f xavier alvarez , andr es cantarero, carla de tom as, daniel m oller, aitor...

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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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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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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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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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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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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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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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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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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Kinetic Regime

sq|scat=

Φq

T

∂fq∂t

∣∣∣∣scat

sq|drift=

j2qκqT 2

κq =j2q

TΦq∂fq∂t

∣∣∣scat

(13)

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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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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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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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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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Dispersion Relations and Density of states (D)

Lattice Dynamics. Bond Charge Model

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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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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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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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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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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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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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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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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:

f xavier alvarez , andr es cantarero, carla de tom as, daniel m oller, aitor...

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