Heat Conduction by Molecular Dynamics Technique

Sebastian Volz

National Engineering School of Mechanics and AerotechnicsLaboratory of Thermal Studies UMR CNRS 6608

Poitiers, France

Denis Lemonnier - Lab. of Thermal Studies - PoitiersJean-Bernard Saulnier - Lab. of Thermal Studies - PoitiersGang Chen - NanoHeat Transfer and Thermoelectrics Lab. - UCLAPierre Beauchamp - Laboratoire de Métallurgie Physique - Poitiers

let

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4. TITLE AND SUBTITLEHeat Conduction by Molecular Dynamics TechniqueUnclassified

5a. CONTRACT NUMBER5b. GRANT NUMBER5c. PROGRAM ELEMENT NUMBER

6. AUTHOR(S)Volz, Sebastian ;Lemonnier, Denis ;Saulnier, Jean-Bernard ;Chen, Gang ;Beauchamp, Pierre ;

5d. PROJECT NUMBER5e. TASK NUMBER5f. WORK UNIT NUMBER

7. PERFORMING ORGANIZATION NAME AND ADDRESSNational Engineering School of Mechanics and AerotechnicsLaboratory of Thermal StudiesUMR CNRS 6608Poitiers, Francexxxxx

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12. DISTRIBUTION/AVAILABILITY STATEMENTAPUBLIC RELEASE,13. SUPPLEMENTARY NOTESSee Also ADM001348, Thermal Materials Workshop 2001, held in Cambridge, UK on May 30-June 1, 2001. Additional papers can bedownloaded from: http://www-mech.eng.cam.ac.uk/onr/14. ABSTRACTUNDERSTANDING AND MONITORING MATERIALS PROPERTIES15. SUBJECT TERMS16. SECURITY CLASSIFICATION OF: 17. LIMITATION

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Standard Form 298 (Rev. 8-98)Prescribed by ANSI Std Z39.18

UNDERSTANDING AND MONITORING MATERIALS PROPERTIES

Si02 Nanoparticles chain

VacuumAbsorbing MicroParticle

5 Å to 1m

NEW MATERIALS contain nano-micro architectured structures

o NANOFIBERS:

multiscale complex materialsultra insulating (0.007 W/mK)

o SUPERLATTICES

monitoring thermal conductivity Bulk/10<< Bulk

o NANOWIRES templates:

monitoring anisotropy 10 nm - 1m

G. Chen - Ni nanowires templatelet

LOW-DIMENSIONAL PHYSICS FOR HEAT CONDUCTION

Tgradq BULK .

Size L< Mean Free Path

BALLISTIC DIFFUSIVEL

o Ballistic Transport of Phonons

o Phonon Confinement

vG and Reduction

SpecularDiffuse

)(. 22222

2

uvvuvtu

tlt

0

1

2

3

4

5

6

7

8

9

0 0.5 1qx (nm-1)

(r

ad.T

Hz)

Si nanowire

Boundary ScatteringMore resistive

Bulk

eff(size)<BULK

let

SITUATION of MD

Classical Heat Conduction

Relaxation time 100psMean Free Path 100nm

Atomic Period 0.1psLattice Constant 0.5nm

Tgradq BULK.

Phonon WaveNo p-p scattering, limited by wlInterface transfer easier

Phonon Particle: Boltzmann TE No interference, Interface transferSmall and Large scales

Phonon Transport: Ballistic Diffusive

Molecular Dynamics - MD

Small Scales

let

MOLECULAR DYNAMICS TECHNIQUE

COMPUTE ALL ATOMIC TRAJECTORIES• STILLINGER-WEBER POTENTIAL • 2nd NEWTON LAW

N

ij1j

ij2i

2

dt

dM F

r3-BODY2-BODY

0.0

u2/

-1.0

a=1.8rij/

jik

rij

rik

-12.9

-12.89

-12.88

-12.87

-12.86

-12.85

-12.84

8.76 8.78 8.8 8.82 8.84 8.86 8.88

Coordonnée X (Angstroem)

Coo

rdon

née

Y (A

ngst

roem

)t=0.3 ps

let

ADVANTAGES OF MD TECHNIQUE

o Phonon Scattering is Difficult to Model: Phonon Particle Approach: Relaxation Time ?? Phonon Wave Approach: No scattering.

MD PROVIDES A COMPLETE DESCRIPTIONOF PHONON SCATTERING

EXAMPLE: NANOWIRE

o Phonon Transport Approach Assumes Fully Periodic Lattices

MD ALLOWS TO INCLUDE ATOMIC DEFAULTS and STRAINS

EXAMPLE: SUPERLATTICE

o Non-Equilibrium Short Time Heat Conduction

MD DESCRIBE HT BEHAVIOUR AT GigaHTz FREQUENCIES

EXAMPLE: IN BULK SI

0100

200300

400500

600

100 200 300 400 500 600

TMD (MD U)

Cor

rect

ed T

(K) BULK Si

TDebye=650K

correctionNo correction

Quantum Effects

(K)

let

HEAT FLUX by MD

N

i

N

ijj

ijijiiieV

t1 1

0 ..211)( rFvvq

o Kinetic and Work termssolids

viei

Kinetic Term

solids WK

Work Termvi

ij

rijFij

let

THERMAL CONDUCTIVITY by MD

o Fluctuation-Dissipation Theorem

0 0020

03

de

TkV i

B

qqAutocorrelation

Silicon at 200K and 500K

=0, Thermal Conductivity

X gradTgradT XThe ForceThe Flux

let

§ 0.2

20 40 60 80 100 120

SILICON NANOWIRE MD MODEL

RIGID BOUNDARY

RIGID BOUNDARY

PERIODIC BOUNDARYCONDITIONS

MD BOX

Phonon Energy Conserved

Free standing Bi Nanowire, M.S.Dresselhaus

let

BOLTZMANN TRANSPORT EQUATION

Bulk

ggt

g

gradv.

o 1D solution to BTE: Boundary Scattering ONLY

)p,(1...cos.)( 0 rr GTg

xTg Bulk

Bulk

Size

Bulk

nw 1

M(r)

S

x

Infinite Length

T

p

S. Volz and G. Chen, Heat and Technology, 18, 37, 2000.

(Ziman -Electrons and Phonons)

q S v g D d dD

r r

1

4 04

. . . . .

Function of G only

let

COMPARISON BETWEEN MD&BTE RESULTS

S. Volz and G. Chen, Applied Physics Letters, 57, 2056, 1999.

Is boundary scattering the only cause forthermal conductivity reduction?

0.8

let

PHONON CONFINEMENT EFFECT ON HEAT CONDUCTION

Tgradq BULK .

Size L< Mean Free Path

BALLISTIC DIFFUSIVEL

o Ballistic Transport of Phonons

o Phonon Confinement

vG and Reduction

SpecularDiffuse

)(. 22222

2

uvvuvtu

tlt

0

1

2

3

4

5

6

7

8

9

0 0.5 1qx (nm-1)

(r

ad.T

Hz)

Si nanowire

Boundary ScatteringMore resistive

Bulk

effective<BULK

let

o S8, discretizing in 48 directions (, wm) for finite length wire

m k m k k m k mL L L. , , , 0

2=300K

RIGID -PERFECTLY REFLECTING BOUNDARY

1=303K

RIGID -PERFECTLY REFLECTING BOUNDARY

Heat Flux qLr

100-800 nm

q LT T

.

1 2

q r w Li j m k m i j mmk

, , , ,,,

.d i

1 481 5

Tij ?

LLL 0.gradΩRPTE - THE DISCRETE ORDINATE METHOD

0,000

0,010

0,020

0,030

0,040

0,050

0,060

0 10 20 30 40 50 60 70

circular frequency, (rad.THz)

(n

m-1

)

p

p

p vrgL

..

=1/

=1/(v

.

Confinement effect

Confinement effect

Bulk

let

PHONON CONFINEMENT vs BOUNDARY SCATTERING

0,00

10,00

20,00

30,00

40,00

50,00

60,00

70,00

80,00

0 200 400 600 800

nanowire length (nm)

ther

mal

con

duct

ivity

(W/m

K)

p = 0

p = 0.5

p = 1

S. Volz and D. Lemonnier, Physics of Low-Dimensional Structures, 5/6, 91, 2000.

Phonon Confinement Effect Only

Boundary Scattering+Ph. C. Effects

Bulk=150 W/mK at 300K

PHONON CONFINEMENT: 50% REDUCTIONBOUNDARY SCATTERING: 70% REDUCTION

let

Si/Ge SUPERLATTICE MD MODELING

PeriodicBoundary Conditions

Ge GeSi

z

x

8 816

28

28

22.2A

38.9A y

SiGe

Ge

let

mm* ♦ •*

• • ;%><« F» S5W3

STRAIN EFFECT ON SUPERLATTICE STRUCTURE

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 5 10 15 20 25 30 3

[001] ATOMIC PLANE NUMBER

DIS

PLA

CEM

ENT

(A)

Ge GeSi

o Starting with mean lattice constant

o Implementing Conjugate Gradient Method

let

SUPERLATTICE THERMAL CONDUCTIVITY

0.1

1

10

100

15 20 25 30 35

LAYER THICKNESS (A)

CR

OSS

-PLA

NE

THER

MA

L C

ON

DU

CTI

VITY

(W

/mK

)

With Minimisation ProcedureWithout Minimisation ProcedureTrend for Experimental ResultsRBTE Solution

S. Volz, J.B Saulnier, G. Chen, P. Beauchamp, Microelectronics Journal 31 (9-10) 815, 2000.let

EFFECTIVE THERMAL CONDUCTIVITY AT GIGAHERTZ FREQUENCIES

q Vk T

q q t e dt gradTB

i t

bg bg bg

z302 0 0

0. .

. . .

e

B

q

gradT

Vk T

q

bg3

0

120

2

2 2. ..

( )

BULK=150W/mK

o Fluctuation Dissipation Theorem

t

eqtqq

2000 00

-1 dependence at Giga frequencies

SiO2 - 900GHtz

Ge - 20GHtz

100

let

CONCLUSION

MOLECULAR DYNAMICS TECHNIQUE:

o COMPLETELY DESCRIBES PHONON SCATTERING

o ALLOWS THE SIMULATION OF DEFAULTS/STRAINS

o GIVES ACCESS TO NON-EQUILIBRIUM REGIMES

- IS VERY HEAVY IN TERMS OF COMPUTATION TIME

- RELIES ON THE INTERACTION POTENTIAL VALIDITY

- DOES NOT INCLUDE QUANTUM EFFECTS

let

ULTRA SHORT TIME HEAT CONDUCTION

let

-1 LAW FOR Si THERMAL CONDUCTIVITY AT GIGAHTZ FREQUENCIES

2.5

(0 n * 2 E

1.5

x

let