇ · J(n+1) = µnJ(n). (23)
allow us to obtain a generalized function for the form factor F
F (Leff) =1
2π2
L2eff
`C `R
(√1 + 4π2
`C `RL2
eff
− 1
)(24)
for thin films Leff = W and for wires Leff = R/√
2.
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Coherent behaviour in thermal transport
Kinetic and Collective Regimes
Dispersion Relations and Density of states (D)
Lattice Dynamics. Bond Charge Model
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Coherent behaviour in thermal transport
Kinetic and Collective Regimes
Relaxation times (τ)
Non-Resistive
Type Expression
Normal τN = 1BNlow
T 3ω2[1−exp(−3T/ΘD)]+ 1
BNhighT
Resistive
Type Expression
Impurities τ−1I = π
6VΓω2Dω
Umklapp τU = exp(ΘU/T )BUω4T [1−exp(−3T/ΘD)]
Boundary τ−1B =
vgLeff
Parameters: BNlow,BNhigh
,BU
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Coherent behaviour in thermal transport
Comparison with data
Comparison with experimental values: Bulk
1
10
100
1000
10000
1 10 100 1000
Therm
al conductivity (
Wm
-1K
-1)
Temperature (K)
Theory Nat
Si28
SiKinCol
Experimental Nat
Si28
Si
BU (s3K−1) BNlow(sK−3) BNhigh
(s−1K−1)
2.8× 10−46 3.9× 10−23 4.0× 108
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Coherent behaviour in thermal transport
Comparison with data
Comparison with experimental values; Thin Films andNanowires
1
10
100
1000
10000
10 100 1000
Therm
al conductivity (
Wm
-1K
-1)
Temperature (K)
Theory TF 1.6µmTF 830nmTF 420nmTF 100nmTF 30nm
Experimental TF 1.6µmTF 830nmTF 420nmTF 100nmTF 30nm
1
10
100
1000
10 100 1000
Therm
al conductivity (
Wm
-1K
-1)
Temperature (K)
Theory NW 115nmNW 56nmNW 37nmNW 22nm
Experimental NW 115nmNW 56nmNW 37nmNW 22nm
Size effects added only by boundary term τ−1B =
vgLeff
Gold impurities not considered (J. E. Allen et al., Nat. Nanotechnol. 3, 168 (2008))
Roughness or specularity not included
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Coherent behaviour in thermal transport
Comparison with data
Comparison with experimental values
1
10
100
1000
10000
1 10 100 1000
Therm
al conductivity (
Wm
-1K
-1)
Temperature (K)
Theory Nat
Si28
Si
NW 115nm
NW 56nm
NW 37nm
NW 22nm
TF 1.6µm
TF 830nm
TF 100nm
TF 30nm
Experim Nat
Si28
Si
NW 115nm
NW 56nm
NW 37nm
NW 22nm
TF 1.6µm
TF 830nm
TF 100nm
TF 30nm
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Coherent behaviour in thermal transport
Extension for Nonequilibrium situations
Non steady-state situation
Guyer Krumhansl equation
τ∂~q
∂t+ ~q = −λ∇T + `C `N∇2~q (25)
Combined with energy conservation
cvdT
dt= −∇ · ~q (26)
gives a diffusion-wave equation
τ∂2T
∂t2+∂T
∂t= −ξ∇2T + α∇ · ∇2~q (27)
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Coherent behaviour in thermal transport
Extension for Nonequilibrium situations
Non steady-state situation
τ∂2T
∂t2+∂T
∂t= −ξ∇2T + α∇ · ∇2~q (28)
Wave front (∂2/∂t2 ∝ ∇2) on a diffusive media (∂/∂t ∝ ∇2)
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Coherent behaviour in thermal transport
Extension for Nonequilibrium situations
Conclusions
GK introduces new phenomenology through its kinetic tocollective transition. It can have implications in thermoelectricefficiency.
GK approach is more suitable to model transport phenomenawhen we are in a nonequilibrium state.
It can be improved by a more accurate calculation of therelaxation time calculations
It is a natural framework for energy transport through itswave-diffusive equation
For sizes lower than 20-30 nm, corrections are needed
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Coherent behaviour in thermal transport
Extension for Nonequilibrium situations
Collaborators
PhD StudentsCarla de Tomas, Daniel Moller
Statistical Physics Group at UABDavid Jou, Javier Bafaluy
Material Science at UABJavier Rodriguez, Aitor Lopeandia, Pablo Ferrando, Gemma Garcia
University of ValenciaAndres Cantarero
Thanks to: