Prediction of the thermal conductivity of a multilayer nanowire

Patrice Chantrenne, Séverine GomésCETHIL UMR 5008 INSA/UCBL1/CNRS

Arnaud Brioude, David CornuLMI UMR 5615 UCBL1/CNRS

MotivationsNanowire descriptionModels

Thanks to Laurent David, CETHIL LyonFlorian Lagrange, LCTS Bordeaux

LAYOUT

Jean-Louis BarratLPMCN UMR 5586 UCBL1/CNRS

Microelectronic components- length scale lower than 30 nm- film thickness less than10 nm

Motivations : applications

Require temperature measurements in order to ensure the reliability of the microsystem

Nanostructured materials (nanoporous, nanosequences, nanolayered)

Nanostructures (nanoparticles, nanotubes, nanowires, nanofilms…)

Require experimental caracterisationslimitation until now : almost one experimental device has been developped for each nanostructure

Motivations : applications

SiC/graphitelike Cnanosequence matérial

SiO2/SiCnanowire

SiC/SiO2/BNnanowire

Temperature measurement

Thermophysical properties measurement

High spatial resolutionbelow 100 nm

Quantitative measurementlower the uncertainty and higher sensitivity

Motivations : development of a new sensor

The most popular commercial sensor actually used with an AFM

Diameter : 5 µm

Length : 200 µm

Curvature radius : 15-20 µm

Motivations : development of a new sensor

Modèle de LefèvreModèle de David

Thermal conductivity- low sensitivity at high thermal conductivity values- uncertainty of about 20 % at low thermal conductivity values

S. Gomès & Dj. Ziane, 2003, Solid State Electronics 47 pp 919-922

L. David Ph D, CETHIL

S. Gomès et al., IEEE Transactions on Components and Packaging Technologies, 2006

Temperature measurement- qualitative values only- quantitative measurement require a calibration - spatial resolution limited by the tip geometry and surface roughness

Interfaces

nanowire

nanolayers

Core : BN, SiCcrystalline / periodic defect (mâcle)

layers :metallicdielectric crystal (SiC)/amorphous (SiO2)

The new sensor : a functionalised multilayer nanowire

The sensor should exhibit a low thermal conductivityin order to a good temperature and thermal conductivity sensitivity

The prediction of the thermal conductivity is essential to optimize the design of the sensor.

Motivations : development of a new sensor

10-50 nmeventually sharpened

Thermal conductivity versus thermal conductance/thermal resistance ?

Length l

Heat transfer across the nanowire depends on heat transfer

- in the core (dielectric crystal)- in metallic nanolayer- in amorphous nanolayer- in dielectric nanolayer- across the interfaces forecoming studies

Thicknesses e1 e2 e3 ...

Radius of the core rc

Tip end

lrcc , lemm ,

leaa , ledd ,

mcR amR mdR

,, R

Model : macroscopic approach

Use the bulk value

Prediction for nanowire

Prediction for nanofilm

Atomic collective vibration modes of energy

Model for dielectric crystals

In dielectric crystaline material, heat carriers are

Wave vector K, polarization p, dispersion curves

number of phonon per vibration mode

K p,1

1e k Tb /

PHONON=

Phonon liftime

These vibration modes may be characterised by

pK,

The total thermal conductivity = sum of individual thermal conductivity of each vibration modes

(K,p)

²1²,

x

x

b eV

exkpKC

Tk

pKx

b

,

dK

pKdv

,

xKK p

x pKpKvpKC ,²cos,,², xKK p

x pKpKvpKC ,²cos,,²,

The kinetic theory of gaz allow to write

xKx pKpKvpKCpK ,2 ²cos,,,,

with Spécific heat

Group velocity

Model for dielectric crystals

Thermal conductivity calculation

require the knowledge of - vibration modes- dispersion curves- relaxation time parameters

main assumption of the modelvibrational properties of a cristalline nanostructure

= vibrational properties of the bulk crystal

Validation of the model for Silicon...

1 1 1 1

K p K p K p K pph ph CL D, , , ,

1

u K p

A

T

B

T,exp

1

CL K p

v K p

F d K,

,

. ( )

1 4

D K pD

,

Model for dielectric crystals

Silicon structurein the real space

diamond structure

the elementary cell contains two atoms

a0

a2

a1a3

a0 = 0.543 nm

x

yz

Model for dielectric crystals

x

y

z

4 0 / a

ba

i j k10

ba

i j k30

ba

i j k20

Vibration modesIn the reciprocal space

- K = linear combination of de b1, b2, b3

- K belong to the first Brillouin ’s zone

- nomber of wave vectors K : number of elementary cells

- Number of polarisations p = 6

i

jk

Model for dielectric crystals

Dispersion curves

B.N. Brockhouse, P.R.L. 2, 256 (1959)

Linear fit of the experimental dispersion curvesin the [1,0,0] direction

S. Wei et M.Y. Chou, PRB, 50, 2221 (1994)

The optical mode contribution to the thermal conductivity is negligible if T < 1000 K

0

0,1

0,2

0,3

0,4

0,5

0,6

0,7

0,8

0,9

0 50 100 150 200 250 300 350

Temperature (K)

Cv/

(3R

)

P. Flubacher et al., Philos. Mag, 4,273 (1959)

0,00E+00

2,50E+12

5,00E+12

7,50E+12

1,00E+13

1,25E+13

1,50E+13

0 0,2 0,4 0,6 0,8 1

k/kmax

f (H

z)

LA

TA

Model for dielectric crystals

Relaxation time parameters determination

M.G. Holland, PR, 132, 2461 (1963)

Fit of the thermal conductivity of a Si crystal (L = 7,16 mm) function of the temperature

Transverse modeA = 7 10-13

B = 0= 1= 4

Longitudinal modeA = 3 10-21

B = 0= 2= 1.5

F = 0.55D = 1.32 10-45 s-3

1

u K p

A

T

B

T,exp

1 4

D K pD

,

LF

pKv

pKCL .

,

,

1

10

100

1000

10000

1 10 100 1000

Temperature (K)

(W

m-1

K-1

)

Model for dielectric crystals

0

5

10

15

20

25

30

35

40

45

50

0 50 100 150 200 250 300 350

Temperature (K)

(W

m-1

K-1

)

22 nm

37 nm

56 nm

115 nm

D. Li, et al., A.P.L, 83, 2934 (2003)

Excellent agreement except for the 22 nm wide

nanowire

Thermal conductivity of Si nanowires

1

10

100

1000

20 60 100 140 180 220 260 300

Temperature (K)

(W

m-1

K-1

)

20 nm

100 nm

0,42 µm

0,83 µm1,6 µm3 µm

M. Asheghi et al., ASME JHT, 120, 30 (1998)M.Z. Bazant, PRB, 56, 8542 (1997)

Excellent agreement with the experimental

resutls

Thermal conductivity of Si nanofilms

0

20

40

60

80

100

120

0 500 1000 1500 2000 2500 3000

film thickness (nm)

(W

/(m

K))

in plane

cross plane

Prediction of the thermal conductivity function of the heat transfer direction

T= 300K

Thermal conductivity of Si nanofilms

CONCLUSION

Thermal conductivity of dielectric nanofilms and nanowires

Thermal conductivity of metallics and amorphous nanofilms

Thermal conctact resistance

Confident to get a accurate value

The bulk value overestimate the real value

Still a Problem, several models may be used However, one need to evaluate the maximun value of the thermal conductivity